WEBVTT 00:00:00.000 --> 00:00:01.440 align:middle line:90% [SQUEAKING] 00:00:01.440 --> 00:00:04.800 align:middle line:90% [RUSTLING] 00:00:04.800 --> 00:00:08.160 align:middle line:90% [CLICKING] 00:00:08.160 --> 00:00:11.678 align:middle line:90% 00:00:11.678 --> 00:00:13.470 align:middle line:84% ERIK DEMAINE: All right, let's get started. 00:00:13.470 --> 00:00:17.790 align:middle line:84% Today we continue probability, the last theme of the class. 00:00:17.790 --> 00:00:20.900 align:middle line:84% So we have three more lectures about probability. 00:00:20.900 --> 00:00:23.370 align:middle line:84% Today's lecture is about expectation, 00:00:23.370 --> 00:00:26.340 align:middle line:90% which is a nice summary. 00:00:26.340 --> 00:00:29.580 align:middle line:84% It's a one-number summary of a random variable. 00:00:29.580 --> 00:00:31.580 align:middle line:84% Remember from last time, a random variable 00:00:31.580 --> 00:00:33.540 align:middle line:90% is neither random nor variable. 00:00:33.540 --> 00:00:38.550 align:middle line:84% It's really a function from your sample space to something, 00:00:38.550 --> 00:00:41.210 align:middle line:90% usually real numbers. 00:00:41.210 --> 00:00:46.830 align:middle line:84% So we use this to represent some computation about an experiment. 00:00:46.830 --> 00:00:50.700 align:middle line:84% We had a similar setup with derived variables, 00:00:50.700 --> 00:00:55.178 align:middle line:84% if you remember back to state machines, which were also not 00:00:55.178 --> 00:00:56.720 align:middle line:84% variables, they were really functions 00:00:56.720 --> 00:00:59.010 align:middle line:84% from the state of the machine to numbers. 00:00:59.010 --> 00:01:01.070 align:middle line:84% So random variable is the same idea. 00:01:01.070 --> 00:01:03.490 align:middle line:84% If I give you-- if I tell you what sample-- 00:01:03.490 --> 00:01:05.120 align:middle line:84% like, what random events happened, 00:01:05.120 --> 00:01:08.690 align:middle line:84% you can compute a single value for this variable. 00:01:08.690 --> 00:01:10.070 align:middle line:90% That's the meaning. 00:01:10.070 --> 00:01:12.190 align:middle line:84% And we had a particular type of random variable 00:01:12.190 --> 00:01:13.815 align:middle line:84% that's going to be really useful today, 00:01:13.815 --> 00:01:16.630 align:middle line:84% which is an indicator random variable where the output is 00:01:16.630 --> 00:01:18.610 align:middle line:84% always 0 or 1, indicating something 00:01:18.610 --> 00:01:22.060 align:middle line:84% happened-- that's the 1-- or it didn't happen. 00:01:22.060 --> 00:01:24.710 align:middle line:90% So functions are complicated. 00:01:24.710 --> 00:01:26.397 align:middle line:84% If I want to write down a function, 00:01:26.397 --> 00:01:27.980 align:middle line:84% I got to write down a lot of numbers-- 00:01:27.980 --> 00:01:31.340 align:middle line:90% one for every possible sample. 00:01:31.340 --> 00:01:35.395 align:middle line:84% So expectation is a way to take that complicated function 00:01:35.395 --> 00:01:36.770 align:middle line:84% and reduce it down to one number. 00:01:36.770 --> 00:01:37.720 align:middle line:84% Obviously, it's not going to tell you 00:01:37.720 --> 00:01:40.210 align:middle line:84% everything about the function, but it tells you 00:01:40.210 --> 00:01:42.620 align:middle line:84% what you expect to happen on average. 00:01:42.620 --> 00:01:44.640 align:middle line:84% What is the-- it's not the most likely thing. 00:01:44.640 --> 00:01:46.640 align:middle line:84% But if you take all the things that could happen 00:01:46.640 --> 00:01:48.860 align:middle line:84% and you average them, that's expectation. 00:01:48.860 --> 00:01:53.110 align:middle line:90% So let's define that precisely. 00:01:53.110 --> 00:01:54.740 align:middle line:90% Match my notation here. 00:01:54.740 --> 00:02:10.199 align:middle line:84% So let's call random variable X over a sample space S. Then 00:02:10.199 --> 00:02:17.010 align:middle line:84% the expectation of X, which we write Ex of X. 00:02:17.010 --> 00:02:19.870 align:middle line:84% Although, often it's written just E of x. 00:02:19.870 --> 00:02:21.730 align:middle line:84% So if you look at other probability sources, 00:02:21.730 --> 00:02:22.900 align:middle line:90% you might see that as well. 00:02:22.900 --> 00:02:24.690 align:middle line:84% Both of these mean the same thing. 00:02:24.690 --> 00:02:27.150 align:middle line:84% It's going to be a weighted average-- weighted 00:02:27.150 --> 00:02:28.330 align:middle line:90% by probabilities. 00:02:28.330 --> 00:02:33.330 align:middle line:84% So we're going to sum over all possible samples, which 00:02:33.330 --> 00:02:38.700 align:middle line:84% we call omega and S of the probability 00:02:38.700 --> 00:02:44.460 align:middle line:90% of that outcome times X-- 00:02:44.460 --> 00:02:47.400 align:middle line:84% the function X evaluated at that income-- 00:02:47.400 --> 00:02:52.080 align:middle line:90% outcome-- omega for outcome. 00:02:52.080 --> 00:02:57.380 align:middle line:84% So if I-- so this is a particular type of average 00:02:57.380 --> 00:03:00.010 align:middle line:84% because the probabilities sum to 1. 00:03:00.010 --> 00:03:02.230 align:middle line:84% And in particular if the probabilities are all equal, 00:03:02.230 --> 00:03:05.590 align:middle line:84% this is literally the average value of X. 00:03:05.590 --> 00:03:07.540 align:middle line:84% But, of course, we would like-- what 00:03:07.540 --> 00:03:09.698 align:middle line:84% we care about is the weighted by the probability. 00:03:09.698 --> 00:03:11.740 align:middle line:84% If there's something that's more likely to occur, 00:03:11.740 --> 00:03:15.320 align:middle line:84% that should get a bigger multiplier in front. 00:03:15.320 --> 00:03:21.190 align:middle line:84% So this is going to be some average of X. 00:03:21.190 --> 00:03:24.700 align:middle line:84% Sometimes the expectation is also called the average. 00:03:24.700 --> 00:03:30.250 align:middle line:84% Let's do a very simple example, which is 00:03:30.250 --> 00:03:35.250 align:middle line:90% we roll a fair six-sided die. 00:03:35.250 --> 00:03:38.710 align:middle line:90% 00:03:38.710 --> 00:03:41.390 align:middle line:84% I happen to have such a die right here. 00:03:41.390 --> 00:03:42.380 align:middle line:90% So we roll it. 00:03:42.380 --> 00:03:45.260 align:middle line:84% It has values 1 to 6, each equally likely. 00:03:45.260 --> 00:03:46.080 align:middle line:90% Here, I got a 1. 00:03:46.080 --> 00:03:48.850 align:middle line:90% 00:03:48.850 --> 00:03:52.540 align:middle line:90% What is the expected value? 00:03:52.540 --> 00:03:57.050 align:middle line:84% So the sample space here is 1 up to 6. 00:03:57.050 --> 00:03:59.730 align:middle line:90% All the probabilities are 1, 6. 00:03:59.730 --> 00:04:04.590 align:middle line:84% Probability of I occurring is 1 over 6. 00:04:04.590 --> 00:04:08.510 align:middle line:90% And so the expected value-- 00:04:08.510 --> 00:04:10.580 align:middle line:84% I should probably give this a name. 00:04:10.580 --> 00:04:15.230 align:middle line:84% Let's call it D. Probability that D equals I 00:04:15.230 --> 00:04:18.990 align:middle line:84% for any particular I between 1 and 6 is 1 over 6. 00:04:18.990 --> 00:04:23.420 align:middle line:84% And so the expected value of D, the die roll 00:04:23.420 --> 00:04:25.020 align:middle line:90% is from that formula. 00:04:25.020 --> 00:04:32.780 align:middle line:84% Well, it's just the probability that we get a 1 times 1 00:04:32.780 --> 00:04:39.595 align:middle line:84% plus the probability that we get a 2 times 2 plus and so on. 00:04:39.595 --> 00:04:41.220 align:middle line:84% I'm going to rewrite that a little bit. 00:04:41.220 --> 00:04:43.290 align:middle line:84% So these probabilities are all 1/6. 00:04:43.290 --> 00:04:50.660 align:middle line:84% So we've got 1/6 times 1 plus 1/6 times 2 plus up to 1/6 times 00:04:50.660 --> 00:04:52.410 align:middle line:90% 6. 00:04:52.410 --> 00:04:58.290 align:middle line:84% These numbers is our favorite summation-- the triangular 00:04:58.290 --> 00:04:59.610 align:middle line:90% number. 00:04:59.610 --> 00:05:03.670 align:middle line:90% This is Gauss, right? 00:05:03.670 --> 00:05:06.340 align:middle line:84% So if I factor out the 1/6 in front, 00:05:06.340 --> 00:05:09.460 align:middle line:84% then we have that times the sum 1 through 6, 00:05:09.460 --> 00:05:15.500 align:middle line:84% which you should remember is 6 times 7 over 2. 00:05:15.500 --> 00:05:16.750 align:middle line:90% It's getting a little crowded. 00:05:16.750 --> 00:05:20.700 align:middle line:84% This is 1 over 6 times 6 times 7 over 2. 00:05:20.700 --> 00:05:23.420 align:middle line:90% Let me get a color chalk. 00:05:23.420 --> 00:05:25.980 align:middle line:90% 00:05:25.980 --> 00:05:31.570 align:middle line:84% It's time for the exciting cancellation moment. 00:05:31.570 --> 00:05:32.970 align:middle line:90% The 6's cancel. 00:05:32.970 --> 00:05:37.290 align:middle line:84% And so we're left with 7 over 2, also known 00:05:37.290 --> 00:05:40.810 align:middle line:84% as 3.5, which is, as you would expect, 00:05:40.810 --> 00:05:43.320 align:middle line:84% the average of numbers 1 through 6. 00:05:43.320 --> 00:05:46.110 align:middle line:84% But notably, a point of this example 00:05:46.110 --> 00:05:51.280 align:middle line:84% is that the expectation is not actually a valid outcome. 00:05:51.280 --> 00:05:53.820 align:middle line:90% You never get a 3.5 on a die. 00:05:53.820 --> 00:05:56.250 align:middle line:84% But you get 1, 2, 3, 4, 5, and 6, 00:05:56.250 --> 00:05:59.220 align:middle line:84% and the average of those is 3 and 1/2. 00:05:59.220 --> 00:06:05.160 align:middle line:84% So don't expect the expectation to be an actual point in your-- 00:06:05.160 --> 00:06:08.610 align:middle line:84% an actual value that X could take on. 00:06:08.610 --> 00:06:09.370 align:middle line:90% All right. 00:06:09.370 --> 00:06:11.913 align:middle line:84% Let's do a more interesting example, 00:06:11.913 --> 00:06:13.205 align:middle line:90% which we're going to use a lot. 00:06:13.205 --> 00:06:15.960 align:middle line:90% 00:06:15.960 --> 00:06:20.370 align:middle line:84% What if I have an indicator random variable, also X-- 00:06:20.370 --> 00:06:22.770 align:middle line:90% no, call it I-- 00:06:22.770 --> 00:06:27.840 align:middle line:84% for event A. So remember this means 00:06:27.840 --> 00:06:32.700 align:middle line:90% I is 1 if event A occurred. 00:06:32.700 --> 00:06:35.850 align:middle line:84% If the sample that happened is in A, 00:06:35.850 --> 00:06:38.730 align:middle line:84% and it's zero if A did not occur. 00:06:38.730 --> 00:06:48.480 align:middle line:84% So let's try to compute the expected value of I. 00:06:48.480 --> 00:06:51.940 align:middle line:84% So this is a little tricky with this formula 00:06:51.940 --> 00:06:56.410 align:middle line:84% because S-- even though I only takes on two values, S can be 00:06:56.410 --> 00:06:57.090 align:middle line:90% size a million. 00:06:57.090 --> 00:06:57.590 align:middle line:90% Who knows? 00:06:57.590 --> 00:06:59.320 align:middle line:84% There's a lot of different random outcomes 00:06:59.320 --> 00:07:00.080 align:middle line:90% that could happen. 00:07:00.080 --> 00:07:03.760 align:middle line:84% And then I kind of projects them all down to this very simple 0,1 00:07:03.760 --> 00:07:04.610 align:middle line:90% space. 00:07:04.610 --> 00:07:08.410 align:middle line:84% So I'm just going to write the definition first. 00:07:08.410 --> 00:07:16.270 align:middle line:84% Omega in S-- probability of omega times I of omega. 00:07:16.270 --> 00:07:19.810 align:middle line:90% But this thing is either 0 or 1. 00:07:19.810 --> 00:07:21.260 align:middle line:90% So this is a summation. 00:07:21.260 --> 00:07:22.972 align:middle line:84% And remember, one of the techniques 00:07:22.972 --> 00:07:25.180 align:middle line:84% for dealing with summations is to rearrange the terms 00:07:25.180 --> 00:07:26.480 align:middle line:90% in some useful way. 00:07:26.480 --> 00:07:28.360 align:middle line:84% And given that every term here is either 00:07:28.360 --> 00:07:30.820 align:middle line:84% multiplied by 0, in which case it disappears, 00:07:30.820 --> 00:07:36.140 align:middle line:84% or multiplied by 1, let's maybe collect like terms. 00:07:36.140 --> 00:07:39.380 align:middle line:84% So we can split this summation into two parts. 00:07:39.380 --> 00:07:47.660 align:middle line:84% There's the sum where W is in a complement probability of omega 00:07:47.660 --> 00:07:54.680 align:middle line:84% times 0, because I of something in a complement here is 0. 00:07:54.680 --> 00:08:00.060 align:middle line:84% Or it could be omega inside A without the complement. 00:08:00.060 --> 00:08:03.920 align:middle line:84% And then we get probability of omega times 1. 00:08:03.920 --> 00:08:08.400 align:middle line:84% So these disappear, and we're just left with-- 00:08:08.400 --> 00:08:12.360 align:middle line:84% and the 1 cancels, or it doesn't do anything. 00:08:12.360 --> 00:08:19.460 align:middle line:84% So we get the sum of omega and A times probability of omega. 00:08:19.460 --> 00:08:25.010 align:middle line:84% And this is our definition of probability of A. 00:08:25.010 --> 00:08:26.880 align:middle line:90% So that's nice and simple. 00:08:26.880 --> 00:08:30.200 align:middle line:84% Another way to write this that's more directly about I is this 00:08:30.200 --> 00:08:33.440 align:middle line:84% is the probability that I equals 1. 00:08:33.440 --> 00:08:36.559 align:middle line:84% So the expected value of an indicator random variable 00:08:36.559 --> 00:08:40.130 align:middle line:84% is always the probability that that indicator random variable 00:08:40.130 --> 00:08:40.860 align:middle line:90% equals 1. 00:08:40.860 --> 00:08:44.640 align:middle line:84% This is just a fun fact because I only takes on 0 and 1. 00:08:44.640 --> 00:08:47.670 align:middle line:84% The zeros don't show up-- they disappear from the expectation 00:08:47.670 --> 00:08:51.960 align:middle line:84% formula, and the 1's basically add up just the probabilities 00:08:51.960 --> 00:08:54.960 align:middle line:90% of the event occurring. 00:08:54.960 --> 00:08:57.430 align:middle line:84% We're going to use this fact, several times today. 00:08:57.430 --> 00:09:00.343 align:middle line:84% So it's an example and a useful lemma. 00:09:00.343 --> 00:09:04.120 align:middle line:90% 00:09:04.120 --> 00:09:07.590 align:middle line:84% Probability, I'm told, was originally 00:09:07.590 --> 00:09:09.730 align:middle line:90% invented to analyze gambling. 00:09:09.730 --> 00:09:13.360 align:middle line:84% So let's put that to good use and do some gambling. 00:09:13.360 --> 00:09:14.620 align:middle line:90% So let me define a game. 00:09:14.620 --> 00:09:16.770 align:middle line:84% And then I'm going to need a volunteer-- 00:09:16.770 --> 00:09:22.155 align:middle line:84% a couple of volunteers-- to play this game with me. 00:09:22.155 --> 00:09:23.530 align:middle line:84% And it doesn't have a fancy name. 00:09:23.530 --> 00:09:27.120 align:middle line:84% We're just going to call it the gambling game. 00:09:27.120 --> 00:09:29.703 align:middle line:84% So we're going to have three players. 00:09:29.703 --> 00:09:34.980 align:middle line:90% 00:09:34.980 --> 00:09:37.370 align:middle line:84% And we're going to follow these sequence of steps. 00:09:37.370 --> 00:09:40.770 align:middle line:90% 00:09:40.770 --> 00:09:42.700 align:middle line:84% Everybody has some pile of money, 00:09:42.700 --> 00:09:46.530 align:middle line:84% which we're going to represent by candy, 00:09:46.530 --> 00:09:48.643 align:middle line:90% but I'll write it as dollars. 00:09:48.643 --> 00:09:51.930 align:middle line:90% 00:09:51.930 --> 00:09:56.970 align:middle line:84% So every round, every player puts $2 into a shared pot. 00:09:56.970 --> 00:10:09.922 align:middle line:84% Then each player picks heads or tails. 00:10:09.922 --> 00:10:12.630 align:middle line:84% You may have played this game in the single-player version, where 00:10:12.630 --> 00:10:14.590 align:middle line:84% you just try to guess the outcome of a coin. 00:10:14.590 --> 00:10:16.790 align:middle line:84% Then we're going to flip a fair coin. 00:10:16.790 --> 00:10:21.940 align:middle line:90% 00:10:21.940 --> 00:10:24.720 align:middle line:84% With multiplayer, it's just more interesting, let's say. 00:10:24.720 --> 00:10:27.640 align:middle line:84% Now we could have multiple winners. 00:10:27.640 --> 00:10:30.420 align:middle line:84% Maybe two people guessed heads, and the outcome of the coin 00:10:30.420 --> 00:10:31.510 align:middle line:90% was heads. 00:10:31.510 --> 00:10:33.870 align:middle line:84% Then what we're going to do is take the pot of money 00:10:33.870 --> 00:10:37.200 align:middle line:84% that everyone added to and evenly distribute it 00:10:37.200 --> 00:10:39.900 align:middle line:84% among the winners, among the people that guessed the coin 00:10:39.900 --> 00:10:42.230 align:middle line:90% outcome correctly. 00:10:42.230 --> 00:10:53.003 align:middle line:84% So we split pot equally among winners. 