WEBVTT 00:00:00.000 --> 00:00:02.435 align:middle line:90% [SQUEAKING] 00:00:02.435 --> 00:00:04.383 align:middle line:90% [RUSTLING] 00:00:04.383 --> 00:00:07.792 align:middle line:90% [CLICKING] 00:00:07.792 --> 00:00:10.552 align:middle line:90% 00:00:10.552 --> 00:00:11.510 align:middle line:90% BRYNMOR CHAPMAN: Hello? 00:00:11.510 --> 00:00:13.190 align:middle line:90% Can people hear me? 00:00:13.190 --> 00:00:15.740 align:middle line:90% Are we good in the back? 00:00:15.740 --> 00:00:16.610 align:middle line:90% Are we good? 00:00:16.610 --> 00:00:17.430 align:middle line:90% Can people hear me? 00:00:17.430 --> 00:00:18.860 align:middle line:90% Good, good, good. 00:00:18.860 --> 00:00:22.970 align:middle line:84% OK, so today, we are going to move on from counting 00:00:22.970 --> 00:00:26.930 align:middle line:84% and start the last unit on probability. 00:00:26.930 --> 00:00:29.278 align:middle line:90% Probability is very important. 00:00:29.278 --> 00:00:31.070 align:middle line:84% It's one of the most important disciplines, 00:00:31.070 --> 00:00:34.820 align:middle line:84% not just in computer science, but in all of the sciences. 00:00:34.820 --> 00:00:36.950 align:middle line:84% Please don't tell your other science instructors 00:00:36.950 --> 00:00:38.780 align:middle line:90% that I said this. 00:00:38.780 --> 00:00:40.550 align:middle line:84% You use it in-- like we've already 00:00:40.550 --> 00:00:44.960 align:middle line:84% seen it in this class with the Miller-Rabin algorithm. 00:00:44.960 --> 00:00:47.900 align:middle line:84% You also see it in other randomized algorithms 00:00:47.900 --> 00:00:53.420 align:middle line:84% that you'll see later in 6 1210, 1220, 5220 if you take that. 00:00:53.420 --> 00:00:55.950 align:middle line:84% We see it in game theory, information theory, 00:00:55.950 --> 00:00:59.920 align:middle line:84% signal processing, cryptography, machine learning, medicine, 00:00:59.920 --> 00:01:01.570 align:middle line:90% stats, forensics. 00:01:01.570 --> 00:01:04.260 align:middle line:90% The list goes on and on. 00:01:04.260 --> 00:01:07.110 align:middle line:84% But unfortunately, it's also one of the least understood 00:01:07.110 --> 00:01:08.760 align:middle line:90% disciplines. 00:01:08.760 --> 00:01:11.970 align:middle line:84% So there's a pretty famous quote attributed 00:01:11.970 --> 00:01:16.950 align:middle line:84% to Mark Twain, Benjamin Disraeli, several other people. 00:01:16.950 --> 00:01:19.090 align:middle line:90% There are three kinds of lies. 00:01:19.090 --> 00:01:25.020 align:middle line:84% There are lies, there are damned lies, and statistics. 00:01:25.020 --> 00:01:28.920 align:middle line:84% So you can kind of lie with probability and statistics 00:01:28.920 --> 00:01:34.290 align:middle line:84% by saying things that are true that are also very misleading. 00:01:34.290 --> 00:01:37.530 align:middle line:84% Common sense is often very demonstrably completely 00:01:37.530 --> 00:01:42.180 align:middle line:84% unreliable, which we'll see in a moment. 00:01:42.180 --> 00:01:43.590 align:middle line:84% There are many graduate students, 00:01:43.590 --> 00:01:45.550 align:middle line:84% like even in our department-- again, 00:01:45.550 --> 00:01:47.880 align:middle line:90% don't tell people I said this-- 00:01:47.880 --> 00:01:50.362 align:middle line:84% they can do all sorts of exciting maths. 00:01:50.362 --> 00:01:52.320 align:middle line:84% But you give them a basic probability question, 00:01:52.320 --> 00:01:55.110 align:middle line:84% and they don't know where to start. 00:01:55.110 --> 00:01:56.850 align:middle line:84% There are many garbage papers that 00:01:56.850 --> 00:01:59.720 align:middle line:84% are based on faulty understanding of probability. 00:01:59.720 --> 00:02:06.470 align:middle line:84% Even some of our colleagues have some, let's say, issues. 00:02:06.470 --> 00:02:09.860 align:middle line:84% Today, we are going to look at an example. 00:02:09.860 --> 00:02:13.130 align:middle line:84% Who has heard of the Monty Hall problem? 00:02:13.130 --> 00:02:14.565 align:middle line:90% Oh, nice. 00:02:14.565 --> 00:02:15.565 align:middle line:90% Decent number of people. 00:02:15.565 --> 00:02:18.164 align:middle line:90% 00:02:18.164 --> 00:02:21.230 align:middle line:84% So, yeah, we'll basically try and use the Monty Hall problem 00:02:21.230 --> 00:02:24.080 align:middle line:84% to illustrate that it's really best 00:02:24.080 --> 00:02:26.210 align:middle line:84% to just throw away your intuition 00:02:26.210 --> 00:02:30.879 align:middle line:84% and just fall back on rigorous step-by-step analyses. 00:02:30.879 --> 00:02:37.040 align:middle line:84% So the Monty Hall problem, it kind of became famous in 1990 00:02:37.040 --> 00:02:40.790 align:middle line:84% when this magazine columnist named 00:02:40.790 --> 00:02:47.300 align:middle line:84% Marilyn vos Savant received a letter from a Craig Whitaker, 00:02:47.300 --> 00:02:49.350 align:middle line:90% which read as follows. 00:02:49.350 --> 00:02:51.860 align:middle line:84% Suppose you're on a game show, and you're 00:02:51.860 --> 00:02:53.910 align:middle line:90% given the choice of three doors. 00:02:53.910 --> 00:02:58.300 align:middle line:84% Behind one door is a car and behind the others, goats. 00:02:58.300 --> 00:02:59.500 align:middle line:90% You pick a door-- 00:02:59.500 --> 00:03:01.830 align:middle line:84% say, number 1-- and the host, who 00:03:01.830 --> 00:03:04.980 align:middle line:84% knows what's behind the doors, opens a different door-- say, 00:03:04.980 --> 00:03:07.110 align:middle line:90% number 3, which has a goat. 00:03:07.110 --> 00:03:10.740 align:middle line:84% He says to you, Do you want to pick door number 2? 00:03:10.740 --> 00:03:13.805 align:middle line:84% Now, the question is, Is it to your advantage to switch? 00:03:13.805 --> 00:03:16.686 align:middle line:90% 00:03:16.686 --> 00:03:21.270 align:middle line:84% And who thinks yes, it is advantageous to switch? 00:03:21.270 --> 00:03:21.888 align:middle line:90% Who thinks no? 00:03:21.888 --> 00:03:23.430 align:middle line:84% Who thinks you're better off sticking 00:03:23.430 --> 00:03:25.800 align:middle line:90% with your original choice? 00:03:25.800 --> 00:03:28.620 align:middle line:90% Who thinks it doesn't matter? 00:03:28.620 --> 00:03:32.610 align:middle line:84% OK, so it looks like most people think that it's 00:03:32.610 --> 00:03:34.290 align:middle line:90% advantageous to switch. 00:03:34.290 --> 00:03:36.540 align:middle line:90% We've had a few abstentions. 00:03:36.540 --> 00:03:40.410 align:middle line:84% I'm going to count those as it doesn't matter. 00:03:40.410 --> 00:03:44.910 align:middle line:84% So Marilyn replied that it's indeed advantageous to switch. 00:03:44.910 --> 00:03:47.200 align:middle line:84% If the car is behind the originally picked door, 00:03:47.200 --> 00:03:48.490 align:middle line:90% then you lose by switching. 00:03:48.490 --> 00:03:51.040 align:middle line:84% But if it's not, which is twice as likely, 00:03:51.040 --> 00:03:53.830 align:middle line:90% then you win by switching. 00:03:53.830 --> 00:03:57.190 align:middle line:84% And, of course, she soon received a torrent of hate mail, 00:03:57.190 --> 00:03:59.680 align:middle line:84% including from professional mathematicians, 00:03:59.680 --> 00:04:02.800 align:middle line:84% adamantly asserting that she was wrong. 00:04:02.800 --> 00:04:05.260 align:middle line:84% So the hypo that was described in this letter 00:04:05.260 --> 00:04:07.900 align:middle line:84% became known as the Monty Hall problem 00:04:07.900 --> 00:04:12.175 align:middle line:84% after the host, Monty Hall, of the game show which 00:04:12.175 --> 00:04:16.480 align:middle line:84% had games like this where contestants 00:04:16.480 --> 00:04:19.329 align:middle line:90% faced very similar situations. 