00:10:53.003 --> 00:10:55.880 align:middle line:90% 00:10:55.880 --> 00:10:58.820 align:middle line:84% There's one special case, which is maybe no one guesses 00:10:58.820 --> 00:10:59.900 align:middle line:90% the correct value. 00:10:59.900 --> 00:11:01.762 align:middle line:84% In that case, we'll also equally split 00:11:01.762 --> 00:11:04.220 align:middle line:84% the pot, which is equivalent to just giving your money back 00:11:04.220 --> 00:11:04.928 align:middle line:90% and trying again. 00:11:04.928 --> 00:11:08.340 align:middle line:84% So we're going to play this game many rounds. 00:11:08.340 --> 00:11:11.420 align:middle line:84% And your goal is to, of course, maximize the amount of money you 00:11:11.420 --> 00:11:12.120 align:middle line:90% get. 00:11:12.120 --> 00:11:13.940 align:middle line:84% Who would like to play this game. 00:11:13.940 --> 00:11:15.770 align:middle line:90% Oh, so many people. 00:11:15.770 --> 00:11:16.700 align:middle line:90% All right. 00:11:16.700 --> 00:11:18.840 align:middle line:84% I'm afraid it's only a three-player game. 00:11:18.840 --> 00:11:22.310 align:middle line:84% So I'm going to pick Jenny, my accomplice. 00:11:22.310 --> 00:11:26.918 align:middle line:90% And how about you? 00:11:26.918 --> 00:11:28.710 align:middle line:84% AUDIENCE: Don't tell them where I got this. 00:11:28.710 --> 00:11:31.030 align:middle line:90% ERIK DEMAINE: [LAUGHS] Yeah. 00:11:31.030 --> 00:11:31.530 align:middle line:90% No. 00:11:31.530 --> 00:11:32.370 align:middle line:90% Totally fair game. 00:11:32.370 --> 00:11:33.650 align:middle line:90% Don't worry. 00:11:33.650 --> 00:11:34.375 align:middle line:90% What's your name? 00:11:34.375 --> 00:11:35.000 align:middle line:90% AUDIENCE: Cleo. 00:11:35.000 --> 00:11:35.670 align:middle line:90% ERIK DEMAINE: Cleo. 00:11:35.670 --> 00:11:36.230 align:middle line:90% All right. 00:11:36.230 --> 00:11:40.440 align:middle line:84% So everyone welcome Cleo and Jenny to the stage. 00:11:40.440 --> 00:11:40.940 align:middle line:90% Yeah. 00:11:40.940 --> 00:11:43.820 align:middle line:90% [APPLAUSE] 00:11:43.820 --> 00:11:49.503 align:middle line:84% I want you to be in like equally hard and good position. 00:11:49.503 --> 00:11:51.170 align:middle line:84% So we're going to put you in the middle. 00:11:51.170 --> 00:11:52.430 align:middle line:90% C-L-E-O? 00:11:52.430 --> 00:11:53.150 align:middle line:90% AUDIENCE: Yeah. 00:11:53.150 --> 00:11:54.150 align:middle line:90% ERIK DEMAINE: All right. 00:11:54.150 --> 00:11:58.730 align:middle line:90% 00:11:58.730 --> 00:12:00.440 align:middle line:90% Try to keep track here. 00:12:00.440 --> 00:12:03.140 align:middle line:90% We've got a coin. 00:12:03.140 --> 00:12:04.410 align:middle line:90% Except I don't have a coin. 00:12:04.410 --> 00:12:05.490 align:middle line:90% I have a giant die. 00:12:05.490 --> 00:12:08.960 align:middle line:84% So we're going to say was it heads, H? 00:12:08.960 --> 00:12:11.923 align:middle line:90% H is huge. 00:12:11.923 --> 00:12:14.090 align:middle line:84% I've got to write this down, or I'm going to forget. 00:12:14.090 --> 00:12:19.940 align:middle line:90% H is huge, and T is tiny. 00:12:19.940 --> 00:12:20.460 align:middle line:90% OK. 00:12:20.460 --> 00:12:22.800 align:middle line:90% So 1, 2, 3 is a tails. 00:12:22.800 --> 00:12:27.440 align:middle line:90% 4, 5, 6 is an H. All right. 00:12:27.440 --> 00:12:30.540 align:middle line:90% So I'm going to guess first. 00:12:30.540 --> 00:12:33.620 align:middle line:84% Then why don't you stand in the middle so we're like in order 00:12:33.620 --> 00:12:34.380 align:middle line:90% here. 00:12:34.380 --> 00:12:37.590 align:middle line:90% So I'll guess tails, my usual. 00:12:37.590 --> 00:12:38.610 align:middle line:90% AUDIENCE: Heads. 00:12:38.610 --> 00:12:39.510 align:middle line:90% AUDIENCE: Tails. 00:12:39.510 --> 00:12:40.770 align:middle line:90% ERIK DEMAINE: Tails. 00:12:40.770 --> 00:12:41.880 align:middle line:90% OK. 00:12:41.880 --> 00:12:44.840 align:middle line:90% Let's roll this dies. 00:12:44.840 --> 00:12:47.080 align:middle line:84% That F doesn't feel very fair, but OK. 00:12:47.080 --> 00:12:49.120 align:middle line:84% Well, I'll do different types of rolls. 00:12:49.120 --> 00:12:52.420 align:middle line:84% So we got a 4, which is H. Ah-ha. 00:12:52.420 --> 00:12:52.930 align:middle line:90% Cleo wins. 00:12:52.930 --> 00:12:54.610 align:middle line:84% Oh, I forgot to distribute the money. 00:12:54.610 --> 00:12:56.560 align:middle line:90% OK, these are all equal. 00:12:56.560 --> 00:13:05.800 align:middle line:84% So pick your pile of cash, which are poker chips. 00:13:05.800 --> 00:13:10.360 align:middle line:84% And so it's, I think, more fun if everyone can see the pile. 00:13:10.360 --> 00:13:12.040 align:middle line:90% So let me-- 00:13:12.040 --> 00:13:17.080 align:middle line:84% So we were supposed to start out by putting two in to the pot. 00:13:17.080 --> 00:13:19.660 align:middle line:84% And then you are the only winner for the pot. 00:13:19.660 --> 00:13:21.390 align:middle line:90% So you get all this money. 00:13:21.390 --> 00:13:22.750 align:middle line:90% That's already looking good. 00:13:22.750 --> 00:13:23.380 align:middle line:90% All right. 00:13:23.380 --> 00:13:25.680 align:middle line:90% So I'm going to flip the-- 00:13:25.680 --> 00:13:28.628 align:middle line:90% I should flip after guessing. 00:13:28.628 --> 00:13:29.920 align:middle line:90% A little too forced, otherwise. 00:13:29.920 --> 00:13:30.420 align:middle line:90% All right. 00:13:30.420 --> 00:13:31.793 align:middle line:90% I'll try heads this time. 00:13:31.793 --> 00:13:32.460 align:middle line:90% AUDIENCE: Tails. 00:13:32.460 --> 00:13:33.293 align:middle line:90% ERIK DEMAINE: Tails. 00:13:33.293 --> 00:13:34.050 align:middle line:90% AUDIENCE: Heads. 00:13:34.050 --> 00:13:35.580 align:middle line:90% ERIK DEMAINE: Heads. 00:13:35.580 --> 00:13:36.480 align:middle line:90% OK. 00:13:36.480 --> 00:13:37.265 align:middle line:90% Now we flip. 00:13:37.265 --> 00:13:39.820 align:middle line:90% 00:13:39.820 --> 00:13:40.320 align:middle line:90% OK. 00:13:40.320 --> 00:13:41.890 align:middle line:90% 6 is high. 00:13:41.890 --> 00:13:44.110 align:middle line:90% Oh, I forgot to do our thing. 00:13:44.110 --> 00:13:47.830 align:middle line:84% Two in, and then we evenly split the pot. 00:13:47.830 --> 00:13:49.360 align:middle line:90% All right. 00:13:49.360 --> 00:13:49.860 align:middle line:90% Great. 00:13:49.860 --> 00:13:51.450 align:middle line:90% AUDIENCE: Again. 00:13:51.450 --> 00:13:54.112 align:middle line:84% ERIK DEMAINE: Very consistent coin. 00:13:54.112 --> 00:13:55.433 align:middle line:90% Play this game properly. 00:13:55.433 --> 00:13:56.850 align:middle line:84% Supposed to put the money-- you're 00:13:56.850 --> 00:13:58.767 align:middle line:84% supposed to put your money where your mouth is 00:13:58.767 --> 00:14:00.780 align:middle line:90% before you play the game. 00:14:00.780 --> 00:14:02.283 align:middle line:90% I'll try heads. 00:14:02.283 --> 00:14:02.950 align:middle line:90% AUDIENCE: Heads. 00:14:02.950 --> 00:14:04.060 align:middle line:90% ERIK DEMAINE: Heads. 00:14:04.060 --> 00:14:05.430 align:middle line:84% AUDIENCE: There's no way it's three heads in a row. 00:14:05.430 --> 00:14:05.930 align:middle line:90% Tails. 00:14:05.930 --> 00:14:08.790 align:middle line:84% ERIK DEMAINE: [LAUGHS] Yeah, three heads in a row 00:14:08.790 --> 00:14:10.980 align:middle line:90% is never useful. 00:14:10.980 --> 00:14:13.480 align:middle line:84% Because whatever happens, we will just take our money back. 00:14:13.480 --> 00:14:13.980 align:middle line:90% All right. 00:14:13.980 --> 00:14:16.260 align:middle line:90% So we flip. 00:14:16.260 --> 00:14:17.380 align:middle line:90% This is quite fun. 00:14:17.380 --> 00:14:18.220 align:middle line:90% Another heads. 00:14:18.220 --> 00:14:22.080 align:middle line:84% Wow, it's a very consistent coin here. 00:14:22.080 --> 00:14:22.660 align:middle line:90% All right. 00:14:22.660 --> 00:14:24.340 align:middle line:90% So here is the distribution. 00:14:24.340 --> 00:14:26.460 align:middle line:90% We put our money back in-- 00:14:26.460 --> 00:14:29.130 align:middle line:90% two each. 00:14:29.130 --> 00:14:30.563 align:middle line:90% I'll do tails. 00:14:30.563 --> 00:14:31.230 align:middle line:90% AUDIENCE: Heads. 00:14:31.230 --> 00:14:32.373 align:middle line:90% ERIK DEMAINE: Heads. 00:14:32.373 --> 00:14:33.040 align:middle line:90% AUDIENCE: Tails. 00:14:33.040 --> 00:14:34.960 align:middle line:90% ERIK DEMAINE: Tails. 00:14:34.960 --> 00:14:35.810 align:middle line:90% I'm very efficient. 00:14:35.810 --> 00:14:36.820 align:middle line:90% This is great. 00:14:36.820 --> 00:14:40.040 align:middle line:84% I'm the slow part here, rolling Oh, yeah. 00:14:40.040 --> 00:14:40.705 align:middle line:90% Another heads. 00:14:40.705 --> 00:14:41.205 align:middle line:90% Great. 00:14:41.205 --> 00:14:43.360 align:middle line:90% [LAUGHS] 00:14:43.360 --> 00:14:43.967 align:middle line:90% I love this. 00:14:43.967 --> 00:14:45.800 align:middle line:84% There's another problem we'll be working on, 00:14:45.800 --> 00:14:46.930 align:middle line:84% which is how many times do you have to flip 00:14:46.930 --> 00:14:48.530 align:middle line:90% a coin until you get tails? 00:14:48.530 --> 00:14:49.560 align:middle line:90% The answer is two. 00:14:49.560 --> 00:14:51.580 align:middle line:90% [LAUGHS] 00:14:51.580 --> 00:14:52.190 align:middle line:90% All right. 00:14:52.190 --> 00:14:57.682 align:middle line:84% So money back in, and I'll do tails. 00:14:57.682 --> 00:14:58.640 align:middle line:90% AUDIENCE: Sorry, heads. 00:14:58.640 --> 00:14:59.865 align:middle line:90% ERIK DEMAINE: Heads. 00:14:59.865 --> 00:15:00.532 align:middle line:90% AUDIENCE: Tails. 00:15:00.532 --> 00:15:01.600 align:middle line:90% ERIK DEMAINE: Tails. 00:15:01.600 --> 00:15:02.280 align:middle line:90% All right. 00:15:02.280 --> 00:15:05.980 align:middle line:90% 00:15:05.980 --> 00:15:11.500 align:middle line:84% Based on how this has been going, good guess. 00:15:11.500 --> 00:15:12.440 align:middle line:90% All right. 00:15:12.440 --> 00:15:13.490 align:middle line:90% Have we played enough? 00:15:13.490 --> 00:15:14.155 align:middle line:90% Do you get the idea? 00:15:14.155 --> 00:15:14.650 align:middle line:90% AUDIENCE: Not yet. 00:15:14.650 --> 00:15:15.733 align:middle line:90% ERIK DEMAINE: Oh, not yet. 00:15:15.733 --> 00:15:17.720 align:middle line:84% You're almost depleted there, Jenny. 00:15:17.720 --> 00:15:18.580 align:middle line:90% All right. 00:15:18.580 --> 00:15:20.350 align:middle line:90% Let's see. 00:15:20.350 --> 00:15:21.048 align:middle line:90% One more. 00:15:21.048 --> 00:15:23.090 align:middle line:84% We should keep going until we get a tails, right. 00:15:23.090 --> 00:15:25.420 align:middle line:90% That can't take forever. 00:15:25.420 --> 00:15:27.370 align:middle line:90% Famous last words. 00:15:27.370 --> 00:15:28.380 align:middle line:90% I've done this before. 00:15:28.380 --> 00:15:28.880 align:middle line:90% All right. 00:15:28.880 --> 00:15:29.383 align:middle line:90% Heads. 00:15:29.383 --> 00:15:30.050 align:middle line:90% AUDIENCE: Tails. 00:15:30.050 --> 00:15:30.923 align:middle line:90% ERIK DEMAINE: Tails. 00:15:30.923 --> 00:15:31.590 align:middle line:90% AUDIENCE: Heads. 00:15:31.590 --> 00:15:32.930 align:middle line:90% ERIK DEMAINE: Heads. 00:15:32.930 --> 00:15:36.750 align:middle line:84% Now, the gambler's ruin would say we're due for a tails now. 00:15:36.750 --> 00:15:40.730 align:middle line:90% It's more likely than 50%. 00:15:40.730 --> 00:15:44.600 align:middle line:90% Oh, gambler is wrong. 00:15:44.600 --> 00:15:47.240 align:middle line:90% It's 50/50 every time. 00:15:47.240 --> 00:15:47.860 align:middle line:90% All right. 00:15:47.860 --> 00:15:51.530 align:middle line:90% 00:15:51.530 --> 00:15:52.370 align:middle line:90% Heads. 00:15:52.370 --> 00:15:53.113 align:middle line:90% AUDIENCE: Tails. 00:15:53.113 --> 00:15:53.780 align:middle line:90% AUDIENCE: Heads. 00:15:53.780 --> 00:16:01.130 align:middle line:90% 00:16:01.130 --> 00:16:02.250 align:middle line:90% ERIK DEMAINE: Yes. 00:16:02.250 --> 00:16:03.190 align:middle line:90% Finally, a tails. 00:16:03.190 --> 00:16:06.440 align:middle line:90% 00:16:06.440 --> 00:16:07.730 align:middle line:90% All right. 00:16:07.730 --> 00:16:10.862 align:middle line:84% Cleo here has won basically all the candy. 00:16:10.862 --> 00:16:13.070 align:middle line:84% You can take as many as you like because I'm actually 00:16:13.070 --> 00:16:15.620 align:middle line:84% lactose intolerant, so I can't eat these. 00:16:15.620 --> 00:16:19.760 align:middle line:84% So both of you are welcome to take as many as you like. 00:16:19.760 --> 00:16:21.590 align:middle line:90% So that's the gambling game. 00:16:21.590 --> 00:16:25.950 align:middle line:84% Now, is it coincidence that Cleo won and I lost? 00:16:25.950 --> 00:16:28.160 align:middle line:90% No. 00:16:28.160 --> 00:16:30.000 align:middle line:84% AUDIENCE: Jenny was picking the other one. 00:16:30.000 --> 00:16:31.810 align:middle line:90% The other from you, indeed. 00:16:31.810 --> 00:16:33.200 align:middle line:90% Very observant. 00:16:33.200 --> 00:16:35.240 align:middle line:84% This is why Jenny was so fast at guessing. 00:16:35.240 --> 00:16:36.740 align:middle line:84% Jenny is one of our TAs, by the way. 00:16:36.740 --> 00:16:40.210 align:middle line:84% That's how I convinced her to cheat. 00:16:40.210 --> 00:16:42.800 align:middle line:84% But she cheated in order to destroy me. 00:16:42.800 --> 00:16:44.560 align:middle line:84% So there's this fun strategy, which 00:16:44.560 --> 00:16:49.010 align:middle line:84% is if you guess the opposite of one of the previous players, 00:16:49.010 --> 00:16:51.920 align:middle line:84% which, if you guess in sequence like this, is easy to do. 00:16:51.920 --> 00:16:53.030 align:middle line:90% Cleo could have done it. 00:16:53.030 --> 00:16:54.770 align:middle line:84% Cleo actually did it most of the time. 00:16:54.770 --> 00:16:57.050 align:middle line:84% I think there were only a couple times-- 00:16:57.050 --> 00:16:59.410 align:middle line:90% one time when we matched. 00:16:59.410 --> 00:17:01.840 align:middle line:90% And that's a good strategy. 00:17:01.840 --> 00:17:06.910 align:middle line:84% In particular-- well, let's analyze. shall we? 00:17:06.910 --> 00:17:09.680 align:middle line:84% How do we analyze a game like this. 00:17:09.680 --> 00:17:13.450 align:middle line:84% So small-- we just draw the decision tree or the probability 00:17:13.450 --> 00:17:14.490 align:middle line:90% tree. 00:17:14.490 --> 00:17:14.990 align:middle line:90% All right. 00:17:14.990 --> 00:17:17.780 align:middle line:84% Now, it's a little bigger than some of the ones we've done. 00:17:17.780 --> 00:17:21.849 align:middle line:84% Luckily, the whole thing is symmetric. 00:17:21.849 --> 00:17:24.042 align:middle line:84% It doesn't really matter which one you do first. 00:17:24.042 --> 00:17:25.250 align:middle line:90% The whole thing is symmetric. 00:17:25.250 --> 00:17:27.333 align:middle line:84% So I'm going to do it in the order of let's branch 00:17:27.333 --> 00:17:29.830 align:middle line:84% on what player one does, player two, player three. 00:17:29.830 --> 00:17:31.510 align:middle line:90% And then what the coin does. 00:17:31.510 --> 00:17:33.930 align:middle line:84% And then we're going to get some outcome, which 00:17:33.930 --> 00:17:36.030 align:middle line:90% I'll define in a second. 00:17:36.030 --> 00:17:38.970 align:middle line:84% So it turns out whether the first player guesses 00:17:38.970 --> 00:17:41.860 align:middle line:84% heads or tails because the coin is random, 00:17:41.860 --> 00:17:43.000 align:middle line:90% it's going to be symmetric. 00:17:43.000 --> 00:17:45.510 align:middle line:84% So let's just ignore the whole right half of the space 00:17:45.510 --> 00:17:47.680 align:middle line:84% and pretend we're in this left half. 00:17:47.680 --> 00:17:51.420 align:middle line:84% So suppose I always guessed heads, 00:17:51.420 --> 00:17:53.470 align:middle line:84% or somebody always guesses heads. 00:17:53.470 --> 00:17:55.770 align:middle line:84% I think it's probably going to be me that I care about. 00:17:55.770 --> 00:17:57.300 align:middle line:90% Yeah. 00:17:57.300 --> 00:18:01.420 align:middle line:84% Then player two guesses heads or tails. 00:18:01.420 --> 00:18:05.520 align:middle line:84% Now, we haven't said what the probabilities are here. 00:18:05.520 --> 00:18:07.870 align:middle line:84% I was trying to be roughly 50/50. 00:18:07.870 --> 00:18:10.080 align:middle line:84% But I wasn't flipping a coin each time 00:18:10.080 --> 00:18:12.360 align:middle line:84% to decide how I should guess the coin. 00:18:12.360 --> 00:18:14.867 align:middle line:84% I was just making some arbitrary choice. 00:18:14.867 --> 00:18:16.450 align:middle line:84% Cleo was making some arbitrary choice. 00:18:16.450 --> 00:18:18.430 align:middle line:84% Jenny was making some less arbitrary choice. 00:18:18.430 --> 00:18:20.230 align:middle line:84% And then the coin-- this part is random. 00:18:20.230 --> 00:18:25.470 align:middle line:84% So this one is one half either way. 00:18:25.470 --> 00:18:28.690 align:middle line:90% So that's helpful. 00:18:28.690 --> 00:18:31.060 align:middle line:84% And then we get some outcome, which 00:18:31.060 --> 00:18:38.260 align:middle line:84% I'm going to measure the net for player one. 00:18:38.260 --> 00:18:39.970 align:middle line:90% That's this number. 00:18:39.970 --> 00:18:41.770 align:middle line:84% And I wrote it down here because it's 00:18:41.770 --> 00:18:43.420 align:middle line:90% very easy to get these wrong. 00:18:43.420 --> 00:18:46.257 align:middle line:84% So the first case is that we all guessed heads. 00:18:46.257 --> 00:18:47.840 align:middle line:84% And then it doesn't matter what we do. 00:18:47.840 --> 00:18:49.510 align:middle line:84% Just our money is going to go back-- 00:18:49.510 --> 00:18:51.890 align:middle line:84% or the pot is going to go back to even distribution. 00:18:51.890 --> 00:18:54.460 align:middle line:84% Whether we win or lose, the money 00:18:54.460 --> 00:18:56.270 align:middle line:90% goes back to where it started. 00:18:56.270 --> 00:18:58.670 align:middle line:84% So the net for player one, in particular, is zero. 00:18:58.670 --> 00:19:01.270 align:middle line:90% So these are boring. 00:19:01.270 --> 00:19:05.720 align:middle line:84% Then we have heads, heads, tails for the three players. 00:19:05.720 --> 00:19:09.770 align:middle line:84% And so then if it's heads, these two players won. 00:19:09.770 --> 00:19:11.360 align:middle line:90% And so they evenly distribute. 00:19:11.360 --> 00:19:13.730 align:middle line:84% So we're always putting 6 in each time. 00:19:13.730 --> 00:19:16.870 align:middle line:90% So they each get 3. 00:19:16.870 --> 00:19:19.160 align:middle line:84% But they started by putting in 2. 00:19:19.160 --> 00:19:20.780 align:middle line:90% And so the net gain is only 1. 00:19:20.780 --> 00:19:23.380 align:middle line:90% This is 3 minus 2. 00:19:23.380 --> 00:19:26.390 align:middle line:84% We gain 3, but we had to pay 2 to get there. 00:19:26.390 --> 00:19:27.950 align:middle line:90% Or we lose. 00:19:27.950 --> 00:19:31.017 align:middle line:84% Whenever we lose-- and all of these tails branches player 1 00:19:31.017 --> 00:19:32.850 align:middle line:84% is going to lose because they guessed heads. 00:19:32.850 --> 00:19:34.340 align:middle line:84% And, in that case, you get minus 2 00:19:34.340 --> 00:19:36.290 align:middle line:90% because you had to put in the 2. 00:19:36.290 --> 00:19:38.070 align:middle line:90% So in this case is the same. 00:19:38.070 --> 00:19:39.060 align:middle line:90% There's 2 heads. 00:19:39.060 --> 00:19:40.710 align:middle line:90% And so you each get 3. 00:19:40.710 --> 00:19:42.030 align:middle line:90% And so it's a net of 1. 00:19:42.030 --> 00:19:43.680 align:middle line:90% And then there's this case. 00:19:43.680 --> 00:19:46.350 align:middle line:84% This is the case you're really aiming for. 00:19:46.350 --> 00:19:52.070 align:middle line:90% This is the best for player one. 00:19:52.070 --> 00:19:55.470 align:middle line:84% Because in this case, player one is the unique winner. 00:19:55.470 --> 00:19:58.430 align:middle line:90% So here, there was heads. 00:19:58.430 --> 00:20:02.503 align:middle line:84% And the other two guessed tails, and the coin came up heads. 00:20:02.503 --> 00:20:03.920 align:middle line:84% And so that's what-- you're really 00:20:03.920 --> 00:20:08.180 align:middle line:84% trying to be the unique winner in this game, the less 00:20:08.180 --> 00:20:09.710 align:middle line:90% common winner. 00:20:09.710 --> 00:20:12.840 align:middle line:84% That's when you get lots of points. 00:20:12.840 --> 00:20:18.320 align:middle line:84% So let's compute the expectation. 00:20:18.320 --> 00:20:20.030 align:middle line:84% To compute the expectation, we need 00:20:20.030 --> 00:20:22.250 align:middle line:84% to know what the probabilities are in these branches 00:20:22.250 --> 00:20:24.300 align:middle line:90% or something about them. 00:20:24.300 --> 00:20:26.830 align:middle line:84% And I'm going to start by saying, 00:20:26.830 --> 00:20:40.170 align:middle line:84% let's suppose that all players are uniform and independent. 00:20:40.170 --> 00:20:46.770 align:middle line:90% So 50-50 on every branch. 00:20:46.770 --> 00:20:50.730 align:middle line:84% Then the probabilities down here-- let me use color-- 00:20:50.730 --> 00:20:56.040 align:middle line:84% are going to be 1/16 in the whole tree or 1/8 in this 00:20:56.040 --> 00:20:56.620 align:middle line:90% subtree. 00:20:56.620 --> 00:20:59.950 align:middle line:84% So I'm going to just condition on the H in the beginning. 00:20:59.950 --> 00:21:02.375 align:middle line:90% And so I'll write 1/8 down here. 00:21:02.375 --> 00:21:04.000 align:middle line:84% But if you were drawing the whole tree, 00:21:04.000 --> 00:21:07.590 align:middle line:90% you'd have to write 1/16. 00:21:07.590 --> 00:21:10.246 align:middle line:84% Each of these things occurs with 1/8 probability. 00:21:10.246 --> 00:21:13.200 align:middle line:90% 00:21:13.200 --> 00:21:14.020 align:middle line:90% You get the idea. 00:21:14.020 --> 00:21:15.190 align:middle line:90% Each of these 1/8. 00:21:15.190 --> 00:21:23.440 align:middle line:84% And so we get the expected value is 1/8 times the sum of these 00:21:23.440 --> 00:21:23.940 align:middle line:90% values-- 00:21:23.940 --> 00:21:32.430 align:middle line:84% 0 plus 0, plus 1, minus 2, plus 1, minus 2, plus 4, minus 2, 00:21:32.430 --> 00:21:34.830 align:middle line:90% which I hope is equal to 0. 00:21:34.830 --> 00:21:39.340 align:middle line:84% Because the negative is 6, and the positive is also 6. 00:21:39.340 --> 00:21:42.790 align:middle line:84% So this is what you might call a fair game. 00:21:42.790 --> 00:21:44.980 align:middle line:90% Your expected winnings are 0. 00:21:44.980 --> 00:21:46.063 align:middle line:90% So you're just as likely-- 00:21:46.063 --> 00:21:47.563 align:middle line:84% well, it doesn't mean you're just as 00:21:47.563 --> 00:21:49.000 align:middle line:90% likely to go above or go below. 00:21:49.000 --> 00:21:51.065 align:middle line:84% But, on average, you will break even. 00:21:51.065 --> 00:21:54.330 align:middle line:90% 00:21:54.330 --> 00:21:57.540 align:middle line:90% But that's not what we did. 00:21:57.540 --> 00:22:04.020 align:middle line:84% We considered the case where, let's say, player two does not 00:22:04.020 --> 00:22:07.740 align:middle line:84% equal player three, which is something that player 00:22:07.740 --> 00:22:08.740 align:middle line:90% three could enforce. 00:22:08.740 --> 00:22:11.940 align:middle line:84% And that's what Jenny was doing, always choosing opposite of Cleo 00:22:11.940 --> 00:22:13.560 align:middle line:90% here. 00:22:13.560 --> 00:22:20.472 align:middle line:84% In that case, it's really bad for me. 00:22:20.472 --> 00:22:22.930 align:middle line:84% Because whatever I guess out here-- if there's both a heads 00:22:22.930 --> 00:22:25.377 align:middle line:84% and tails out here, if they're always guessing opposite, 00:22:25.377 --> 00:22:26.210 align:middle line:90% one of them's heads. 00:22:26.210 --> 00:22:27.080 align:middle line:90% One of them's tails. 00:22:27.080 --> 00:22:31.490 align:middle line:84% So that means I always match one of them guaranteed. 00:22:31.490 --> 00:22:35.510 align:middle line:84% And the other of them is in the unique pile. 00:22:35.510 --> 00:22:37.580 align:middle line:90% And so they have a 50% chance. 00:22:37.580 --> 00:22:39.790 align:middle line:84% Like in this situation, Cleo has a 50% chance 00:22:39.790 --> 00:22:42.520 align:middle line:90% of getting that plus 4. 00:22:42.520 --> 00:22:47.300 align:middle line:84% There's still a 50% chance that it came up tails. 00:22:47.300 --> 00:22:48.560 align:middle line:90% Of course, not with this die. 00:22:48.560 --> 00:22:53.270 align:middle line:84% But, in principle, there's a 50% chance that it comes up tails. 00:22:53.270 --> 00:22:54.920 align:middle line:84% But then we each only get 1 point. 00:22:54.920 --> 00:22:57.140 align:middle line:84% So if you're focusing on my winnings, 00:22:57.140 --> 00:23:04.330 align:middle line:84% I will never get $4 in a round with Jenny's strategy. 00:23:04.330 --> 00:23:08.230 align:middle line:84% I will always get either 1 or minus 2. 00:23:08.230 --> 00:23:10.370 align:middle line:84% That doesn't sound like a very good game for me. 00:23:10.370 --> 00:23:13.990 align:middle line:84% If I randomly get-- a 50% chance I get a 1, 50% chance 00:23:13.990 --> 00:23:17.510 align:middle line:90% I get a minus 2. 00:23:17.510 --> 00:23:24.155 align:middle line:84% So this was, I guess, expected net for player 1. 00:23:24.155 --> 00:23:25.530 align:middle line:84% That's what we've been computing. 00:23:25.530 --> 00:23:31.280 align:middle line:84% So in this situation the expected net for player 1, 00:23:31.280 --> 00:23:39.440 align:middle line:84% If we want to be precise, you can say it's this subtree here. 00:23:39.440 --> 00:23:42.200 align:middle line:84% This is when the last two guesses 00:23:42.200 --> 00:23:45.650 align:middle line:84% are heads tails or tails heads is the middle part 00:23:45.650 --> 00:23:46.770 align:middle line:90% of this subtree. 00:23:46.770 --> 00:23:49.760 align:middle line:84% And you can see my possible outcomes 00:23:49.760 --> 00:23:52.370 align:middle line:90% are limited to these guys. 00:23:52.370 --> 00:23:57.480 align:middle line:84% And it turns out they're equally likely. 00:23:57.480 --> 00:24:00.030 align:middle line:84% In fact, independent of what player two does. 00:24:00.030 --> 00:24:01.530 align:middle line:90% I won't try to prove that here. 00:24:01.530 --> 00:24:06.890 align:middle line:84% But because all that matters is this last coin flip, if it's 00:24:06.890 --> 00:24:08.730 align:middle line:90% heads, then I get a plus 1. 00:24:08.730 --> 00:24:10.850 align:middle line:84% If it's tails, I get a minus 2 every time 00:24:10.850 --> 00:24:12.360 align:middle line:90% in this half of the tree. 00:24:12.360 --> 00:24:15.420 align:middle line:90% And so it ends up being half-- 00:24:15.420 --> 00:24:19.080 align:middle line:84% in other words, the average of 1 minus 2, also 00:24:19.080 --> 00:24:22.350 align:middle line:90% known as minus a half. 00:24:22.350 --> 00:24:27.490 align:middle line:84% So, given how Jenny played, in fact, I expect to lose. 00:24:27.490 --> 00:24:28.990 align:middle line:90% And you saw that happening. 00:24:28.990 --> 00:24:30.610 align:middle line:90% Jenny also happened to lose. 00:24:30.610 --> 00:24:32.760 align:middle line:84% The way this is practical, if you're a gambler, 00:24:32.760 --> 00:24:36.340 align:middle line:84% is if you have two players that collude, 00:24:36.340 --> 00:24:38.880 align:middle line:84% they decide whatever happens, we're 00:24:38.880 --> 00:24:42.240 align:middle line:84% going to take our total winnings and split it evenly between us 00:24:42.240 --> 00:24:44.710 align:middle line:84% at the end of the game after all the rounds are done. 00:24:44.710 --> 00:24:46.930 align:middle line:84% Then this is not quite a guarantee, 00:24:46.930 --> 00:24:52.380 align:middle line:84% but it's a pretty good bet that together, Cleo and Jenny would 00:24:52.380 --> 00:24:57.090 align:middle line:84% get most of the coins, most of the money, 00:24:57.090 --> 00:25:03.090 align:middle line:84% because their expected outcome is plus a half, I believe, 00:25:03.090 --> 00:25:05.250 align:middle line:90% should be the complement. 00:25:05.250 --> 00:25:06.390 align:middle line:90% Plus-- yeah. 00:25:06.390 --> 00:25:07.920 align:middle line:90% Anyway, it's positive. 00:25:07.920 --> 00:25:11.470 align:middle line:90% And my expectation is negative. 00:25:11.470 --> 00:25:15.720 align:middle line:84% So yeah, don't play this game if you don't trust players two 00:25:15.720 --> 00:25:17.910 align:middle line:90% and three not to collude. 00:25:17.910 --> 00:25:19.950 align:middle line:90% All right. 00:25:19.950 --> 00:25:20.790 align:middle line:90% Cool. 00:25:20.790 --> 00:25:23.230 align:middle line:84% You might think this is a silly game. 00:25:23.230 --> 00:25:24.520 align:middle line:90% Who would ever play this game? 00:25:24.520 --> 00:25:28.710 align:middle line:84% Well, the Massachusetts lottery played this game. 00:25:28.710 --> 00:25:30.570 align:middle line:90% It's a little more complicated. 00:25:30.570 --> 00:25:32.620 align:middle line:84% I think you had up to four digits, 00:25:32.620 --> 00:25:35.550 align:middle line:84% and you tried to guess a sequence of four digits that 00:25:35.550 --> 00:25:36.790 align:middle line:90% would come up randomly. 00:25:36.790 --> 00:25:38.748 align:middle line:84% I mean, it's pretty similar to lotteries today. 00:25:38.748 --> 00:25:40.200 align:middle line:90% They just have more numbers. 00:25:40.200 --> 00:25:43.240 align:middle line:84% And there's this great MIT professor, Herman Chernoff, 00:25:43.240 --> 00:25:45.540 align:middle line:84% who we'll talk about some of his probability results 00:25:45.540 --> 00:25:47.385 align:middle line:90% later in the class. 00:25:47.385 --> 00:25:50.010 align:middle line:84% And he wrote this paper, "How to Beat the Massachusetts Numbers 00:25:50.010 --> 00:25:52.890 align:middle line:84% Game," which is what the lottery was called originally. 00:25:52.890 --> 00:25:54.550 align:middle line:90% And he just looked at some data. 00:25:54.550 --> 00:25:56.403 align:middle line:90% And it's a similar situation. 00:25:56.403 --> 00:25:57.820 align:middle line:84% If there are many lottery winners, 00:25:57.820 --> 00:25:59.380 align:middle line:90% they evenly split the pot. 00:25:59.380 --> 00:26:02.380 align:middle line:84% Also, the state takes like 40% or 60% cut. 00:26:02.380 --> 00:26:04.110 align:middle line:90% I forget which way it goes. 00:26:04.110 --> 00:26:07.763 align:middle line:84% But even then, he made money off of the lottery. 00:26:07.763 --> 00:26:09.930 align:middle line:84% And there have been more recent versions of this run 00:26:09.930 --> 00:26:13.630 align:middle line:84% by MIT students, I hear, to make a lot of money 00:26:13.630 --> 00:26:15.202 align:middle line:90% off the Massachusetts lottery. 00:26:15.202 --> 00:26:16.660 align:middle line:84% And they've since changed the rules 00:26:16.660 --> 00:26:19.180 align:middle line:84% to make those strategies hard to run. 00:26:19.180 --> 00:26:22.850 align:middle line:84% But, in particular, he executed this, made a lot of money. 00:26:22.850 --> 00:26:25.060 align:middle line:84% The whole-- the idea is if you can 00:26:25.060 --> 00:26:28.060 align:middle line:84% guess numbers that are less likely to be guessed 00:26:28.060 --> 00:26:31.370 align:middle line:84% by other people, and all numbers are equally likely, 00:26:31.370 --> 00:26:33.170 align:middle line:90% then you expect to win. 00:26:33.170 --> 00:26:36.280 align:middle line:84% And it turns out people are so predictable. 00:26:36.280 --> 00:26:39.130 align:middle line:84% Let's just say zeros and 9's were unpopular. 00:26:39.130 --> 00:26:41.980 align:middle line:84% Smaller digits were popular, and so he ended up just 00:26:41.980 --> 00:26:49.270 align:middle line:84% going with random numbers from 5 to 8, I think and 0 to 9 maybe. 00:26:49.270 --> 00:26:53.943 align:middle line:84% And he won not a lot of money, but he won some money. 00:26:53.943 --> 00:26:55.610 align:middle line:84% You can read the paper for more details. 00:26:55.610 --> 00:26:56.860 align:middle line:90% It's a pretty fun read. 00:26:56.860 --> 00:26:59.620 align:middle line:84% It goes through all the probability and stuff. 00:26:59.620 --> 00:27:01.760 align:middle line:84% This is like a simplified version. 00:27:01.760 --> 00:27:04.030 align:middle line:90% So cool, practical. 00:27:04.030 --> 00:27:07.300 align:middle line:90% Now you know how to make money. 00:27:07.300 --> 00:27:11.930 align:middle line:84% Let's go on to some other ways to compute 00:27:11.930 --> 00:27:14.245 align:middle line:84% expectation and some more examples and so on. 00:27:14.245 --> 00:27:16.755 align:middle line:90% 00:27:16.755 --> 00:27:17.255 align:middle line:90% Cool. 00:27:17.255 --> 00:27:19.880 align:middle line:90% 00:27:19.880 --> 00:27:20.780 align:middle line:90% All right. 00:27:20.780 --> 00:27:24.680 align:middle line:84% We talked about-- so here, we computed 00:27:24.680 --> 00:27:27.420 align:middle line:84% if we roll a single six-sided die, 00:27:27.420 --> 00:27:30.540 align:middle line:84% the expected value of that single die is 3.5. 00:27:30.540 --> 00:27:34.010 align:middle line:84% What if I rolled two dice, A and B, 00:27:34.010 --> 00:27:36.560 align:middle line:84% and I want to know the expectation of dice A, 00:27:36.560 --> 00:27:40.350 align:middle line:84% die A. Should also be 3.5, right? 00:27:40.350 --> 00:27:42.540 align:middle line:84% If I roll two dice, it shouldn't change anything. 00:27:42.540 --> 00:27:45.060 align:middle line:84% But if I computed it explicitly in this way, 00:27:45.060 --> 00:27:47.210 align:middle line:84% it would be really tedious because the sample 00:27:47.210 --> 00:27:50.940 align:middle line:84% space would go from 1 through 6 to 1 through 6 squared. 00:27:50.940 --> 00:27:54.900 align:middle line:84% So now there's 36 different possible outcomes. 