00:04:19.329 --> 00:04:21.410 align:middle line:84% So today, we're going to analyze it. 00:04:21.410 --> 00:04:24.430 align:middle line:84% But first, can I get a handful of volunteers to come 00:04:24.430 --> 00:04:27.370 align:middle line:90% play Let's Make a Deal? 00:04:27.370 --> 00:04:30.760 align:middle line:84% OK, so empirically, at least, it looks like it's better 00:04:30.760 --> 00:04:31.670 align:middle line:90% to switch, right? 00:04:31.670 --> 00:04:35.320 align:middle line:84% Most people switched, and most people won the prize 00:04:35.320 --> 00:04:36.940 align:middle line:90% when they did that. 00:04:36.940 --> 00:04:39.320 align:middle line:84% So even though there's only one door with the prize, 00:04:39.320 --> 00:04:41.650 align:middle line:84% it seems to be pretty promising, right? 00:04:41.650 --> 00:04:46.050 align:middle line:84% So let's see if we can analyze this mathematically. 00:04:46.050 --> 00:04:49.360 align:middle line:84% So we're going to solve the Monty Hall problem using 00:04:49.360 --> 00:04:53.700 align:middle line:84% what we call the tree method, which is a very simple 00:04:53.700 --> 00:04:58.028 align:middle line:84% and elementary approach that doesn't rely on intuition. 00:04:58.028 --> 00:04:59.945 align:middle line:84% It's just going to rely on mathematical rigor. 00:04:59.945 --> 00:05:02.640 align:middle line:90% 00:05:02.640 --> 00:05:09.263 align:middle line:84% So the tree method uses four-ish steps. 00:05:09.263 --> 00:05:10.930 align:middle line:84% We used to call it the four-step method. 00:05:10.930 --> 00:05:12.305 align:middle line:84% But I couldn't for the life of me 00:05:12.305 --> 00:05:14.280 align:middle line:84% tell you what the four steps are if they 00:05:14.280 --> 00:05:15.990 align:middle line:90% weren't written in front of me. 00:05:15.990 --> 00:05:19.380 align:middle line:84% So I prefer to call it the tree method. 00:05:19.380 --> 00:05:21.070 align:middle line:90% Oh, I can't even count. 00:05:21.070 --> 00:05:22.270 align:middle line:90% It kind of has five steps. 00:05:22.270 --> 00:05:23.630 align:middle line:90% We've got a step zero. 00:05:23.630 --> 00:05:26.400 align:middle line:90% 00:05:26.400 --> 00:05:29.490 align:middle line:84% Before you even start, you really 00:05:29.490 --> 00:05:31.900 align:middle line:84% want to make sure that you know what question 00:05:31.900 --> 00:05:32.900 align:middle line:90% you're trying to answer. 00:05:32.900 --> 00:05:40.098 align:middle line:90% 00:05:40.098 --> 00:05:42.750 align:middle line:84% So before we can even think about solving 00:05:42.750 --> 00:05:47.670 align:middle line:84% a mathematical problem, we need to know what the problem is. 00:05:47.670 --> 00:05:50.720 align:middle line:84% We need to understand the setup, make 00:05:50.720 --> 00:05:53.540 align:middle line:84% sure we know what we're trying to ask before we 00:05:53.540 --> 00:05:54.870 align:middle line:90% can start analyzing it. 00:05:54.870 --> 00:05:58.760 align:middle line:84% So the letter from this Mr. Whitaker 00:05:58.760 --> 00:06:01.440 align:middle line:90% was not entirely precise. 00:06:01.440 --> 00:06:05.910 align:middle line:84% So we've got to figure out what axioms we're working with. 00:06:05.910 --> 00:06:09.540 align:middle line:84% So we're going to assume the following. 00:06:09.540 --> 00:06:12.462 align:middle line:90% 00:06:12.462 --> 00:06:23.815 align:middle line:84% A prize is equally likely to be in any box. 00:06:23.815 --> 00:06:28.444 align:middle line:90% 00:06:28.444 --> 00:06:31.730 align:middle line:84% So, in principle, it's possible that Monty Hall 00:06:31.730 --> 00:06:35.180 align:middle line:84% tries to be tricky and tries to get into the contestants' minds, 00:06:35.180 --> 00:06:36.930 align:middle line:84% figure out what box they're going to pick, 00:06:36.930 --> 00:06:39.830 align:middle line:84% and put the prize somewhere else or, alternatively, put 00:06:39.830 --> 00:06:41.663 align:middle line:84% the prize somewhere where they will pick it, 00:06:41.663 --> 00:06:43.760 align:middle line:84% because it's an exciting game show. 00:06:43.760 --> 00:06:46.730 align:middle line:84% But we're going to assume that Monty is just 00:06:46.730 --> 00:06:48.150 align:middle line:90% picking the boxes randomly. 00:06:48.150 --> 00:06:51.920 align:middle line:84% So the prize is going to go into a randomly selected box, 00:06:51.920 --> 00:06:56.960 align:middle line:84% completely uniform, and then the other two will have the goats. 00:06:56.960 --> 00:07:02.640 align:middle line:84% We're also going to assume that Marilyn or your colleagues, 00:07:02.640 --> 00:07:14.060 align:middle line:84% whoever is doing the picking, is equally likely to pick 00:07:14.060 --> 00:07:20.123 align:middle line:90% any box for the first pick. 00:07:20.123 --> 00:07:22.290 align:middle line:84% She doesn't know anything about what Monty is doing. 00:07:22.290 --> 00:07:24.650 align:middle line:84% She's just going to toss some random coins in her head, 00:07:24.650 --> 00:07:27.750 align:middle line:84% and uniformly, at random, she'll just pick one of the boxes. 00:07:27.750 --> 00:07:32.012 align:middle line:84% It doesn't really matter where the prize is. 00:07:32.012 --> 00:07:37.280 align:middle line:84% Now, after Marilyn picks her box, 00:07:37.280 --> 00:07:46.880 align:middle line:90% Monty must reveal a non-prize. 00:07:46.880 --> 00:07:50.020 align:middle line:90% 00:07:50.020 --> 00:07:53.970 align:middle line:84% And moreover, it cannot be in Marilyn's box. 00:07:53.970 --> 00:08:05.032 align:middle line:90% 00:08:05.032 --> 00:08:08.090 align:middle line:84% So all that Mr. Whitaker said was 00:08:08.090 --> 00:08:10.270 align:middle line:84% Monty did reveal a non-prize that 00:08:10.270 --> 00:08:13.390 align:middle line:90% wasn't behind the chosen door. 00:08:13.390 --> 00:08:16.520 align:middle line:84% It's not entirely clear that this must happen. 00:08:16.520 --> 00:08:18.400 align:middle line:84% And, in fact, I believe on the game show, 00:08:18.400 --> 00:08:23.028 align:middle line:84% there were instances where he didn't. 00:08:23.028 --> 00:08:26.500 align:middle line:84% So we're just going to assume that Monty 00:08:26.500 --> 00:08:29.710 align:middle line:84% is going to reveal a prize, regardless of whether Marilyn 00:08:29.710 --> 00:08:32.115 align:middle line:84% picked the correct box to begin with. 00:08:32.115 --> 00:08:35.813 align:middle line:90% 00:08:35.813 --> 00:08:47.620 align:middle line:84% And, lastly, if Monty has a choice, he picks randomly. 00:08:47.620 --> 00:08:55.520 align:middle line:90% 00:08:55.520 --> 00:09:00.780 align:middle line:84% So if Marilyn has picked a non-prize to begin with, 00:09:00.780 --> 00:09:02.840 align:middle line:84% there's only one other non-prize. 00:09:02.840 --> 00:09:04.730 align:middle line:90% So Monty has to reveal that one. 00:09:04.730 --> 00:09:09.660 align:middle line:84% But if Marilyn has picked the prize box to begin with, 00:09:09.660 --> 00:09:11.960 align:middle line:84% then Monty could, in principle, open 00:09:11.960 --> 00:09:13.250 align:middle line:90% either of the other two boxes. 00:09:13.250 --> 00:09:16.250 align:middle line:84% And that would still follow this rule. 00:09:16.250 --> 00:09:18.120 align:middle line:84% So when Monty has a choice like this, 00:09:18.120 --> 00:09:19.850 align:middle line:84% we're going to say that he tosses a coin, 00:09:19.850 --> 00:09:22.060 align:middle line:84% and he's going to open one of them at random. 00:09:22.060 --> 00:09:26.352 align:middle line:90% 00:09:26.352 --> 00:09:30.990 align:middle line:84% So note that these are all axioms. 00:09:30.990 --> 00:09:33.360 align:middle line:84% We are just taking them on faith. 00:09:33.360 --> 00:09:35.760 align:middle line:90% They may or may not have merit. 