00:27:54.900 --> 00:27:57.230 align:middle line:84% And for each of them, I need to write down a term 00:27:57.230 --> 00:27:58.500 align:middle line:90% and add it all up. 00:27:58.500 --> 00:28:01.380 align:middle line:84% It will come out to 3.5, but that's super annoying to do. 00:28:01.380 --> 00:28:03.350 align:middle line:84% So here's a nice theorem that makes 00:28:03.350 --> 00:28:09.300 align:middle line:84% it easy to do this with six terms instead of 36 terms. 00:28:09.300 --> 00:28:13.360 align:middle line:84% If I have a random variable x, and I want its expectation, 00:28:13.360 --> 00:28:22.800 align:middle line:84% this is equal to the sum where x is in the range of big X 00:28:22.800 --> 00:28:30.840 align:middle line:84% of probability X equals X times little x. 00:28:30.840 --> 00:28:31.760 align:middle line:90% Let me erase this-- 00:28:31.760 --> 00:28:37.020 align:middle line:90% 00:28:37.020 --> 00:28:39.840 align:middle line:90% my terrible losing game. 00:28:39.840 --> 00:28:41.430 align:middle line:90% How many coin flips till tails? 00:28:41.430 --> 00:28:43.870 align:middle line:90% 1, 2, 3, 4, 5, 6, 7. 00:28:43.870 --> 00:28:49.560 align:middle line:84% All right, Keep that in mind-- seven coin flips to get a tails. 00:28:49.560 --> 00:28:50.260 align:middle line:90% All right. 00:28:50.260 --> 00:28:52.230 align:middle line:90% So here's a formula. 00:28:52.230 --> 00:28:55.980 align:middle line:84% It's really exactly what we did here. 00:28:55.980 --> 00:28:59.410 align:middle line:84% So we wrote down, here are all the possible outcomes, 00:28:59.410 --> 00:29:02.890 align:middle line:84% and we multiply them by what the random variable could be. 00:29:02.890 --> 00:29:04.890 align:middle line:84% And when there was an indicator, random variable 00:29:04.890 --> 00:29:05.800 align:middle line:90% is either 0 or 1. 00:29:05.800 --> 00:29:07.030 align:middle line:90% So there were two groups. 00:29:07.030 --> 00:29:12.750 align:middle line:84% In general, there's whatever the range of x groups. 00:29:12.750 --> 00:29:17.190 align:middle line:90% So if I-- yeah. 00:29:17.190 --> 00:29:19.500 align:middle line:84% So this is exactly this kind of clustering where 00:29:19.500 --> 00:29:20.920 align:middle line:90% we split it into two sums. 00:29:20.920 --> 00:29:22.680 align:middle line:84% Each of them was the probability of being 00:29:22.680 --> 00:29:25.360 align:middle line:84% in that group times the outcome of that group. 00:29:25.360 --> 00:29:30.330 align:middle line:84% This sum is exactly probability of not naught A. 00:29:30.330 --> 00:29:37.950 align:middle line:84% And this sum is probability of A. 00:29:37.950 --> 00:29:41.020 align:middle line:84% And in general, if we split into not two groups, 00:29:41.020 --> 00:29:43.500 align:middle line:84% but all the different possible values of big X-- 00:29:43.500 --> 00:29:47.070 align:middle line:84% it can be little x for every little x in the range of big X, 00:29:47.070 --> 00:29:52.180 align:middle line:84% then this is-- we get a sum of probability of outcomes, 00:29:52.180 --> 00:29:54.653 align:middle line:84% which is exactly the probability that big X equals small x. 00:29:54.653 --> 00:29:56.070 align:middle line:84% And then we multiply that by small 00:29:56.070 --> 00:30:01.810 align:middle line:84% x because that's what we have in the definition of expectation. 00:30:01.810 --> 00:30:03.370 align:middle line:90% So this is an easy theorem. 00:30:03.370 --> 00:30:05.110 align:middle line:84% I won't write down a proof because it's 00:30:05.110 --> 00:30:06.870 align:middle line:84% this thing generalized a little bit. 00:30:06.870 --> 00:30:09.880 align:middle line:90% 00:30:09.880 --> 00:30:14.835 align:middle line:84% And let's do an example maybe on a different board. 00:30:14.835 --> 00:30:25.900 align:middle line:90% 00:30:25.900 --> 00:30:32.693 align:middle line:84% Suppose I want to know the number of hearts in a 3-card 00:30:32.693 --> 00:30:33.193 align:middle line:90% hand. 00:30:33.193 --> 00:30:36.670 align:middle line:90% 00:30:36.670 --> 00:30:40.160 align:middle line:84% So what that means is I have 52 cards in a deck. 00:30:40.160 --> 00:30:43.100 align:middle line:84% I randomly choose 3 cards without replacement. 00:30:43.100 --> 00:30:44.770 align:middle line:84% So just like you're drawing from the top 00:30:44.770 --> 00:30:46.130 align:middle line:90% of a randomly shuffled deck. 00:30:46.130 --> 00:30:49.420 align:middle line:84% I want to know how many hearts do I get in expectation? 00:30:49.420 --> 00:30:54.760 align:middle line:90% Let's call this-- yeah-- 00:30:54.760 --> 00:31:04.970 align:middle line:84% expected number of hearts is, if we just use the definition, 00:31:04.970 --> 00:31:11.930 align:middle line:84% probability of w times the number of hearts in omega. 00:31:11.930 --> 00:31:13.280 align:middle line:90% This would be the definition. 00:31:13.280 --> 00:31:19.920 align:middle line:90% And S, here, is size 52 cubed. 00:31:19.920 --> 00:31:20.870 align:middle line:90% Not quite. 00:31:20.870 --> 00:31:23.460 align:middle line:90% 52 times 51 times 50. 00:31:23.460 --> 00:31:26.675 align:middle line:90% It's really big. 00:31:26.675 --> 00:31:28.800 align:middle line:84% So I don't want to have to compute this explicitly. 00:31:28.800 --> 00:31:31.320 align:middle line:84% Instead, I'm going to use this formula and say, well look, 00:31:31.320 --> 00:31:34.500 align:middle line:84% the number of hearts in a 3-card hand is 0, 1, 2, or 3. 00:31:34.500 --> 00:31:38.750 align:middle line:84% So I can write this as the probability 00:31:38.750 --> 00:31:44.010 align:middle line:84% that number of hearts equals 0 times 0. 00:31:44.010 --> 00:31:45.740 align:middle line:90% OK, that's not very interesting. 00:31:45.740 --> 00:31:51.800 align:middle line:84% We can take the probability number of hearts equals 1 00:31:51.800 --> 00:31:56.180 align:middle line:84% and multiply that by 1, plus the probability the number of hearts 00:31:56.180 --> 00:32:01.470 align:middle line:84% equals 2 times 2, plus probability number of hearts 00:32:01.470 --> 00:32:05.280 align:middle line:90% equals 3 times 3. 00:32:05.280 --> 00:32:08.070 align:middle line:90% And so now I have three terms-- 00:32:08.070 --> 00:32:09.250 align:middle line:90% relatively easy. 00:32:09.250 --> 00:32:12.750 align:middle line:84% And these are, hopefully, things you know how to compute. 00:32:12.750 --> 00:32:16.380 align:middle line:84% For example, probability I get three hearts-- 00:32:16.380 --> 00:32:18.750 align:middle line:90% I'm going to cheat. 00:32:18.750 --> 00:32:27.553 align:middle line:84% This is like 13 choose 3 divided by 52 choose 3. 00:32:27.553 --> 00:32:31.500 align:middle line:90% 00:32:31.500 --> 00:32:34.420 align:middle line:84% Because there are 13 different heart cards. 00:32:34.420 --> 00:32:37.710 align:middle line:84% So the number of ways to choose three hearts out of those 13 00:32:37.710 --> 00:32:38.883 align:middle line:90% heart cards is 13 choose 3. 00:32:38.883 --> 00:32:40.800 align:middle line:84% The total number of ways to choose three cards 00:32:40.800 --> 00:32:42.040 align:middle line:90% is 52 choose 3. 00:32:42.040 --> 00:32:43.420 align:middle line:90% And so that's a probability. 00:32:43.420 --> 00:32:47.260 align:middle line:84% And you need to multiply it by 3 to get this formula. 00:32:47.260 --> 00:32:49.620 align:middle line:84% And there's some similar things for these terms. 00:32:49.620 --> 00:32:51.875 align:middle line:84% I think I maybe won't write them down. 00:32:51.875 --> 00:32:54.480 align:middle line:90% 00:32:54.480 --> 00:32:57.130 align:middle line:84% Maybe I'll write down the middle one just for kicks. 00:32:57.130 --> 00:33:01.080 align:middle line:84% I went through the effort of figuring things out. 00:33:01.080 --> 00:33:03.660 align:middle line:84% This point should be something that you could do 00:33:03.660 --> 00:33:05.075 align:middle line:90% just like on your problem set. 00:33:05.075 --> 00:33:07.980 align:middle line:90% 00:33:07.980 --> 00:33:11.730 align:middle line:84% These are combinatorics problems you know how to solve. 00:33:11.730 --> 00:33:14.350 align:middle line:84% So you get some sum of binomial terms. 00:33:14.350 --> 00:33:17.230 align:middle line:84% But hey, it has some reasonably small size. 00:33:17.230 --> 00:33:20.220 align:middle line:84% Of course, these have huge factorials in them. 00:33:20.220 --> 00:33:22.600 align:middle line:90% But it's only choose 1, 2, or 3. 00:33:22.600 --> 00:33:23.710 align:middle line:90% So they're not too big. 00:33:23.710 --> 00:33:28.770 align:middle line:84% You plug this into Wolfram Alpha, say, and you get 3/4-- 00:33:28.770 --> 00:33:31.595 align:middle line:90% very nice number. 00:33:31.595 --> 00:33:32.720 align:middle line:90% We might come back to this. 00:33:32.720 --> 00:33:37.980 align:middle line:90% 00:33:37.980 --> 00:33:41.440 align:middle line:84% A lot of the time, because this is 6 1,200, 00:33:41.440 --> 00:33:45.060 align:middle line:84% we're working with integers, natural numbers. 00:33:45.060 --> 00:33:49.620 align:middle line:84% All of the random variables I've defined so far, I believe, 00:33:49.620 --> 00:33:52.350 align:middle line:90% are integers. 00:33:52.350 --> 00:33:55.390 align:middle line:84% We happen to get an expectation that was not an integer. 00:33:55.390 --> 00:33:58.660 align:middle line:84% But all of the values that D could take on 00:33:58.660 --> 00:34:01.910 align:middle line:84% or hearts could take on was an integer. 00:34:01.910 --> 00:34:10.330 align:middle line:84% So let's consider the case where I'll say the codomain of X 00:34:10.330 --> 00:34:12.040 align:middle line:90% is the natural number-- 00:34:12.040 --> 00:34:14.560 align:middle line:90% 0, 1, 2, 3. 00:34:14.560 --> 00:34:19.150 align:middle line:84% Remember, codomain is just what you declare this function could 00:34:19.150 --> 00:34:20.003 align:middle line:90% take on for values. 00:34:20.003 --> 00:34:21.670 align:middle line:84% It doesn't have to take on those values. 00:34:21.670 --> 00:34:24.550 align:middle line:84% Range here-- the intent is these are just the values that 00:34:24.550 --> 00:34:26.360 align:middle line:90% are taken on by this function. 00:34:26.360 --> 00:34:28.292 align:middle line:90% And so I didn't have to write-- 00:34:28.292 --> 00:34:30.500 align:middle line:84% even though the number of hearts could, I don't know, 00:34:30.500 --> 00:34:34.370 align:middle line:84% be 100 in some other card, in this sample space, 00:34:34.370 --> 00:34:35.870 align:middle line:90% we know it's just 0, 1, 2, or 3. 00:34:35.870 --> 00:34:39.350 align:middle line:90% And so we just need four terms. 00:34:39.350 --> 00:34:41.800 align:middle line:84% But let's suppose potentially all the natural numbers 00:34:41.800 --> 00:34:44.500 align:middle line:84% are there, but not necessarily all of them. 00:34:44.500 --> 00:34:49.360 align:middle line:84% Then expected value of X has a nice form. 00:34:49.360 --> 00:34:52.656 align:middle line:84% And this is a formula I encourage you not to memorize. 00:34:52.656 --> 00:34:54.489 align:middle line:84% You should write it down on your cheat sheet 00:34:54.489 --> 00:34:58.580 align:middle line:84% and always look it up from there because it's easy to get wrong, 00:34:58.580 --> 00:34:59.665 align:middle line:90% especially the next one. 00:34:59.665 --> 00:35:03.410 align:middle line:90% 00:35:03.410 --> 00:35:09.800 align:middle line:84% 1 to infinity, probability x equals I 00:35:09.800 --> 00:35:14.670 align:middle line:84% times I. This is just exactly this theorem. 00:35:14.670 --> 00:35:15.590 align:middle line:90% Nothing exciting. 00:35:15.590 --> 00:35:17.700 align:middle line:84% Here, I wrote a sum over a range. 00:35:17.700 --> 00:35:19.860 align:middle line:84% Now I'm writing a sum over an explicit range. 00:35:19.860 --> 00:35:23.690 align:middle line:84% The one clever idea I had is skip 0 because 0 00:35:23.690 --> 00:35:26.310 align:middle line:84% is going to get-- the 0 case is going to get multiplied by 0, 00:35:26.310 --> 00:35:27.720 align:middle line:90% so that never matters. 00:35:27.720 --> 00:35:29.070 align:middle line:90% So we get to start at 1. 00:35:29.070 --> 00:35:31.550 align:middle line:84% Wow, I saved this one step for evermore. 00:35:31.550 --> 00:35:33.560 align:middle line:84% And then sum up to infinity if we 00:35:33.560 --> 00:35:39.230 align:middle line:84% might have to go that high So that is true. 00:35:39.230 --> 00:35:42.260 align:middle line:84% But there's an even cooler form, which 00:35:42.260 --> 00:35:50.600 align:middle line:84% is sum I equals 1 to infinity of probability of X greater than 00:35:50.600 --> 00:35:57.570 align:middle line:84% or equal to I. Or there's no I here-- 00:35:57.570 --> 00:36:06.690 align:middle line:84% I equals 0 to infinity probability X greater than I. 00:36:06.690 --> 00:36:11.010 align:middle line:84% So this part is just this theorem. 00:36:11.010 --> 00:36:14.640 align:middle line:84% This step is I just changed greater than or equal to greater 00:36:14.640 --> 00:36:20.020 align:middle line:84% than and changed where I started I. So here, it started at I,1. 00:36:20.020 --> 00:36:22.060 align:middle line:90% And here, it started at I, 0. 00:36:22.060 --> 00:36:25.120 align:middle line:84% And so here, I'm effectively adding 1 to I. 00:36:25.120 --> 00:36:26.250 align:middle line:90% And so I get to do-- 00:36:26.250 --> 00:36:28.240 align:middle line:84% or subtracting 1 from I. So I do greater than 00:36:28.240 --> 00:36:29.740 align:middle line:84% instead of greater than or equal to. 00:36:29.740 --> 00:36:32.890 align:middle line:84% So these two arrow equalities should be obvious. 00:36:32.890 --> 00:36:35.320 align:middle line:84% And this one is the interesting one. 00:36:35.320 --> 00:36:37.925 align:middle line:90% So let's prove that. 00:36:37.925 --> 00:36:46.050 align:middle line:90% 00:36:46.050 --> 00:36:47.433 align:middle line:90% OK, proof. 00:36:47.433 --> 00:37:04.440 align:middle line:90% 00:37:04.440 --> 00:37:05.200 align:middle line:90% OK. 00:37:05.200 --> 00:37:09.150 align:middle line:84% I want to compute this sum and claim that it equals this one. 00:37:09.150 --> 00:37:13.350 align:middle line:84% And I'm going to do it by writing it in an obvious way. 00:37:13.350 --> 00:37:15.600 align:middle line:84% First, I'm just going to write out the infinite series 00:37:15.600 --> 00:37:16.630 align:middle line:90% with dot, dot, dot. 00:37:16.630 --> 00:37:26.100 align:middle line:84% So we've got X equal to 1 plus X gradient equal to 2 plus X 00:37:26.100 --> 00:37:27.147 align:middle line:90% gradient equal to 3. 00:37:27.147 --> 00:37:29.730 align:middle line:84% Just give me a moment, and I'll write all the remaining terms. 00:37:29.730 --> 00:37:32.040 align:middle line:90% Might take a while. 00:37:32.040 --> 00:37:39.330 align:middle line:84% And now I want to expand these out into separate pieces 00:37:39.330 --> 00:37:46.407 align:middle line:84% in terms of X equal I. So this is, what is the probability 00:37:46.407 --> 00:37:47.740 align:middle line:90% that X is greater or equal to 1. 00:37:47.740 --> 00:37:52.120 align:middle line:84% Given that, again, that they're all natural numbers, 00:37:52.120 --> 00:37:56.320 align:middle line:84% we know this is probability X equals 1 plus probability X 00:37:56.320 --> 00:38:01.360 align:middle line:84% equals 2 plus probability X equals 3 plus and so 00:38:01.360 --> 00:38:05.770 align:middle line:90% on up to infinity. 00:38:05.770 --> 00:38:08.590 align:middle line:90% What's this one? 00:38:08.590 --> 00:38:10.760 align:middle line:84% Well, same thing, but starting here. 00:38:10.760 --> 00:38:15.070 align:middle line:84% Probability X equals 2 plus probability 00:38:15.070 --> 00:38:19.270 align:middle line:90% X equals 3 plus and so on. 00:38:19.270 --> 00:38:23.360 align:middle line:90% This one starts over here. 00:38:23.360 --> 00:38:27.670 align:middle line:90% Probability X equals 3. 00:38:27.670 --> 00:38:34.600 align:middle line:84% So if I now sum this way down the columns, 00:38:34.600 --> 00:38:41.110 align:middle line:84% I get 1 times probability X equals 1 plus 2 times 00:38:41.110 --> 00:38:44.050 align:middle line:90% probability X equals 2. 00:38:44.050 --> 00:38:46.520 align:middle line:84% Because there are two values here-- 00:38:46.520 --> 00:38:49.620 align:middle line:84% there are two ranges here for X that include X equals 2-- 00:38:49.620 --> 00:38:50.870 align:middle line:90% these two. 00:38:50.870 --> 00:38:53.550 align:middle line:84% 2 is not greater than or equal to 3, 4, 5, and so on. 00:38:53.550 --> 00:38:55.220 align:middle line:90% So I get exactly two of them. 00:38:55.220 --> 00:39:01.440 align:middle line:84% I get exactly three probability X equals 3, and so on. 00:39:01.440 --> 00:39:03.740 align:middle line:84% And if you work out the dot, dot, dots via proof 00:39:03.740 --> 00:39:07.640 align:middle line:84% by induction, you get some I probability X equals I. 00:39:07.640 --> 00:39:11.120 align:middle line:90% And that's what we wanted. 00:39:11.120 --> 00:39:13.550 align:middle line:90% 1 to infinity. 00:39:13.550 --> 00:39:18.350 align:middle line:84% So that's why these two sums are the same. 00:39:18.350 --> 00:39:20.270 align:middle line:84% So the point is to do this work once, 00:39:20.270 --> 00:39:23.090 align:middle line:84% so that in the future if you have a problem, 00:39:23.090 --> 00:39:25.650 align:middle line:84% you can use any of these formulas you want. 00:39:25.650 --> 00:39:27.870 align:middle line:84% You could use the definition of expectation, 00:39:27.870 --> 00:39:31.560 align:middle line:84% which is a sum over outcomes in the sample space. 00:39:31.560 --> 00:39:33.770 align:middle line:84% Or you could use this formula-- sum 00:39:33.770 --> 00:39:38.070 align:middle line:84% over all the things in the range with equality here. 00:39:38.070 --> 00:39:39.900 align:middle line:90% And then you multiply it by x. 00:39:39.900 --> 00:39:43.203 align:middle line:84% Or if in the special case where your function only takes 00:39:43.203 --> 00:39:45.620 align:middle line:84% on integer values-- so this doesn't work if you have other 00:39:45.620 --> 00:39:47.400 align:middle line:90% real numbers-- 00:39:47.400 --> 00:39:50.082 align:middle line:84% then you can use one of these three formulas. 00:39:50.082 --> 00:39:52.540 align:middle line:84% Probably the last one or the first one are the useful ones. 00:39:52.540 --> 00:39:55.540 align:middle line:84% If it's easy to compute the probability X equals something, 00:39:55.540 --> 00:39:57.450 align:middle line:90% then maybe use this formula. 00:39:57.450 --> 00:40:00.570 align:middle line:84% If it's easy to compute the probability that X is bigger 00:40:00.570 --> 00:40:03.630 align:middle line:84% than something-- this was CDF and this was PMF, remember, 00:40:03.630 --> 00:40:04.560 align:middle line:90% from last time. 00:40:04.560 --> 00:40:07.900 align:middle line:84% It's easy, if you have a nice form for PMF, then use that. 00:40:07.900 --> 00:40:11.100 align:middle line:84% If you have a nice form for CDF, use that, 00:40:11.100 --> 00:40:13.300 align:middle line:90% provided you work with integers. 00:40:13.300 --> 00:40:16.120 align:middle line:84% Do not try to mix these formulas together. 00:40:16.120 --> 00:40:19.920 align:middle line:84% Don't compute the sum of I equals 1 00:40:19.920 --> 00:40:22.140 align:middle line:84% to infinity, probability X greater than or equal to I 00:40:22.140 --> 00:40:24.000 align:middle line:84% times I. That would give you something 00:40:24.000 --> 00:40:25.290 align:middle line:90% that is not expectation. 00:40:25.290 --> 00:40:28.060 align:middle line:90% But surprisingly, students try. 00:40:28.060 --> 00:40:29.530 align:middle line:90% So don't do that. 00:40:29.530 --> 00:40:34.480 align:middle line:84% Use this one, or this one, or this one, or the definition. 00:40:34.480 --> 00:40:39.390 align:middle line:90% 00:40:39.390 --> 00:40:40.630 align:middle line:90% Cool. 00:40:40.630 --> 00:40:42.370 align:middle line:84% Let's do some more interesting examples. 00:40:42.370 --> 00:40:50.918 align:middle line:90% 00:40:50.918 --> 00:40:51.854 align:middle line:90% Lonely brace. 00:40:51.854 --> 00:41:01.620 align:middle line:90% 00:41:01.620 --> 00:41:03.600 align:middle line:90% Is there questions? 00:41:03.600 --> 00:41:04.120 align:middle line:90% All right. 00:41:04.120 --> 00:41:13.650 align:middle line:90% 00:41:13.650 --> 00:41:16.890 align:middle line:84% Next problem is called mean time to failure. 00:41:16.890 --> 00:41:18.040 align:middle line:90% It's a little negative. 00:41:18.040 --> 00:41:21.840 align:middle line:84% You can use it to compute successful things as well. 00:41:21.840 --> 00:41:23.982 align:middle line:84% It's related to this problem of how many times-- 00:41:23.982 --> 00:41:26.190 align:middle line:84% it is essentially this problem of how many times do I 00:41:26.190 --> 00:41:29.650 align:middle line:84% have to flip a coin until I get tails, or until I get heads? 00:41:29.650 --> 00:41:32.387 align:middle line:90% Either way. 00:41:32.387 --> 00:41:34.720 align:middle line:84% I'll write it as heads because that's what my notes say. 00:41:34.720 --> 00:41:42.070 align:middle line:90% 00:41:42.070 --> 00:41:43.870 align:middle line:84% It's less interesting-- it's more 00:41:43.870 --> 00:41:45.350 align:middle line:90% interesting for a biased coin. 00:41:45.350 --> 00:41:54.550 align:middle line:84% So, let's say, the probability of getting heads is p. 00:41:54.550 --> 00:42:05.530 align:middle line:84% I want to know what is the expected number of flips 00:42:05.530 --> 00:42:10.270 align:middle line:90% until I get heads. 00:42:10.270 --> 00:42:13.030 align:middle line:84% So you could use this if you're analyzing coin flipping. 00:42:13.030 --> 00:42:16.690 align:middle line:84% But you could also use it for things like my laptop, 00:42:16.690 --> 00:42:20.650 align:middle line:84% every day there's a 1 in 1,000,000 chance that the hard 00:42:20.650 --> 00:42:22.160 align:middle line:90% drive dies or something. 00:42:22.160 --> 00:42:27.430 align:middle line:84% So how many days do I expect until the hard drive dies? 00:42:27.430 --> 00:42:30.010 align:middle line:84% Hard drives are usually designed-- engineered to last, 00:42:30.010 --> 00:42:31.580 align:middle line:90% I don't know, five years or so. 00:42:31.580 --> 00:42:34.150 align:middle line:84% And then all your hard drives-- that's for spinning disks. 00:42:34.150 --> 00:42:36.710 align:middle line:90% SSDs are probably different. 00:42:36.710 --> 00:42:39.070 align:middle line:90% They depend on usage. 00:42:39.070 --> 00:42:43.165 align:middle line:84% So that's something you might care about. 00:42:43.165 --> 00:42:45.290 align:middle line:84% And that's why this is called mean time to failure. 00:42:45.290 --> 00:42:47.240 align:middle line:84% It could be something more positive, 00:42:47.240 --> 00:42:49.290 align:middle line:84% like maybe you have an algorithm. 00:42:49.290 --> 00:42:51.390 align:middle line:84% You're doing randomized algorithms in the future, 00:42:51.390 --> 00:42:54.470 align:middle line:84% and you have an algorithm that either solves your problem 00:42:54.470 --> 00:42:59.700 align:middle line:84% with some probability 1 minus P. With probability P, 00:42:59.700 --> 00:43:00.780 align:middle line:90% it solves your problem. 00:43:00.780 --> 00:43:03.920 align:middle line:84% And with probability 1 minus P it says, I don't know. 00:43:03.920 --> 00:43:07.190 align:middle line:84% There are lots of algorithms like this out there. 00:43:07.190 --> 00:43:10.190 align:middle line:84% How many times do I have to run this algorithm until it actually 00:43:10.190 --> 00:43:11.192 align:middle line:90% gives me an answer? 00:43:11.192 --> 00:43:13.650 align:middle line:84% So I can run it, and it gives-- it might say, I don't know. 00:43:13.650 --> 00:43:15.150 align:middle line:84% But if I run it again, it might then 00:43:15.150 --> 00:43:17.570 align:middle line:84% solve the problem, depending on the random coin 00:43:17.570 --> 00:43:18.720 align:middle line:90% flips that it does. 00:43:18.720 --> 00:43:21.360 align:middle line:84% So then it's mean time to success. 00:43:21.360 --> 00:43:23.360 align:middle line:84% How many times do I have to flip this coin, do I 00:43:23.360 --> 00:43:25.970 align:middle line:84% have to run this algorithm, given some probability of it 00:43:25.970 --> 00:43:28.430 align:middle line:84% succeeding-- that's P. How many times do 00:43:28.430 --> 00:43:31.200 align:middle line:84% I have to keep going until I get a successful answer? 00:43:31.200 --> 00:43:33.300 align:middle line:90% Lots of other examples. 00:43:33.300 --> 00:43:37.490 align:middle line:90% We will get to them in a moment. 00:43:37.490 --> 00:43:40.710 align:middle line:84% Now, our traditional approach to solving tough problems like this 00:43:40.710 --> 00:43:41.980 align:middle line:90% is to draw a tree. 00:43:41.980 --> 00:43:43.960 align:middle line:84% In this case, the tree is infinite. 00:43:43.960 --> 00:43:46.500 align:middle line:84% So that's going to be a little harder to draw. 00:43:46.500 --> 00:43:52.260 align:middle line:84% We have P versus 1 minus P. Whenever we get a heads, 00:43:52.260 --> 00:43:53.440 align:middle line:90% we stop. 00:43:53.440 --> 00:43:57.540 align:middle line:84% But this tree keeps going off to the right. 00:43:57.540 --> 00:44:00.900 align:middle line:84% So that's-- maybe let's not look at the infinite tree. 00:44:00.900 --> 00:44:03.570 align:middle line:90% Let's use some formulas. 00:44:03.570 --> 00:44:09.090 align:middle line:84% So I'm going to call this number-- 00:44:09.090 --> 00:44:11.830 align:middle line:84% the number of flips I'm going to call F. 00:44:11.830 --> 00:44:14.920 align:middle line:84% And then we want to compute-- remember, F is a function. 00:44:14.920 --> 00:44:17.500 align:middle line:84% So there are various outcomes here 00:44:17.500 --> 00:44:21.280 align:middle line:84% which correspond to the executions-- 00:44:21.280 --> 00:44:23.110 align:middle line:90% the various numbers of flips. 00:44:23.110 --> 00:44:26.130 align:middle line:84% It's going to be tails, tails, tails, tails, tails, tails I 00:44:26.130 --> 00:44:27.552 align:middle line:90% times and then heads. 00:44:27.552 --> 00:44:29.010 align:middle line:84% Those are the various outcomes that 00:44:29.010 --> 00:44:31.218 align:middle line:84% could happen because we stop whenever we get a heads. 00:44:31.218 --> 00:44:33.040 align:middle line:84% So it could be 0 tails and then heads, 00:44:33.040 --> 00:44:35.880 align:middle line:90% 1 tails then heads, and so on. 00:44:35.880 --> 00:44:37.810 align:middle line:90% So that's our sample space. 00:44:37.810 --> 00:44:39.510 align:middle line:84% And then F is a function that maps 00:44:39.510 --> 00:44:42.850 align:middle line:84% that sample space to count how many total flips were there-- 00:44:42.850 --> 00:44:44.170 align:middle line:90% heads and tails? 00:44:44.170 --> 00:44:45.840 align:middle line:84% So it's always going to be at least one. 00:44:45.840 --> 00:44:48.900 align:middle line:84% And then expect-- so F is a function-- 00:44:48.900 --> 00:44:53.770 align:middle line:84% can take on some integer value between 0 and 1, I guess, 00:44:53.770 --> 00:44:55.343 align:middle line:90% and infinity. 00:44:55.343 --> 00:44:56.760 align:middle line:84% And now we're going to reduce this 00:44:56.760 --> 00:44:58.390 align:middle line:84% to a single number via expectation. 00:44:58.390 --> 00:45:01.780 align:middle line:84% What is the average number of flips that we expect to take? 00:45:01.780 --> 00:45:04.900 align:middle line:90% So that's expectation of F. 00:45:04.900 --> 00:45:09.540 align:middle line:84% Now, we will use our latest and greatest formula-- 00:45:09.540 --> 00:45:11.640 align:middle line:90% this one. 00:45:11.640 --> 00:45:13.560 align:middle line:90% Let me memorize it for a second. 00:45:13.560 --> 00:45:21.520 align:middle line:84% It's just sum X greater than I. Already forgotten it. 00:45:21.520 --> 00:45:24.510 align:middle line:84% It's too far from over there to over here. 00:45:24.510 --> 00:45:30.015 align:middle line:84% Was it probability F is greater than I, I hope. 00:45:30.015 --> 00:45:30.515 align:middle line:90% Yes. 00:45:30.515 --> 00:45:33.190 align:middle line:90% 00:45:33.190 --> 00:45:35.590 align:middle line:90% So this is that CDF. 00:45:35.590 --> 00:45:40.330 align:middle line:84% And I claim this is not too hard to figure out 00:45:40.330 --> 00:45:41.960 align:middle line:90% using what we know. 00:45:41.960 --> 00:45:44.740 align:middle line:84% Because we're assuming each flip is 00:45:44.740 --> 00:45:46.520 align:middle line:84% independent from subsequent flips, 00:45:46.520 --> 00:45:48.925 align:middle line:84% like the flips of this coin are mutually independent. 00:45:48.925 --> 00:45:51.550 align:middle line:84% It's not some magical coin that always comes up heads or always 00:45:51.550 --> 00:45:52.610 align:middle line:90% comes up tails or something. 00:45:52.610 --> 00:45:54.318 align:middle line:84% Well, that would be represented by the P. 00:45:54.318 --> 00:45:58.850 align:middle line:84% But there's no correlation between the different flips. 00:45:58.850 --> 00:46:07.480 align:middle line:84% So for there to be at least greater than I flips 00:46:07.480 --> 00:46:10.240 align:middle line:84% before I get a heads, that means that the first I 00:46:10.240 --> 00:46:11.780 align:middle line:90% flips are tails. 00:46:11.780 --> 00:46:14.680 align:middle line:90% These are equivalent events. 00:46:14.680 --> 00:46:25.933 align:middle line:84% This F greater than I is the same as first I flips are tails. 00:46:25.933 --> 00:46:28.413 align:middle line:90% 00:46:28.413 --> 00:46:29.830 align:middle line:84% And then after that, I don't know, 00:46:29.830 --> 00:46:31.122 align:middle line:90% I might get a heads right away. 00:46:31.122 --> 00:46:32.630 align:middle line:90% Then F is I plus 1. 00:46:32.630 --> 00:46:34.080 align:middle line:90% Or it might take longer. 00:46:34.080 --> 00:46:37.530 align:middle line:84% But F being greater than I is exactly this situation. 00:46:37.530 --> 00:46:40.660 align:middle line:84% So what is the probability that the first I flips are tails? 00:46:40.660 --> 00:46:45.140 align:middle line:90% 00:46:45.140 --> 00:46:47.665 align:middle line:84% 1 minus P to the I. I saw you whispering it. 00:46:47.665 --> 00:46:51.140 align:middle line:90% 00:46:51.140 --> 00:46:53.880 align:middle line:90% I feel like-- yeah, cool. 00:46:53.880 --> 00:46:56.480 align:middle line:84% Now, the hard part is working out this sum, 00:46:56.480 --> 00:46:58.110 align:middle line:90% or remembering what this sum is. 00:46:58.110 --> 00:47:00.290 align:middle line:90% This is a geometric sum. 00:47:00.290 --> 00:47:02.150 align:middle line:84% So 1 minus P to the I because this is 00:47:02.150 --> 00:47:03.450 align:middle line:90% the chance of getting a tails. 00:47:03.450 --> 00:47:04.530 align:middle line:84% And then they're each independent, 00:47:04.530 --> 00:47:05.738 align:middle line:90% so we just take the product-- 00:47:05.738 --> 00:47:08.210 align:middle line:90% via product rule. 00:47:08.210 --> 00:47:10.020 align:middle line:90% This is a geometric thing. 00:47:10.020 --> 00:47:14.990 align:middle line:84% Sum of X to the I from I equals 0 to infinity is 1 over 1 00:47:14.990 --> 00:47:21.830 align:middle line:84% minus X. So this is 1 over 1 minus X here is 1 minus p. 00:47:21.830 --> 00:47:25.310 align:middle line:84% So 1 minus 1 minus p, that's also known as p. 00:47:25.310 --> 00:47:27.450 align:middle line:90% So this whole thing is 1 over p. 00:47:27.450 --> 00:47:31.555 align:middle line:84% Amazingly simple formula for the mean time to failure. 00:47:31.555 --> 00:47:33.930 align:middle line:84% This is why mean time to failure is such a famous example 00:47:33.930 --> 00:47:36.490 align:middle line:84% because it has such a beautiful solution. 00:47:36.490 --> 00:47:40.686 align:middle line:84% So if I'm flipping a fair coin, 50/50. 00:47:40.686 --> 00:47:42.160 align:middle line:90% P is a half. 00:47:42.160 --> 00:47:46.180 align:middle line:84% I expect to have to flip it twice to get a heads or a tails. 00:47:46.180 --> 00:47:50.440 align:middle line:84% Now, reality is not equal to the average, 00:47:50.440 --> 00:47:54.660 align:middle line:84% sadly, so I had to flip seven times to get a heads or tails. 00:47:54.660 --> 00:47:57.960 align:middle line:84% But if I was gaming for heads, it would be very fast. 00:47:57.960 --> 00:48:00.060 align:middle line:84% It happened in the very first step. 00:48:00.060 --> 00:48:03.940 align:middle line:90% So that's the expectation. 00:48:03.940 --> 00:48:06.190 align:middle line:84% It doesn't tell you everything about the distribution. 00:48:06.190 --> 00:48:13.560 align:middle line:84% This distribution has a name Alluded to 00:48:13.560 --> 00:48:16.120 align:middle line:90% by this geometric series. 00:48:16.120 --> 00:48:18.720 align:middle line:90% This was the CDF. 00:48:18.720 --> 00:48:21.426 align:middle line:84% We call this the geometric distribution. 00:48:21.426 --> 00:48:32.190 align:middle line:90% 00:48:32.190 --> 00:48:37.830 align:middle line:84% It comes up somewhat frequently in real life distributions. 00:48:37.830 --> 00:48:44.700 align:middle line:84% If I were to write down the PMF, it's 1 minus p to the I times p. 00:48:44.700 --> 00:48:47.940 align:middle line:84% This is the chance that I get I tails and then a heads. 00:48:47.940 --> 00:48:50.400 align:middle line:90% That's-- well, OK. 00:48:50.400 --> 00:48:53.070 align:middle line:84% It's either that or probability-- the way 00:48:53.070 --> 00:48:57.150 align:middle line:84% I defined it before, it was F equals I plus 1. 00:48:57.150 --> 00:48:58.900 align:middle line:90% So one of these is true. 00:48:58.900 --> 00:48:59.400 align:middle line:90% Pick one. 00:48:59.400 --> 00:49:01.442 align:middle line:84% These are both called the geometric distribution. 00:49:01.442 --> 00:49:03.120 align:middle line:84% If you look at the Wikipedia definition 00:49:03.120 --> 00:49:04.380 align:middle line:84% of geometric distribution, it says 00:49:04.380 --> 00:49:06.838 align:middle line:84% there are two distributions called geometric distribution-- 00:49:06.838 --> 00:49:08.080 align:middle line:90% this one and this one. 00:49:08.080 --> 00:49:11.130 align:middle line:90% So pick your poison. 