00:09:35.760 --> 00:09:38.270 align:middle line:84% But if you accept them, then you also 00:09:38.270 --> 00:09:43.395 align:middle line:84% have to accept whatever conclusions we draw from them. 00:09:43.395 --> 00:09:44.770 align:middle line:84% But I should point out that there 00:09:44.770 --> 00:09:48.460 align:middle line:84% are other perfectly reasonable sets of axioms that would lead 00:09:48.460 --> 00:09:49.810 align:middle line:90% us to different conclusions. 00:09:49.810 --> 00:09:52.690 align:middle line:90% Like, maybe Monty is very rude. 00:09:52.690 --> 00:09:57.560 align:middle line:84% And if Marilyn has picked the prize, 00:09:57.560 --> 00:09:59.780 align:middle line:84% then he offers her the option to switch. 00:09:59.780 --> 00:10:02.630 align:middle line:84% And if Marilyn has not picked the prize, 00:10:02.630 --> 00:10:06.860 align:middle line:84% then he forces her to go with whatever she picked. 00:10:06.860 --> 00:10:10.960 align:middle line:84% This is a totally reasonable interpretation. 00:10:10.960 --> 00:10:14.290 align:middle line:84% And that would lead us to a different conclusion. 00:10:14.290 --> 00:10:15.980 align:middle line:84% Or, conversely, maybe it's benevolent. 00:10:15.980 --> 00:10:19.580 align:middle line:84% Maybe if Marilyn didn't pick the prize initially, 00:10:19.580 --> 00:10:22.850 align:middle line:84% he'll always try and encourage her to switch. 00:10:22.850 --> 00:10:25.100 align:middle line:84% But if she picked the prize, he'll just be like, nope, 00:10:25.100 --> 00:10:26.225 align:middle line:90% you got to stick with that. 00:10:26.225 --> 00:10:31.410 align:middle line:90% 00:10:31.410 --> 00:10:34.360 align:middle line:84% So now that we've got these axioms, 00:10:34.360 --> 00:10:37.420 align:middle line:84% we can now pose the mathematical question, 00:10:37.420 --> 00:10:52.860 align:middle line:84% What is the probability that Marilyn wins by switching? 00:10:52.860 --> 00:10:59.810 align:middle line:90% 00:10:59.810 --> 00:11:04.560 align:middle line:84% So yes, this is why I singled you out. 00:11:04.560 --> 00:11:06.890 align:middle line:84% I don't know where you have gone. 00:11:06.890 --> 00:11:10.190 align:middle line:84% Your colleague who did not switch, 00:11:10.190 --> 00:11:13.268 align:middle line:84% we're trying to analyze this probability, which 00:11:13.268 --> 00:11:14.310 align:middle line:90% is why I singled you out. 00:11:14.310 --> 00:11:17.480 align:middle line:84% So I apologize for that, but it's for the maths. 00:11:17.480 --> 00:11:20.540 align:middle line:90% Don't take it personally. 00:11:20.540 --> 00:11:27.380 align:middle line:84% OK, so now we have a well-defined question. 00:11:27.380 --> 00:11:29.980 align:middle line:84% So now we can analyze it using the four-step method. 00:11:29.980 --> 00:11:34.730 align:middle line:90% 00:11:34.730 --> 00:11:36.770 align:middle line:90% The first of our four steps-- 00:11:36.770 --> 00:11:40.770 align:middle line:84% well, the tree method, which has four steps-- 00:11:40.770 --> 00:11:43.445 align:middle line:84% we need to figure out what we call the sample space. 00:11:43.445 --> 00:11:57.840 align:middle line:90% 00:11:57.840 --> 00:11:59.940 align:middle line:84% So whenever we're thinking about a probability 00:11:59.940 --> 00:12:03.570 align:middle line:84% problem, underlying it is always what 00:12:03.570 --> 00:12:06.610 align:middle line:90% we call a probability space. 00:12:06.610 --> 00:12:09.095 align:middle line:84% And a probability space consists of two things. 00:12:09.095 --> 00:12:27.060 align:middle line:90% 00:12:27.060 --> 00:12:32.180 align:middle line:84% These two elements are a set, which we call the sample space. 00:12:32.180 --> 00:12:44.471 align:middle line:90% 00:12:44.471 --> 00:12:46.380 align:middle line:84% And for the purposes of this class, 00:12:46.380 --> 00:12:49.620 align:middle line:84% we're only going to be looking at discrete probability spaces. 00:12:49.620 --> 00:12:56.880 align:middle line:84% So S is going to be a non-empty and countable, or finite, set. 00:12:56.880 --> 00:13:00.696 align:middle line:84% We're not going to look at sets like R or something like that. 00:13:00.696 --> 00:13:04.770 align:middle line:84% So this could be a set of three boxes, 00:13:04.770 --> 00:13:06.710 align:middle line:84% or it could be the natural numbers, something 00:13:06.710 --> 00:13:11.180 align:middle line:84% like that, something that's discrete and countable. 00:13:11.180 --> 00:13:15.180 align:middle line:84% And then we also have a function, 00:13:15.180 --> 00:13:21.530 align:middle line:84% which we're going to denote by the letters Pr, which maps S 00:13:21.530 --> 00:13:23.960 align:middle line:90% into the unit interval 0 to 1. 00:13:23.960 --> 00:13:28.550 align:middle line:90% 00:13:28.550 --> 00:13:37.312 align:middle line:84% So this is going to be a total function with-- 00:13:37.312 --> 00:13:43.500 align:middle line:90% 00:13:43.500 --> 00:13:44.920 align:middle line:90% I probably shouldn't use S here. 00:13:44.920 --> 00:13:45.770 align:middle line:90% Let's call it x. 00:13:45.770 --> 00:13:53.328 align:middle line:90% 00:13:53.328 --> 00:13:55.840 align:middle line:84% And, slightly oddly, when we're talking about probabilities, 00:13:55.840 --> 00:13:59.100 align:middle line:84% we use square brackets instead of parentheses 00:13:59.100 --> 00:14:02.040 align:middle line:90% to enclose the argument. 00:14:02.040 --> 00:14:06.040 align:middle line:90% But this is just a function. 00:14:06.040 --> 00:14:09.110 align:middle line:84% You can think of it as, like, Pr of x. 00:14:09.110 --> 00:14:11.625 align:middle line:90% 00:14:11.625 --> 00:14:13.440 align:middle line:84% so We're going to have the requirement 00:14:13.440 --> 00:14:18.390 align:middle line:84% that if you sum up Pr of x over all of the x in your sample 00:14:18.390 --> 00:14:21.652 align:middle line:90% space, you should get 1. 00:14:21.652 --> 00:14:24.110 align:middle line:84% That's what it means for this to be a probability function. 00:14:24.110 --> 00:14:32.400 align:middle line:90% 00:14:32.400 --> 00:14:35.650 align:middle line:84% Oh, actually, why don't we leave both of these 00:14:35.650 --> 00:14:38.700 align:middle line:90% up and move onto this board? 00:14:38.700 --> 00:14:45.850 align:middle line:90% 00:14:45.850 --> 00:14:49.770 align:middle line:84% OK, so this is called our probability function. 00:14:49.770 --> 00:14:55.990 align:middle line:90% 00:14:55.990 --> 00:14:59.972 align:middle line:90% And elements of s are what-- 00:14:59.972 --> 00:15:01.430 align:middle line:84% we're going to call those outcomes. 00:15:01.430 --> 00:15:10.780 align:middle line:90% 00:15:10.780 --> 00:15:16.370 align:middle line:84% So, we've got a sample space S, which is a set of outcomes. 00:15:16.370 --> 00:15:18.310 align:middle line:84% And we've got a probability function, 00:15:18.310 --> 00:15:23.290 align:middle line:84% which maps each outcome to a number between 0 and 1. 00:15:23.290 --> 00:15:25.870 align:middle line:84% And it should be the case that all of those probabilities 00:15:25.870 --> 00:15:27.850 align:middle line:90% should sum to 1. 00:15:27.850 --> 00:15:30.760 align:middle line:84% Does that make sense to everybody? 00:15:30.760 --> 00:15:33.070 align:middle line:84% People reasonably Happy with this? 00:15:33.070 --> 00:15:35.715 align:middle line:90% Questions, comments, concerns? 00:15:35.715 --> 00:15:38.275 align:middle line:90% 00:15:38.275 --> 00:15:38.775 align:middle line:90% OK. 00:15:38.775 --> 00:15:42.340 align:middle line:90% 00:15:42.340 --> 00:15:46.180 align:middle line:84% So our first objective is going to be to figure out what 00:15:46.180 --> 00:15:47.515 align:middle line:90% our sample space should be. 00:15:47.515 --> 00:15:50.360 align:middle line:90% 00:15:50.360 --> 00:15:54.540 align:middle line:84% So we're going to do that using what we call a tree diagram. 