00:49:11.130 --> 00:49:13.530 align:middle line:90% Both are called that. 00:49:13.530 --> 00:49:18.060 align:middle line:84% So here is an application, another application 00:49:18.060 --> 00:49:20.490 align:middle line:90% of mean time to failure. 00:49:20.490 --> 00:49:24.450 align:middle line:84% I don't know about you, but I like collecting things. 00:49:24.450 --> 00:49:26.760 align:middle line:84% Everyone likes collecting something. 00:49:26.760 --> 00:49:30.675 align:middle line:84% To me, the epitome of collecting things is gacha. 00:49:30.675 --> 00:49:32.050 align:middle line:84% Oh, that's the video game version 00:49:32.050 --> 00:49:33.425 align:middle line:84% of Gacha, where you're collecting 00:49:33.425 --> 00:49:35.560 align:middle line:90% characters in a video game. 00:49:35.560 --> 00:49:38.060 align:middle line:90% Please do not play these games. 00:49:38.060 --> 00:49:39.910 align:middle line:90% They're annoyingly addictive-- 00:49:39.910 --> 00:49:41.950 align:middle line:84% I spend most of my nights these days. 00:49:41.950 --> 00:49:45.940 align:middle line:84% But gacha is named after this notion of-- 00:49:45.940 --> 00:49:50.140 align:middle line:84% you can see it in the top right there-- is a gashapon, 00:49:50.140 --> 00:49:54.070 align:middle line:90% which is like these little-- 00:49:54.070 --> 00:49:57.310 align:middle line:84% if you grew up in the US, you have vending machines 00:49:57.310 --> 00:49:58.520 align:middle line:90% that give you candy. 00:49:58.520 --> 00:50:02.420 align:middle line:84% Well, these are like the really cool vending machines in Japan. 00:50:02.420 --> 00:50:05.000 align:middle line:90% This was a Gachapon festival. 00:50:05.000 --> 00:50:08.470 align:middle line:84% And you pick your favorite weird thing 00:50:08.470 --> 00:50:11.420 align:middle line:84% that you can get in a little ball out of this machine. 00:50:11.420 --> 00:50:15.310 align:middle line:84% You put in your 200 yen, and you get a randomly selected ball 00:50:15.310 --> 00:50:18.380 align:middle line:90% from those sets of weird toys. 00:50:18.380 --> 00:50:21.980 align:middle line:84% So this is like a toy delivery-- or not delivery service. 00:50:21.980 --> 00:50:25.250 align:middle line:84% It's a toy random selection service. 00:50:25.250 --> 00:50:27.800 align:middle line:84% And if you want to collect all the different things 00:50:27.800 --> 00:50:31.130 align:middle line:84% on this list, how many times do you have to do it? 00:50:31.130 --> 00:50:33.530 align:middle line:84% This is a problem called coupon collectors, 00:50:33.530 --> 00:50:36.200 align:middle line:84% or the Gotcha problem, I would say. 00:50:36.200 --> 00:50:41.600 align:middle line:84% But we'll solve it in general in recitation tomorrow. 00:50:41.600 --> 00:50:44.360 align:middle line:84% But I'll solve a very simple version here, 00:50:44.360 --> 00:50:47.485 align:middle line:84% which is a nice application of this-- 00:50:47.485 --> 00:50:51.800 align:middle line:90% 00:50:51.800 --> 00:50:53.080 align:middle line:90% of mean time to failure. 00:50:53.080 --> 00:50:57.800 align:middle line:90% 00:50:57.800 --> 00:51:01.093 align:middle line:84% Let's solve the special case where there are only two toys. 00:51:01.093 --> 00:51:07.460 align:middle line:90% 00:51:07.460 --> 00:51:10.410 align:middle line:84% gacha-- gacha is the sound that when you crank the thing, 00:51:10.410 --> 00:51:12.260 align:middle line:84% it's like, gacha, gacha, gacha, puh, 00:51:12.260 --> 00:51:15.890 align:middle line:90% is when the ball drops down-- 00:51:15.890 --> 00:51:20.960 align:middle line:84% gacha with two toys or characters 00:51:20.960 --> 00:51:22.880 align:middle line:90% or whatever you're collecting. 00:51:22.880 --> 00:51:25.030 align:middle line:90% And so you get a random choice. 00:51:25.030 --> 00:51:26.930 align:middle line:84% Let's say it's 50/50 between the two. 00:51:26.930 --> 00:51:31.090 align:middle line:84% So in your first pull is what we usually call these, at least 00:51:31.090 --> 00:51:33.590 align:middle line:84% in the video game context, our first crank 00:51:33.590 --> 00:51:38.710 align:middle line:84% in the physical context, you get one of the toys. 00:51:38.710 --> 00:51:41.470 align:middle line:84% So now it's really just, how many steps 00:51:41.470 --> 00:51:44.620 align:middle line:84% do I need until I get the second toy? 00:51:44.620 --> 00:51:45.460 align:middle line:90% Two. 00:51:45.460 --> 00:51:47.300 align:middle line:84% This is the coin-flipping problem. 00:51:47.300 --> 00:52:00.640 align:middle line:84% So how many more pulls until you get a distinct toy? 00:52:00.640 --> 00:52:05.090 align:middle line:84% And in this case, it's going to be 50/50 each pull. 00:52:05.090 --> 00:52:14.530 align:middle line:84% And so you expect two more, which is a total of three. 00:52:14.530 --> 00:52:17.400 align:middle line:84% Now, when you have k toys you want to collect, it's annoying. 00:52:17.400 --> 00:52:19.150 align:middle line:84% Because in the beginning, it's really easy 00:52:19.150 --> 00:52:22.412 align:middle line:84% to get lots of distinct toys because you're very likely. 00:52:22.412 --> 00:52:24.370 align:middle line:84% But when you're trying to collect the last toy, 00:52:24.370 --> 00:52:26.333 align:middle line:84% it's very unlikely that you pull the last toy 00:52:26.333 --> 00:52:28.750 align:middle line:84% because there's only like a 1 in k chance that you get it. 00:52:28.750 --> 00:52:30.660 align:middle line:84% So you end up with some harmonic series, 00:52:30.660 --> 00:52:32.080 align:middle line:90% and you get a log factor. 00:52:32.080 --> 00:52:34.230 align:middle line:90% You'll see that tomorrow. 00:52:34.230 --> 00:52:36.270 align:middle line:90% OK. 00:52:36.270 --> 00:52:39.660 align:middle line:84% Let's do a little more theory and then some more examples. 00:52:39.660 --> 00:52:41.520 align:middle line:90% Where should I go? 00:52:41.520 --> 00:52:42.395 align:middle line:90% Let's erase gambling. 00:52:42.395 --> 00:52:47.435 align:middle line:90% 00:52:47.435 --> 00:52:49.060 align:middle line:84% I feel like I should have a disclaimer. 00:52:49.060 --> 00:52:51.810 align:middle line:90% Please don't gamble. 00:52:51.810 --> 00:52:52.910 align:middle line:90% It's not good for you. 00:52:52.910 --> 00:52:59.760 align:middle line:90% 00:52:59.760 --> 00:53:02.122 align:middle line:90% Please just simulate gambling. 00:53:02.122 --> 00:53:02.705 align:middle line:90% That's better. 00:53:02.705 --> 00:53:06.570 align:middle line:90% 00:53:06.570 --> 00:53:09.870 align:middle line:84% I was writing a simulation of this game 00:53:09.870 --> 00:53:12.720 align:middle line:84% this morning to see what were my chances of actually 00:53:12.720 --> 00:53:15.730 align:middle line:84% winning despite the odds being against my favor? 00:53:15.730 --> 00:53:20.170 align:middle line:84% And with 10 rounds, I had a 20% chance of winning. 00:53:20.170 --> 00:53:22.190 align:middle line:84% So luckily, the demo went in my favor. 00:53:22.190 --> 00:53:24.250 align:middle line:84% But there was a decent chance that I 00:53:24.250 --> 00:53:29.620 align:middle line:84% could have won even though the expectation was against me. 00:53:29.620 --> 00:53:30.570 align:middle line:90% So I guess. 00:53:30.570 --> 00:53:44.920 align:middle line:90% 00:53:44.920 --> 00:53:48.520 align:middle line:84% If there is one thing in this class 00:53:48.520 --> 00:53:50.480 align:middle line:84% that you remember about probability, 00:53:50.480 --> 00:53:53.570 align:middle line:84% I suggest linearity of expectation. 00:53:53.570 --> 00:53:56.330 align:middle line:84% It is simultaneously the coolest thing, in my opinion, 00:53:56.330 --> 00:53:58.510 align:middle line:84% about probability and the most useful thing 00:53:58.510 --> 00:53:59.840 align:middle line:90% for solving problems. 00:53:59.840 --> 00:54:03.400 align:middle line:90% So what is it? 00:54:03.400 --> 00:54:05.690 align:middle line:84% The expectation function is linear. 00:54:05.690 --> 00:54:08.380 align:middle line:90% Let me formalize. 00:54:08.380 --> 00:54:11.050 align:middle line:84% Suppose you have two random variables, X and Y. 00:54:11.050 --> 00:54:14.890 align:middle line:84% And you take their sum, and then you take their expectation. 00:54:14.890 --> 00:54:18.426 align:middle line:84% It turns out that equals the sum of the expectations. 00:54:18.426 --> 00:54:25.040 align:middle line:90% 00:54:25.040 --> 00:54:27.950 align:middle line:84% Or if you take a random variable and you multiply it 00:54:27.950 --> 00:54:33.830 align:middle line:84% by a constant, you can pull that constant out. 00:54:33.830 --> 00:54:37.950 align:middle line:84% And so, more generally, this is the full linearity form. 00:54:37.950 --> 00:54:41.120 align:middle line:90% If I have a-- 00:54:41.120 --> 00:54:42.210 align:middle line:90% where am I here-- 00:54:42.210 --> 00:54:48.050 align:middle line:84% sum I equals 1 to n of Ci times Xi, 00:54:48.050 --> 00:54:55.370 align:middle line:84% this is the sum I equals 1 to n of Ci 00:54:55.370 --> 00:54:57.610 align:middle line:90% times the expected value of Xi. 00:54:57.610 --> 00:55:01.100 align:middle line:90% 00:55:01.100 --> 00:55:03.930 align:middle line:84% It takes almost longer to write it down than to prove it. 00:55:03.930 --> 00:55:08.150 align:middle line:84% I mean, it's actually not hard to prove if you stare 00:55:08.150 --> 00:55:10.370 align:middle line:90% at this definition enough. 00:55:10.370 --> 00:55:13.230 align:middle line:84% It's because this is a linear function. 00:55:13.230 --> 00:55:16.070 align:middle line:84% If you look at how it depends on the X's, it's 00:55:16.070 --> 00:55:17.770 align:middle line:90% a linear function in X. 00:55:17.770 --> 00:55:19.990 align:middle line:84% You've got some constant weights. 00:55:19.990 --> 00:55:23.680 align:middle line:84% I mean, they depend on your probability setup. 00:55:23.680 --> 00:55:26.340 align:middle line:84% But if you mess around with X and don't mess around 00:55:26.340 --> 00:55:28.510 align:middle line:84% with your sample space or its probabilities, 00:55:28.510 --> 00:55:31.320 align:middle line:84% then this is just a sum of constants 00:55:31.320 --> 00:55:34.240 align:middle line:90% times X applied to things. 00:55:34.240 --> 00:55:37.890 align:middle line:84% So, in particular, if I replace X here with X plus Y, 00:55:37.890 --> 00:55:39.760 align:middle line:84% and I just jam that into the sum, 00:55:39.760 --> 00:55:41.370 align:middle line:90% I can split this into two sums. 00:55:41.370 --> 00:55:43.410 align:middle line:84% And lo and behold, I get expectation of X 00:55:43.410 --> 00:55:45.330 align:middle line:84% plus expectation of Y. And these other things 00:55:45.330 --> 00:55:46.450 align:middle line:90% follow in the same way. 00:55:46.450 --> 00:55:47.700 align:middle line:90% So I won't write down a proof. 00:55:47.700 --> 00:55:50.520 align:middle line:84% There's one in the notes if you want to look. 00:55:50.520 --> 00:55:54.630 align:middle line:84% So it's almost an obvious thing once I tell it to you. 00:55:54.630 --> 00:55:57.930 align:middle line:84% But it's extremely powerful, mainly 00:55:57.930 --> 00:56:00.930 align:middle line:90% because it has no conditions. 00:56:00.930 --> 00:56:07.530 align:middle line:84% You don't need that the Xi's are independent. 00:56:07.530 --> 00:56:09.670 align:middle line:84% You don't need that they're disjoint. 00:56:09.670 --> 00:56:13.170 align:middle line:84% You don't need anything except that they are functions 00:56:13.170 --> 00:56:15.250 align:middle line:90% over the same sample space. 00:56:15.250 --> 00:56:17.910 align:middle line:84% I guess that's the one assumption you need. 00:56:17.910 --> 00:56:22.280 align:middle line:84% Compare this with-- remember, there's the product rule. 00:56:22.280 --> 00:56:28.740 align:middle line:90% 00:56:28.740 --> 00:56:33.780 align:middle line:84% This only works if A and B are independent. 00:56:33.780 --> 00:56:35.600 align:middle line:90% Or there's the sum rule. 00:56:35.600 --> 00:56:39.750 align:middle line:90% 00:56:39.750 --> 00:56:41.410 align:middle line:90% We use these all the time. 00:56:41.410 --> 00:56:43.600 align:middle line:84% But you have to be very careful when you use them. 00:56:43.600 --> 00:56:44.350 align:middle line:90% You have to check. 00:56:44.350 --> 00:56:46.050 align:middle line:90% Are those really independent? 00:56:46.050 --> 00:56:49.393 align:middle line:84% For the sum rule it only works if A and B are disjoint. 00:56:49.393 --> 00:56:52.110 align:middle line:90% 00:56:52.110 --> 00:56:54.240 align:middle line:84% So when we do law of total probability, 00:56:54.240 --> 00:56:56.080 align:middle line:90% we're using this all the time. 00:56:56.080 --> 00:56:59.590 align:middle line:84% But you have to be very careful that you use it correctly. 00:56:59.590 --> 00:57:01.110 align:middle line:84% Whereas, linearity of expectation, 00:57:01.110 --> 00:57:04.170 align:middle line:84% you don't need to be careful other than making sure you 00:57:04.170 --> 00:57:07.770 align:middle line:84% define your random variables well. 00:57:07.770 --> 00:57:10.520 align:middle line:90% Let's do an example-- 00:57:10.520 --> 00:57:15.130 align:middle line:90% 00:57:15.130 --> 00:57:18.410 align:middle line:84% common one in games, gambling games 00:57:18.410 --> 00:57:20.390 align:middle line:90% and other games, dice games. 00:57:20.390 --> 00:57:27.370 align:middle line:84% Let's say we roll 2 fair two-side-- or sorry-- 00:57:27.370 --> 00:57:29.363 align:middle line:90% six-sided dice. 00:57:29.363 --> 00:57:33.370 align:middle line:90% 00:57:33.370 --> 00:57:38.295 align:middle line:84% What is the expected value of the sum? 00:57:38.295 --> 00:57:42.760 align:middle line:90% 00:57:42.760 --> 00:57:45.910 align:middle line:84% Well, the expected value of the sum 00:57:45.910 --> 00:57:49.430 align:middle line:84% is the expected value of the individual parts. 00:57:49.430 --> 00:57:52.270 align:middle line:90% Let me rewrite this as a-- 00:57:52.270 --> 00:57:54.680 align:middle line:84% instead of words, let's use some notation. 00:57:54.680 --> 00:57:57.310 align:middle line:84% Let's say D1 is the random variable that just gives me what 00:57:57.310 --> 00:57:59.180 align:middle line:90% is the value of the first die. 00:57:59.180 --> 00:58:01.510 align:middle line:84% And D2 is the random variable that gives 00:58:01.510 --> 00:58:02.948 align:middle line:90% me what's the second value. 00:58:02.948 --> 00:58:04.990 align:middle line:84% The sample space here is a pair, an ordered pair, 00:58:04.990 --> 00:58:06.740 align:middle line:84% of the first die, the second die. 00:58:06.740 --> 00:58:08.680 align:middle line:84% So this extracts the first part, and this 00:58:08.680 --> 00:58:10.180 align:middle line:90% extracts the second part. 00:58:10.180 --> 00:58:14.320 align:middle line:84% The expected value of this sum by linearity of expectation 00:58:14.320 --> 00:58:16.086 align:middle line:90% is the sum of the expectations. 00:58:16.086 --> 00:58:18.780 align:middle line:90% 00:58:18.780 --> 00:58:20.620 align:middle line:90% And we already did this example. 00:58:20.620 --> 00:58:26.040 align:middle line:90% This was 3.5, 3.5. 00:58:26.040 --> 00:58:29.970 align:middle line:90% So two total is 7. 00:58:29.970 --> 00:58:31.260 align:middle line:90% Easy. 00:58:31.260 --> 00:58:33.960 align:middle line:84% And what's cool is this didn't even need 00:58:33.960 --> 00:58:35.620 align:middle line:90% that the dice were independent. 00:58:35.620 --> 00:58:37.380 align:middle line:84% You could glue the dice together so 00:58:37.380 --> 00:58:38.980 align:middle line:90% that they always roll together. 00:58:38.980 --> 00:58:40.980 align:middle line:84% As long as that doesn't change the probabilities 00:58:40.980 --> 00:58:44.260 align:middle line:84% of the outcomes, which may be a little questionable with gluing. 00:58:44.260 --> 00:58:45.690 align:middle line:84% But imagine, I don't know, they're 00:58:45.690 --> 00:58:47.270 align:middle line:90% there quantumly entangled. 00:58:47.270 --> 00:58:49.020 align:middle line:84% And however I roll this die, the other one 00:58:49.020 --> 00:58:52.800 align:middle line:84% comes out the same way, this will still be true. 00:58:52.800 --> 00:58:55.050 align:middle line:84% It changes the distribution, but it does not change 00:58:55.050 --> 00:58:57.575 align:middle line:90% the expectation, which is cool. 00:58:57.575 --> 00:59:01.810 align:middle line:90% 00:59:01.810 --> 00:59:04.080 align:middle line:84% So we just needed the probabilities of each outcome 00:59:04.080 --> 00:59:06.420 align:middle line:90% were equal to get the 3.5. 00:59:06.420 --> 00:59:07.330 align:middle line:90% Question? 00:59:07.330 --> 00:59:10.830 align:middle line:90% AUDIENCE: [INAUDIBLE]. 00:59:10.830 --> 00:59:13.470 align:middle line:84% ERIK DEMAINE: Gluing dice together. 00:59:13.470 --> 00:59:16.840 align:middle line:84% I'm imagining-- like if I take two dice. 00:59:16.840 --> 00:59:18.220 align:middle line:90% I glue them together. 00:59:18.220 --> 00:59:24.730 align:middle line:84% Then when I roll that unit, however the first die comes up, 00:59:24.730 --> 00:59:27.760 align:middle line:84% it forces the other one to come up in a particular orientation. 