00:15:54.540 --> 00:16:01.540 align:middle line:90% 00:16:01.540 --> 00:16:05.680 align:middle line:84% So, for the moment, we're going to disregard the probability 00:16:05.680 --> 00:16:08.760 align:middle line:90% function and just focus on S. 00:16:08.760 --> 00:16:10.510 align:middle line:84% And then once we've figured out what S is, 00:16:10.510 --> 00:16:14.390 align:middle line:84% we can figure out how to assign probabilities to everything. 00:16:14.390 --> 00:16:18.160 align:middle line:84% So the tree diagram, as you might expect, 00:16:18.160 --> 00:16:21.700 align:middle line:84% it's going to be a diagram in the shape of a tree, where 00:16:21.700 --> 00:16:24.700 align:middle line:84% each level of the tree represents 00:16:24.700 --> 00:16:28.380 align:middle line:84% some step in the random process we're trying to model. 00:16:28.380 --> 00:16:31.036 align:middle line:90% 00:16:31.036 --> 00:16:34.430 align:middle line:84% So in the case of the Monty Hall problem, 00:16:34.430 --> 00:16:38.200 align:middle line:90% what might be the first step? 00:16:38.200 --> 00:16:40.240 align:middle line:84% What's the first thing that happens 00:16:40.240 --> 00:16:45.850 align:middle line:84% in the Monty Hall problem, the first source of randomness? 00:16:45.850 --> 00:16:47.110 align:middle line:90% Yeah? 00:16:47.110 --> 00:16:49.915 align:middle line:90% AUDIENCE: You pick a first door. 00:16:49.915 --> 00:16:51.790 align:middle line:84% BRYNMOR CHAPMAN: OK, there's something random 00:16:51.790 --> 00:16:53.120 align:middle line:90% that happens even before that. 00:16:53.120 --> 00:16:54.680 align:middle line:90% The answer was, you pick a door. 00:16:54.680 --> 00:16:55.780 align:middle line:90% Yeah? 00:16:55.780 --> 00:16:57.843 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:16:57.843 --> 00:16:58.760 align:middle line:90% BRYNMOR CHAPMAN: Yeah. 00:16:58.760 --> 00:17:03.700 align:middle line:84% So the first random thing that happens is I, or Monty, 00:17:03.700 --> 00:17:08.000 align:middle line:84% picks which door the prize is behind. 00:17:08.000 --> 00:17:11.410 align:middle line:84% So that's going to be the first level of our tree diagram. 00:17:11.410 --> 00:17:14.540 align:middle line:84% So let's start drawing out our tree diagram. 00:17:14.540 --> 00:17:16.270 align:middle line:84% We're going to have our root here. 00:17:16.270 --> 00:17:21.989 align:middle line:84% And let's label this with Monty hides prize. 00:17:21.989 --> 00:17:29.297 align:middle line:90% 00:17:29.297 --> 00:17:33.260 align:middle line:84% And we're going to have three edges, which 00:17:33.260 --> 00:17:37.982 align:middle line:84% go to the three possibilities for what happens here. 00:17:37.982 --> 00:17:41.060 align:middle line:84% So it could be in door A, door B-- actually, 00:17:41.060 --> 00:17:43.640 align:middle line:84% why don't I call these purple, gold, and silver? 00:17:43.640 --> 00:17:44.485 align:middle line:90% Where's the eraser? 00:17:44.485 --> 00:17:47.930 align:middle line:90% 00:17:47.930 --> 00:17:48.640 align:middle line:90% Here we go. 00:17:48.640 --> 00:17:53.570 align:middle line:90% 00:17:53.570 --> 00:17:58.240 align:middle line:84% So we've got our purple box, our gold box, and our silver box. 00:17:58.240 --> 00:18:02.810 align:middle line:90% 00:18:02.810 --> 00:18:04.720 align:middle line:84% What's the next random thing that happens? 00:18:04.720 --> 00:18:10.460 align:middle line:90% 00:18:10.460 --> 00:18:11.380 align:middle line:90% Yeah? 00:18:11.380 --> 00:18:12.630 align:middle line:90% AUDIENCE: You choose the door. 00:18:12.630 --> 00:18:13.130 align:middle line:90% BRYNMOR CHAPMAN: Yeah. 00:18:13.130 --> 00:18:14.056 align:middle line:90% AUDIENCE: Same as last time. 00:18:14.056 --> 00:18:14.973 align:middle line:90% BRYNMOR CHAPMAN: Yeah. 00:18:14.973 --> 00:18:17.383 align:middle line:84% So as your colleague said last time, 00:18:17.383 --> 00:18:18.800 align:middle line:84% the next random thing that happens 00:18:18.800 --> 00:18:24.680 align:middle line:84% is Marilyn, or the contestant, picks a door. 00:18:24.680 --> 00:18:28.040 align:middle line:90% So M-- oh, dear. 00:18:28.040 --> 00:18:31.132 align:middle line:84% Monty and Marilyn both begin with the same letter, OK. 00:18:31.132 --> 00:18:32.340 align:middle line:90% Let's just call this Student. 