00:59:27.760 --> 00:59:32.040 align:middle line:84% So they're literally tied together so that the-- 00:59:32.040 --> 00:59:35.610 align:middle line:84% D1 determines D2 if they're glued together. 00:59:35.610 --> 00:59:38.050 align:middle line:84% But this formula applies in that situation, 00:59:38.050 --> 00:59:39.480 align:middle line:84% provided it's still equally likely 00:59:39.480 --> 00:59:43.570 align:middle line:84% that you get 1, 2, 3, 4, 5, or 6 for each die individually. 00:59:43.570 --> 00:59:47.660 align:middle line:90% 00:59:47.660 --> 00:59:49.510 align:middle line:84% It's maybe a little clearer with coins. 00:59:49.510 --> 00:59:51.510 align:middle line:84% If I glue two coins together, then whichever 00:59:51.510 --> 00:59:52.950 align:middle line:84% one comes up heads, the other one 00:59:52.950 --> 00:59:56.760 align:middle line:84% also comes up heads or comes up tails, 00:59:56.760 --> 00:59:58.935 align:middle line:84% depending if I glue the head sides together or heads 00:59:58.935 --> 00:59:59.435 align:middle line:90% to tails. 00:59:59.435 --> 01:00:04.050 align:middle line:90% 01:00:04.050 --> 01:00:08.003 align:middle line:84% So I want to do more cool examples. 01:00:08.003 --> 01:00:10.840 align:middle line:90% 01:00:10.840 --> 01:00:18.000 align:middle line:84% And they illustrate how to use linearity of expectation 01:00:18.000 --> 01:00:20.620 align:middle line:90% with a-- 01:00:20.620 --> 01:00:25.990 align:middle line:84% the key to using linearity of expectation is this technique. 01:00:25.990 --> 01:00:28.360 align:middle line:84% If you want to compute the expectation 01:00:28.360 --> 01:00:32.830 align:middle line:84% of some random variable, decompose that random variable 01:00:32.830 --> 01:00:46.780 align:middle line:84% into a sum of other random variables, possibly 01:00:46.780 --> 01:00:50.570 align:middle line:90% indicator random variables. 01:00:50.570 --> 01:00:51.860 align:middle line:90% You see it here already. 01:00:51.860 --> 01:00:53.990 align:middle line:84% I mean, we said, what's the expectation of the sum? 01:00:53.990 --> 01:00:55.510 align:middle line:84% Well, let's decompose that into the first die 01:00:55.510 --> 01:00:56.400 align:middle line:90% and the second die. 01:00:56.400 --> 01:00:59.120 align:middle line:84% And then the sum is the first one plus the second one. 01:00:59.120 --> 01:01:01.965 align:middle line:84% And then we could split it up using linearity of expectation. 01:01:01.965 --> 01:01:04.090 align:middle line:84% And then compute each of these individually instead 01:01:04.090 --> 01:01:06.173 align:middle line:84% of having to look at the giant sample space, which 01:01:06.173 --> 01:01:07.410 align:middle line:90% is the pairs of die rolls. 01:01:07.410 --> 01:01:10.393 align:middle line:90% 01:01:10.393 --> 01:01:16.390 align:middle line:84% So this is the technique we're going 01:01:16.390 --> 01:01:20.510 align:middle line:84% to use now several times, whatever remains of our time. 01:01:20.510 --> 01:01:21.525 align:middle line:90% So first example. 01:01:21.525 --> 01:01:55.420 align:middle line:90% 01:01:55.420 --> 01:01:58.240 align:middle line:84% So previously, we analyzed how many times do I 01:01:58.240 --> 01:02:03.040 align:middle line:84% flip a biased coin until I get a heads? 01:02:03.040 --> 01:02:07.130 align:middle line:84% Now, let's suppose I want two heads-- 01:02:07.130 --> 01:02:09.200 align:middle line:90% natural generalization. 01:02:09.200 --> 01:02:13.820 align:middle line:84% This seems quite a bit harder because you keep flipping. 01:02:13.820 --> 01:02:15.840 align:middle line:84% And then at some point, you get the first heads. 01:02:15.840 --> 01:02:16.590 align:middle line:90% You keep flipping. 01:02:16.590 --> 01:02:19.440 align:middle line:84% At some point, you get the second heads. 01:02:19.440 --> 01:02:22.380 align:middle line:84% Well, that sounds kind of like two steps. 01:02:22.380 --> 01:02:25.460 align:middle line:84% There's how many times do I flip until I get the first head? 01:02:25.460 --> 01:02:26.960 align:middle line:84% And then there's how many more times 01:02:26.960 --> 01:02:29.120 align:middle line:84% I have to flip until I get the second heads? 01:02:29.120 --> 01:02:32.510 align:middle line:84% Both of those problems sound like mean time to failure. 01:02:32.510 --> 01:02:36.950 align:middle line:84% So let's write that out carefully with random variables. 01:02:36.950 --> 01:02:43.100 align:middle line:84% So let's call this number of flips F. 01:02:43.100 --> 01:02:54.920 align:middle line:84% Then F equals F1 plus F2, where F1 01:02:54.920 --> 01:03:00.170 align:middle line:84% is the number of flips from the beginning 01:03:00.170 --> 01:03:06.150 align:middle line:90% until we get the first head. 01:03:06.150 --> 01:03:18.240 align:middle line:84% And F2 is the number of flips starting after the first head 01:03:18.240 --> 01:03:22.010 align:middle line:90% until the second head. 01:03:22.010 --> 01:03:24.700 align:middle line:90% 01:03:24.700 --> 01:03:26.910 align:middle line:84% So to be precise, the sample space 01:03:26.910 --> 01:03:32.280 align:middle line:84% here is strings of H's and T's, where 01:03:32.280 --> 01:03:36.270 align:middle line:84% there's exactly two H's, and the last character is 01:03:36.270 --> 01:03:39.180 align:middle line:84% an H. Because I keep flipping heads, tails, heads, tails, 01:03:39.180 --> 01:03:42.130 align:middle line:84% heads, tails until I get the second H, then I stop. 01:03:42.130 --> 01:03:44.140 align:middle line:90% So that's my sample space. 01:03:44.140 --> 01:03:47.370 align:middle line:84% And F1 is just counting how many flips are there up 01:03:47.370 --> 01:03:49.030 align:middle line:90% to including the first head? 01:03:49.030 --> 01:03:52.980 align:middle line:84% And F2 is counting how many flips are there starting right 01:03:52.980 --> 01:03:54.780 align:middle line:84% after the first head and counting up to 01:03:54.780 --> 01:03:57.370 align:middle line:84% and including the second head, which is the end of the string? 01:03:57.370 --> 01:04:02.430 align:middle line:84% So together, F1 and F2 count the total number of flips. 01:04:02.430 --> 01:04:06.520 align:middle line:84% But crucially, F1 and F2, if you think about it, 01:04:06.520 --> 01:04:10.093 align:middle line:84% are exactly the mean time to failure problems 01:04:10.093 --> 01:04:11.260 align:middle line:90% because they were counting-- 01:04:11.260 --> 01:04:12.910 align:middle line:90% I mean, F1 is most obvious. 01:04:12.910 --> 01:04:16.240 align:middle line:84% It's how many flips do I do until the first head? 01:04:16.240 --> 01:04:29.280 align:middle line:84% So we know here the expected value of F 01:04:29.280 --> 01:04:31.176 align:middle line:90% Is the sum of the expectations. 01:04:31.176 --> 01:04:37.230 align:middle line:90% 01:04:37.230 --> 01:04:40.860 align:middle line:84% F1 is almost literally the definition of mean time 01:04:40.860 --> 01:04:41.620 align:middle line:90% to failure. 01:04:41.620 --> 01:04:45.990 align:middle line:90% So it's 1 over P in expectation. 01:04:45.990 --> 01:04:48.420 align:middle line:84% F2, if you think about it a little bit, 01:04:48.420 --> 01:04:51.460 align:middle line:84% this is the same experiment, but starting after the first heads. 01:04:51.460 --> 01:04:53.290 align:middle line:84% And because all the flips are independent, 01:04:53.290 --> 01:04:55.120 align:middle line:84% it doesn't matter what happened before now. 01:04:55.120 --> 01:04:59.740 align:middle line:84% It's still going to be the same situation for F2. 01:04:59.740 --> 01:05:05.210 align:middle line:84% And so the expectation is 2 over P. Almost as simple. 01:05:05.210 --> 01:05:09.460 align:middle line:84% And, in general, if I want K heads, 01:05:09.460 --> 01:05:12.910 align:middle line:84% then the expected number of flips I need to do 01:05:12.910 --> 01:05:16.700 align:middle line:84% is K over P by linearity of expectation. 01:05:16.700 --> 01:05:17.290 align:middle line:90% Beautiful. 01:05:17.290 --> 01:05:18.030 align:middle line:90% So simple. 01:05:18.030 --> 01:05:23.860 align:middle line:90% 01:05:23.860 --> 01:05:25.770 align:middle line:84% Let's do another coin-flipping example. 01:05:25.770 --> 01:05:33.310 align:middle line:90% 01:05:33.310 --> 01:05:35.020 align:middle line:84% So here, I stopped the experiment 01:05:35.020 --> 01:05:36.650 align:middle line:84% when I got a desired number of heads. 01:05:36.650 --> 01:05:39.200 align:middle line:84% What if I just flip a coin n times? 01:05:39.200 --> 01:05:41.510 align:middle line:84% No matter what, I always flip n times. 01:05:41.510 --> 01:05:45.280 align:middle line:84% How many heads do I expect to get? 01:05:45.280 --> 01:05:48.210 align:middle line:90% That's kind of the dual problem. 01:05:48.210 --> 01:05:57.800 align:middle line:90% 01:05:57.800 --> 01:06:00.470 align:middle line:90% What do se want to do? 01:06:00.470 --> 01:06:01.970 align:middle line:90% Let's do biased again. 01:06:01.970 --> 01:06:06.110 align:middle line:84% n biased coins-- again, independent. 01:06:06.110 --> 01:06:18.290 align:middle line:90% 01:06:18.290 --> 01:06:21.440 align:middle line:84% Let H be the number of heads in those n coin flips. 01:06:21.440 --> 01:06:29.060 align:middle line:84% We can rewrite this as a sum of n indicator random variables, 01:06:29.060 --> 01:06:40.310 align:middle line:84% where Hi is indicator random variable for the ith coin coming 01:06:40.310 --> 01:06:40.930 align:middle line:90% up heads. 01:06:40.930 --> 01:06:50.978 align:middle line:90% 01:06:50.978 --> 01:06:56.320 align:middle line:84% So expected value of H by linearity is the sum 01:06:56.320 --> 01:06:58.313 align:middle line:84% of the expected values of the Hi's. 01:06:58.313 --> 01:07:04.450 align:middle line:90% 01:07:04.450 --> 01:07:07.360 align:middle line:84% But the coins are all identical because they're independent, 01:07:07.360 --> 01:07:08.570 align:middle line:90% and they have the same bias. 01:07:08.570 --> 01:07:10.900 align:middle line:84% I didn't mention it, but we're supposing, again, 01:07:10.900 --> 01:07:12.285 align:middle line:90% probability of heads is p. 01:07:12.285 --> 01:07:15.970 align:middle line:90% 01:07:15.970 --> 01:07:18.970 align:middle line:84% So what's the-- so this is just going 01:07:18.970 --> 01:07:22.420 align:middle line:84% to be n times the expectation of Hi, 01:07:22.420 --> 01:07:24.670 align:middle line:84% because they all behave the same. 01:07:24.670 --> 01:07:27.392 align:middle line:84% And what's the expectation of Hi? 01:07:27.392 --> 01:07:28.850 align:middle line:84% Well, we worked that out over here. 01:07:28.850 --> 01:07:31.750 align:middle line:84% If you have an indicator random variable, the expectation of Hi 01:07:31.750 --> 01:07:34.510 align:middle line:84% is the probability that it equals 1. 01:07:34.510 --> 01:07:36.520 align:middle line:84% So the probability of the event, which here 01:07:36.520 --> 01:07:37.940 align:middle line:90% was the probability of heads. 01:07:37.940 --> 01:07:40.570 align:middle line:90% So that's just p. 01:07:40.570 --> 01:07:48.670 align:middle line:84% So this is n times probability Hi equals 1, which is n times p. 01:07:48.670 --> 01:07:51.613 align:middle line:90% OK, maybe obvious, but-- 01:07:51.613 --> 01:07:55.100 align:middle line:90% 01:07:55.100 --> 01:07:55.600 align:middle line:90% Yeah. 01:07:55.600 --> 01:07:57.600 align:middle line:84% We didn't even need that these were independent. 01:07:57.600 --> 01:08:00.220 align:middle line:90% That's where it's less obvious. 01:08:00.220 --> 01:08:02.980 align:middle line:84% It just-- all we need is that for each coin, 01:08:02.980 --> 01:08:07.860 align:middle line:84% the probability of it coming up heads is p. 01:08:07.860 --> 01:08:10.450 align:middle line:84% But it could be all the coins always flip the same. 01:08:10.450 --> 01:08:12.340 align:middle line:84% They're always all heads or all tails, 01:08:12.340 --> 01:08:14.440 align:middle line:90% but heads with probability p. 01:08:14.440 --> 01:08:16.479 align:middle line:84% Or it could be either independent. 01:08:16.479 --> 01:08:18.220 align:middle line:90% These are different setups. 01:08:18.220 --> 01:08:20.910 align:middle line:84% But the expected number of heads in all cases 01:08:20.910 --> 01:08:24.630 align:middle line:84% will be n times p by linearity of expectation. 01:08:24.630 --> 01:08:25.834 align:middle line:90% So that's pretty cool. 01:08:25.834 --> 01:08:31.439 align:middle line:90% 01:08:31.439 --> 01:08:35.340 align:middle line:90% You can do a similar trick for-- 01:08:35.340 --> 01:08:36.660 align:middle line:90% did I erase it-- 01:08:36.660 --> 01:08:38.760 align:middle line:90% the hearts problem. 01:08:38.760 --> 01:08:39.680 align:middle line:90% I think I erased it. 01:08:39.680 --> 01:08:43.374 align:middle line:90% 01:08:43.374 --> 01:08:43.874 align:middle line:90% Yeah. 01:08:43.874 --> 01:08:46.439 align:middle line:90% 01:08:46.439 --> 01:08:48.450 align:middle line:90% I'll just tell you briefly. 01:08:48.450 --> 01:08:53.740 align:middle line:84% So with the hearts, you can say the number of hearts 01:08:53.740 --> 01:09:02.420 align:middle line:84% is a sum of 13 different indicator random variables. 01:09:02.420 --> 01:09:10.142 align:middle line:84% Part i is the ith heart card chosen. 01:09:10.142 --> 01:09:13.330 align:middle line:90% 01:09:13.330 --> 01:09:15.770 align:middle line:84% Again, this is a setup where we have 52 cards. 01:09:15.770 --> 01:09:17.615 align:middle line:84% We choose three without replacement, 01:09:17.615 --> 01:09:19.490 align:middle line:84% and I want to know how many hearts there are. 01:09:19.490 --> 01:09:21.340 align:middle line:90% So this is number of hearts. 01:09:21.340 --> 01:09:24.220 align:middle line:84% And this is just did I choose the ith heart 01:09:24.220 --> 01:09:26.361 align:middle line:90% card in this setup? 01:09:26.361 --> 01:09:27.819 align:middle line:84% So, of course, the number of hearts 01:09:27.819 --> 01:09:33.370 align:middle line:84% is exactly the sum of 1 for every heart card that I chose. 01:09:33.370 --> 01:09:36.220 align:middle line:84% So one-- this is like Ace, and this is King. 01:09:36.220 --> 01:09:37.790 align:middle line:84% And I've got all the 1's in between. 01:09:37.790 --> 01:09:41.060 align:middle line:84% So this is-- whereas, here was the ith coin flip or whatever. 01:09:41.060 --> 01:09:44.140 align:middle line:84% Here's the ith heart that we potentially choose. 01:09:44.140 --> 01:09:49.979 align:middle line:84% And the probability of choosing a given heart is easy. 01:09:49.979 --> 01:09:53.899 align:middle line:90% It's-- it's not so easy. 01:09:53.899 --> 01:09:56.990 align:middle line:90% It's some small expression. 01:09:56.990 --> 01:09:59.760 align:middle line:84% It could be you choose it in the first step. 01:09:59.760 --> 01:10:02.840 align:middle line:84% Or you don't choose it in the first step, 01:10:02.840 --> 01:10:07.640 align:middle line:84% but then you choose it in the second step. 01:10:07.640 --> 01:10:11.330 align:middle line:84% Or you don't choose it in the second step, 01:10:11.330 --> 01:10:15.180 align:middle line:84% and then you'd have to choose it in the third step. 01:10:15.180 --> 01:10:17.690 align:middle line:84% Those are the three ways you can get it. 01:10:17.690 --> 01:10:23.315 align:middle line:84% And conveniently, all the hard numbers here cancel. 01:10:23.315 --> 01:10:24.565 align:middle line:90% That's exciting. 01:10:24.565 --> 01:10:27.440 align:middle line:90% 01:10:27.440 --> 01:10:32.450 align:middle line:90% And you get 2 over 52? 01:10:32.450 --> 01:10:33.620 align:middle line:90% 3 over 52. 01:10:33.620 --> 01:10:34.300 align:middle line:90% I can count. 01:10:34.300 --> 01:10:38.450 align:middle line:90% 01:10:38.450 --> 01:10:41.810 align:middle line:90% So you add them up, 13 of them. 01:10:41.810 --> 01:10:45.623 align:middle line:84% So you get 13 times 3 over 52, and that's 3/4. 01:10:45.623 --> 01:10:49.260 align:middle line:90% 01:10:49.260 --> 01:10:52.470 align:middle line:90% So kind of nice. 01:10:52.470 --> 01:10:56.460 align:middle line:84% No binomial coefficients required. 01:10:56.460 --> 01:10:58.713 align:middle line:90% One more pair of examples. 01:10:58.713 --> 01:11:07.450 align:middle line:90% 01:11:07.450 --> 01:11:07.950 align:middle line:90% Yeah. 01:11:07.950 --> 01:11:16.320 align:middle line:90% 01:11:16.320 --> 01:11:19.080 align:middle line:84% I have in my notes that this is a situation that 01:11:19.080 --> 01:11:20.910 align:middle line:84% happens in All of Us are Dead, which 01:11:20.910 --> 01:11:27.210 align:middle line:84% is a great zombie high school TV show. 01:11:27.210 --> 01:11:29.380 align:middle line:84% I imagine it just happens in lots of schools. 01:11:29.380 --> 01:11:31.332 align:middle line:84% But I never went to school, so I don't know. 01:11:31.332 --> 01:11:33.040 align:middle line:84% You can tell me whether this is the case. 01:11:33.040 --> 01:11:34.280 align:middle line:90% Probably less so in the US. 01:11:34.280 --> 01:11:45.390 align:middle line:90% 01:11:45.390 --> 01:11:47.560 align:middle line:90% Maybe less so when you're older. 01:11:47.560 --> 01:11:49.757 align:middle line:84% Maybe when you're young and you have a phone, 01:11:49.757 --> 01:11:52.090 align:middle line:84% you have to submit your phone when you arrive at school. 01:11:52.090 --> 01:11:53.320 align:middle line:90% And here's the weird part. 01:11:53.320 --> 01:11:55.710 align:middle line:84% And this is maybe only in a zombie scenario. 01:11:55.710 --> 01:11:57.450 align:middle line:84% At the end of the day, phones are 01:11:57.450 --> 01:11:59.320 align:middle line:90% returned uniformly at random. 01:11:59.320 --> 01:12:03.020 align:middle line:84% [LAUGHS] I hope this didn't happen to you. 01:12:03.020 --> 01:12:06.300 align:middle line:90% 01:12:06.300 --> 01:12:10.290 align:middle line:84% How many people get the right phone? 01:12:10.290 --> 01:12:11.075 align:middle line:90% Natural question. 01:12:11.075 --> 01:12:15.330 align:middle line:90% 01:12:15.330 --> 01:12:17.538 align:middle line:84% AUDIENCE: The natural question is, do I get my phone? 01:12:17.538 --> 01:12:19.872 align:middle line:84% ERIK DEMAINE: You care about whether you get your phone? 01:12:19.872 --> 01:12:20.590 align:middle line:90% That's also true. 01:12:20.590 --> 01:12:22.780 align:middle line:84% That's a closely-- we're going to have to solve that problem, 01:12:22.780 --> 01:12:23.280 align:middle line:90% in fact. 01:12:23.280 --> 01:12:28.650 align:middle line:90% 01:12:28.650 --> 01:12:30.552 align:middle line:90% So we're assuming here each-- 01:12:30.552 --> 01:12:32.010 align:middle line:84% there's a one-to-one correspondence 01:12:32.010 --> 01:12:34.000 align:middle line:90% between phones and people. 01:12:34.000 --> 01:12:36.570 align:middle line:90% No co-ownership. 01:12:36.570 --> 01:12:41.220 align:middle line:84% So the number of kids that get their own phone we 01:12:41.220 --> 01:12:42.540 align:middle line:90% can write it as a sum. 01:12:42.540 --> 01:12:45.560 align:middle line:90% Over all kids, we had n kids. 01:12:45.560 --> 01:12:47.170 align:middle line:90% So k for kids. 01:12:47.170 --> 01:12:55.065 align:middle line:84% 1 up to n of the probability that kid k gets their phone. 01:12:55.065 --> 01:12:58.590 align:middle line:90% 01:12:58.590 --> 01:12:59.090 align:middle line:90% OK. 01:12:59.090 --> 01:13:00.940 align:middle line:84% I'm just writing this because I've 01:13:00.940 --> 01:13:02.148 align:middle line:90% preworked out these examples. 01:13:02.148 --> 01:13:03.648 align:middle line:84% Of course, when you're solving them, 01:13:03.648 --> 01:13:05.650 align:middle line:84% you'll probably know in this part of the class 01:13:05.650 --> 01:13:07.390 align:middle line:84% that you should use this technique. 01:13:07.390 --> 01:13:10.070 align:middle line:84% Always-- if you're asked for an expectation, 01:13:10.070 --> 01:13:13.060 align:middle line:84% probably you might need to decompose the random variable 01:13:13.060 --> 01:13:14.843 align:middle line:84% into sums of other random variables. 01:13:14.843 --> 01:13:15.760 align:middle line:90% You might not have to. 01:13:15.760 --> 01:13:18.050 align:middle line:84% This hard problem, we solved in two different ways. 01:13:18.050 --> 01:13:20.210 align:middle line:84% But if you can, it's going to be easier. 01:13:20.210 --> 01:13:21.770 align:middle line:84% So I encourage you to try to do this. 01:13:21.770 --> 01:13:24.560 align:middle line:84% But how do you decompose into sum of random variables? 01:13:24.560 --> 01:13:26.090 align:middle line:84% I mean, that's divine inspiration. 01:13:26.090 --> 01:13:27.632 align:middle line:84% You just got to try a bunch of things 01:13:27.632 --> 01:13:30.190 align:middle line:84% until you find something that works, that gives you-- 01:13:30.190 --> 01:13:32.890 align:middle line:84% think about what you know how to compute, and then 01:13:32.890 --> 01:13:33.730 align:middle line:90% try to build it up. 01:13:33.730 --> 01:13:37.150 align:middle line:84% Or think about how to decompose, top down, whatever. 01:13:37.150 --> 01:13:39.460 align:middle line:84% But as Brinmore said, what I care about 01:13:39.460 --> 01:13:42.600 align:middle line:84% is whether I get my phone, and I'm kid seven. 01:13:42.600 --> 01:13:43.580 align:middle line:90% I don't know. 01:13:43.580 --> 01:13:46.970 align:middle line:84% What's the probability that kid seven gets their phone, or kid 01:13:46.970 --> 01:13:48.650 align:middle line:90% one or whatever. 01:13:48.650 --> 01:13:53.030 align:middle line:84% It turns out they're all the same because-- 01:13:53.030 --> 01:13:54.620 align:middle line:90% oh. 01:13:54.620 --> 01:13:55.460 align:middle line:90% Yeah. 01:13:55.460 --> 01:13:57.720 align:middle line:90% I did two steps here. 01:13:57.720 --> 01:13:59.700 align:middle line:84% This is how many kids get their phone. 01:13:59.700 --> 01:14:06.170 align:middle line:84% We can call this X. And this is going to be the sum-- 01:14:06.170 --> 01:14:06.810 align:middle line:90% sorry. 01:14:06.810 --> 01:14:08.790 align:middle line:84% I skipped way too many steps here. 01:14:08.790 --> 01:14:12.310 align:middle line:84% The expectation of x is going to be this. 01:14:12.310 --> 01:14:14.990 align:middle line:90% 01:14:14.990 --> 01:14:17.480 align:middle line:84% But the number of kids that get their phone 01:14:17.480 --> 01:14:20.840 align:middle line:84% is going to be equal to the sum of Xk. 01:14:20.840 --> 01:14:27.290 align:middle line:84% That equals 1 to n, where Xk is an indicator random variable 01:14:27.290 --> 01:14:34.375 align:middle line:84% for whether kid k got their phone. 01:14:34.375 --> 01:14:39.930 align:middle line:90% 01:14:39.930 --> 01:14:40.430 align:middle line:90% OK. 01:14:40.430 --> 01:14:43.120 align:middle line:84% This is important because this was a random variable. 01:14:43.120 --> 01:14:44.700 align:middle line:84% So if I'm going to write equals, I 01:14:44.700 --> 01:14:47.400 align:middle line:84% should have some random variables over there. 01:14:47.400 --> 01:14:50.590 align:middle line:84% When I wrote a probability, that means I'm working with a number. 01:14:50.590 --> 01:14:53.470 align:middle line:84% And so that means the best I'm hoping for is the expectation. 01:14:53.470 --> 01:14:55.840 align:middle line:84% So this first equation I wrote is wrong. 01:14:55.840 --> 01:14:57.730 align:middle line:84% Please do not do that in your homework. 01:14:57.730 --> 01:14:59.140 align:middle line:90% It's easy to do, as you can see. 01:14:59.140 --> 01:15:01.750 align:middle line:84% But you should first set up the random variables, 01:15:01.750 --> 01:15:02.830 align:middle line:90% which are functions. 01:15:02.830 --> 01:15:04.950 align:middle line:84% Decompose them into sums of other functions. 01:15:04.950 --> 01:15:07.420 align:middle line:84% And then say, OK, let's compute the expectation. 01:15:07.420 --> 01:15:08.980 align:middle line:90% Use linearity of expectation. 01:15:08.980 --> 01:15:13.410 align:middle line:84% This is equal to by linearity, the sum of the expectations 01:15:13.410 --> 01:15:15.150 align:middle line:90% of the Xi's. 01:15:15.150 --> 01:15:18.347 align:middle line:84% And then by the properties of indicator random variables, 01:15:18.347 --> 01:15:20.430 align:middle line:84% the expected value of an indicator random variable 01:15:20.430 --> 01:15:22.400 align:middle line:84% is the probability of that event. 01:15:22.400 --> 01:15:25.122 align:middle line:90% 01:15:25.122 --> 01:15:31.020 align:middle line:84% So then, what's the probability that kid k gets their phone? 01:15:31.020 --> 01:15:33.420 align:middle line:84% Well, how many different distributions 01:15:33.420 --> 01:15:35.760 align:middle line:90% are there of the n phones? 01:15:35.760 --> 01:15:36.335 align:middle line:90% Help me out. 01:15:36.335 --> 01:15:41.820 align:middle line:90% 01:15:41.820 --> 01:15:42.453 align:middle line:90% AUDIENCE: n. 01:15:42.453 --> 01:15:43.120 align:middle line:90% ERIK DEMAINE: n. 01:15:43.120 --> 01:15:43.710 align:middle line:90% Great answer. 01:15:43.710 --> 01:15:50.710 align:middle line:90% [LAUGHS] n different ways-- 01:15:50.710 --> 01:15:53.400 align:middle line:84% there's n different possibilities for the first kid 01:15:53.400 --> 01:15:56.850 align:middle line:90% to get their phone back. 01:15:56.850 --> 01:16:00.034 align:middle line:90% And then-- 01:16:00.034 --> 01:16:00.976 align:middle line:90% AUDIENCE: [INAUDIBLE]. 01:16:00.976 --> 01:16:01.447 align:middle line:90% ERIK DEMAINE: Hmm? 01:16:01.447 --> 01:16:01.920 align:middle line:90% STUDENT: A factorial. 01:16:01.920 --> 01:16:02.830 align:middle line:90% ERIK DEMAINE: A factorial. 01:16:02.830 --> 01:16:03.330 align:middle line:90% Yeah. 01:16:03.330 --> 01:16:06.112 align:middle line:84% There's n factorial different distributions of the phones. 01:16:06.112 --> 01:16:07.320 align:middle line:90% AUDIENCE: That's what I said. 01:16:07.320 --> 01:16:08.028 align:middle line:90% ERIK DEMAINE: Oh. 01:16:08.028 --> 01:16:10.630 align:middle line:84% Oh, you said n really loud, which is n factorial. 01:16:10.630 --> 01:16:12.120 align:middle line:90% Thank you. 01:16:12.120 --> 01:16:14.070 align:middle line:90% Bad pun. 01:16:14.070 --> 01:16:15.600 align:middle line:90% So that's the denominator. 01:16:15.600 --> 01:16:22.050 align:middle line:84% Those are all the possible rearrangements 01:16:22.050 --> 01:16:23.640 align:middle line:90% of the phones to the kids. 01:16:23.640 --> 01:16:26.550 align:middle line:90% n, very emphatically. 01:16:26.550 --> 01:16:29.680 align:middle line:84% And then how many are there where kid k gets their phone? 01:16:29.680 --> 01:16:30.827 align:middle line:90% Well this is easy, right? 01:16:30.827 --> 01:16:32.910 align:middle line:84% This is just saying, well, kid k gets their phone. 01:16:32.910 --> 01:16:36.130 align:middle line:84% And then among all the other n minus 1 phones, 01:16:36.130 --> 01:16:37.700 align:middle line:90% they're whatever they are. 01:16:37.700 --> 01:16:39.075 align:middle line:84% How many different ways are there 01:16:39.075 --> 01:16:40.930 align:middle line:90% to distribute those phones? 01:16:40.930 --> 01:16:42.225 align:middle line:90% n minus 1. 01:16:42.225 --> 01:16:45.610 align:middle line:90% [LAUGHTER] 01:16:45.610 --> 01:16:48.850 align:middle line:84% Hopefully, I didn't max out the mic. 01:16:48.850 --> 01:16:49.540 align:middle line:90% OK. 01:16:49.540 --> 01:16:52.370 align:middle line:84% N minus 1 factorial over n factorial. 01:16:52.370 --> 01:16:53.180 align:middle line:90% Oh, my gosh. 01:16:53.180 --> 01:16:54.410 align:middle line:90% Such big factorials. 01:16:54.410 --> 01:16:56.080 align:middle line:84% But basically, everything cancels. 01:16:56.080 --> 01:16:57.390 align:middle line:90% And this is 1 over n. 01:16:57.390 --> 01:17:00.980 align:middle line:90% 01:17:00.980 --> 01:17:04.630 align:middle line:90% 1 over n summed n times-- 01:17:04.630 --> 01:17:07.420 align:middle line:90% that's 1. 01:17:07.420 --> 01:17:08.620 align:middle line:90% Ha, ha, ha. 01:17:08.620 --> 01:17:11.505 align:middle line:84% One kid expects to get their phone back. 01:17:11.505 --> 01:17:15.008 align:middle line:90% 01:17:15.008 --> 01:17:16.050 align:middle line:90% Hopefully, it's Brinmore. 01:17:16.050 --> 01:17:19.355 align:middle line:90% [LAUGHS] Who is the-- 01:17:19.355 --> 01:17:20.980 align:middle line:84% I mean, it doesn't mean that one person 01:17:20.980 --> 01:17:22.790 align:middle line:90% will get their phone back. 01:17:22.790 --> 01:17:24.020 align:middle line:90% Maybe none of them do. 01:17:24.020 --> 01:17:25.610 align:middle line:90% Here's a related game. 01:17:25.610 --> 01:17:34.420 align:middle line:84% Suppose n diners put their phones-- 01:17:34.420 --> 01:17:43.040 align:middle line:84% put their phone singular on a turntable, sometimes called 01:17:43.040 --> 01:17:46.790 align:middle line:84% a Lazy Susan, for distributing food, 01:17:46.790 --> 01:17:48.620 align:middle line:90% if you're eating family style. 01:17:48.620 --> 01:17:55.075 align:middle line:84% And then you spin the table, spin uniformly at random. 01:17:55.075 --> 01:18:04.640 align:middle line:90% 01:18:04.640 --> 01:18:06.625 align:middle line:84% And then you take the phone in front of you. 01:18:06.625 --> 01:18:10.010 align:middle line:90% 01:18:10.010 --> 01:18:12.930 align:middle line:84% Let's assume that this is a discrete table. 01:18:12.930 --> 01:18:15.530 align:middle line:84% So when you spin it, there's only n different rotations 01:18:15.530 --> 01:18:16.650 align:middle line:90% that could come out. 01:18:16.650 --> 01:18:19.410 align:middle line:84% And so there's always a phone immediately in front of you. 01:18:19.410 --> 01:18:23.180 align:middle line:84% This is a different way to be terrible with phones. 01:18:23.180 --> 01:18:25.142 align:middle line:84% In this one, anything could happen. 01:18:25.142 --> 01:18:26.850 align:middle line:84% It could be no one gets their phone back. 01:18:26.850 --> 01:18:28.863 align:middle line:84% That's maybe-- it's not quite what you expect. 01:18:28.863 --> 01:18:30.780 align:middle line:84% You expect one person to get their phone back. 01:18:30.780 --> 01:18:33.030 align:middle line:84% Or it could be everyone gets exactly their phone back. 01:18:33.030 --> 01:18:35.020 align:middle line:84% If you do it randomly, anything's possible. 01:18:35.020 --> 01:18:36.600 align:middle line:84% It's just a 1 over n factorial chance 01:18:36.600 --> 01:18:39.210 align:middle line:84% that any particular outcome happens. 01:18:39.210 --> 01:18:41.400 align:middle line:90% Here, it's actually guaranteed. 01:18:41.400 --> 01:18:45.367 align:middle line:84% Either everyone gets their phone back or no one does. 01:18:45.367 --> 01:18:47.950 align:middle line:84% Because if I rotate by 0, then everyone gets their phone back. 01:18:47.950 --> 01:18:50.490 align:middle line:84% If I rotate by any other amount, then everyone 01:18:50.490 --> 01:18:51.820 align:middle line:90% gets a different phone. 01:18:51.820 --> 01:18:53.770 align:middle line:84% So it's a different distribution. 01:18:53.770 --> 01:18:56.830 align:middle line:84% It's a different probability setup. 01:18:56.830 --> 01:19:00.540 align:middle line:90% But the expectation is the same. 01:19:00.540 --> 01:19:07.380 align:middle line:84% Expected number of matching people phones, 01:19:07.380 --> 01:19:09.185 align:middle line:90% whatever-- matching phones. 01:19:09.185 --> 01:19:12.390 align:middle line:90% 01:19:12.390 --> 01:19:14.227 align:middle line:90% It's still 1. 01:19:14.227 --> 01:19:16.560 align:middle line:84% Here, it's really easy to compute because there are only 01:19:16.560 --> 01:19:18.040 align:middle line:90% n different outcomes. 01:19:18.040 --> 01:19:22.980 align:middle line:84% And in one of those scenarios, you get-- 01:19:22.980 --> 01:19:27.540 align:middle line:84% so 1 over n chance everyone gets their phone back. 01:19:27.540 --> 01:19:31.930 align:middle line:84% And in all other scenarios, no one gets their phone back. 01:19:31.930 --> 01:19:33.728 align:middle line:90% So that disappears. 01:19:33.728 --> 01:19:34.770 align:middle line:90% You get 1 over n times n. 01:19:34.770 --> 01:19:36.180 align:middle line:90% You get 1. 01:19:36.180 --> 01:19:39.880 align:middle line:84% That's by the definition of expectation, 01:19:39.880 --> 01:19:40.930 align:middle line:90% how to compute this. 01:19:40.930 --> 01:19:43.060 align:middle line:84% But you can also set it up in exactly the same way. 01:19:43.060 --> 01:19:45.330 align:middle line:84% You can say, well, you could sum over all the diners 01:19:45.330 --> 01:19:48.000 align:middle line:84% and check what's the probability that a particular diner gets 01:19:48.000 --> 01:19:48.880 align:middle line:90% their phone back. 01:19:48.880 --> 01:19:51.810 align:middle line:84% But now you don't have to do this n minus 1 factorial over n 01:19:51.810 --> 01:19:52.720 align:middle line:90% factorial. 01:19:52.720 --> 01:19:54.570 align:middle line:90% It's just 1 over n by the setup. 01:19:54.570 --> 01:19:56.880 align:middle line:90% That was this term here. 01:19:56.880 --> 01:20:03.600 align:middle line:84% So this is also the probability of Xi. 01:20:03.600 --> 01:20:06.100 align:middle line:84% And so you can also do it with linearity of expectation. 01:20:06.100 --> 01:20:07.530 align:middle line:84% So this is a funny scenario where 01:20:07.530 --> 01:20:09.210 align:middle line:84% I have two rather different setups 01:20:09.210 --> 01:20:11.190 align:middle line:90% for redistributing phones. 01:20:11.190 --> 01:20:12.850 align:middle line:84% They have very different behaviors. 01:20:12.850 --> 01:20:14.800 align:middle line:84% But in expectation, they're the same. 01:20:14.800 --> 01:20:16.680 align:middle line:84% So expectation doesn't tell you everything 01:20:16.680 --> 01:20:17.710 align:middle line:90% about the distribution. 01:20:17.710 --> 01:20:19.420 align:middle line:84% And in the next couple of lectures, 01:20:19.420 --> 01:20:21.060 align:middle line:84% we'll talk more about understanding 01:20:21.060 --> 01:20:24.570 align:middle line:84% the rest of the distribution beyond just the mean. 01:20:24.570 --> 01:20:26.240 align:middle line:90% All right. 01:20:26.240 --> 01:20:31.000 align:middle line:90%