00:18:32.340 --> 00:18:38.800 align:middle line:90% 00:18:38.800 --> 00:18:42.700 align:middle line:84% So now each of these is going to split 00:18:42.700 --> 00:18:46.214 align:middle line:90% into three more possibilities. 00:18:46.214 --> 00:18:52.240 align:middle line:84% So our contestant can pick purple, gold, or silver 00:18:52.240 --> 00:18:54.620 align:middle line:90% for each of these three things. 00:18:54.620 --> 00:18:57.320 align:middle line:84% Regardless of where Monty has hidden the prize, 00:18:57.320 --> 00:19:06.645 align:middle line:84% the contestant can pick any of the doors. 00:19:06.645 --> 00:19:22.760 align:middle line:90% 00:19:22.760 --> 00:19:26.300 align:middle line:84% What's the next random thing that could happen? 00:19:26.300 --> 00:19:26.800 align:middle line:90% Yeah? 00:19:26.800 --> 00:19:31.300 align:middle line:90% 00:19:31.300 --> 00:19:32.735 align:middle line:90% Were you raising your hand? 00:19:32.735 --> 00:19:33.860 align:middle line:90% AUDIENCE: Sorry about that. 00:19:33.860 --> 00:19:34.860 align:middle line:90% BRYNMOR CHAPMAN: Oh, no. 00:19:34.860 --> 00:19:37.640 align:middle line:84% AUDIENCE: If Monty has a choice of which door to reveal 00:19:37.640 --> 00:19:41.150 align:middle line:90% [INAUDIBLE]. 00:19:41.150 --> 00:19:43.670 align:middle line:84% BRYNMOR CHAPMAN: Yeah, so if Monty has the choice of door 00:19:43.670 --> 00:19:46.490 align:middle line:84% to reveal, then he picks one of the two remaining 00:19:46.490 --> 00:19:47.925 align:middle line:90% doors randomly. 00:19:47.925 --> 00:19:50.415 align:middle line:84% So, so far, we've got three branches 00:19:50.415 --> 00:19:51.415 align:middle line:90% where that could happen. 00:19:51.415 --> 00:19:54.642 align:middle line:90% 00:19:54.642 --> 00:20:01.350 align:middle line:84% We've got both Monty and Marilyn pick door P. So in that case, 00:20:01.350 --> 00:20:11.990 align:middle line:84% Monty can reveal either G or S. So let's make a branch for that. 00:20:11.990 --> 00:20:18.240 align:middle line:84% Similarly, if they both pick P, we could have either P or S. 00:20:18.240 --> 00:20:21.440 align:middle line:84% And if they both pick S, then here, 00:20:21.440 --> 00:20:26.085 align:middle line:90% we could have either P or G. 00:20:26.085 --> 00:20:28.000 align:middle line:90% 00:20:28.000 --> 00:20:33.010 align:middle line:84% Now, for the remaining ones, we could 00:20:33.010 --> 00:20:36.130 align:middle line:84% say that there's a random process there, 00:20:36.130 --> 00:20:39.500 align:middle line:84% where Monty is randomly picking from a single element set. 00:20:39.500 --> 00:20:41.420 align:middle line:84% So it's always going to be the same thing. 00:20:41.420 --> 00:20:44.050 align:middle line:84% We could consider it a random outcome, though. 00:20:44.050 --> 00:20:47.385 align:middle line:84% So we can also draw these arrows here. 00:20:47.385 --> 00:20:59.920 align:middle line:90% 00:20:59.920 --> 00:21:08.320 align:middle line:84% Kind of running out of space here, so let's put this up here, 00:21:08.320 --> 00:21:10.520 align:middle line:84% and let's label it on this board. 00:21:10.520 --> 00:21:19.690 align:middle line:90% So Monty reveals a zonk. 00:21:19.690 --> 00:21:23.080 align:middle line:84% So this is the technical term for a non-prize. 00:21:23.080 --> 00:21:26.850 align:middle line:84% I think the non-prizes on Monty's show were called zonks. 00:21:26.850 --> 00:21:28.230 align:middle line:90% They weren't always goats. 00:21:28.230 --> 00:21:31.950 align:middle line:84% Sometimes it was some other ridiculous prize, 00:21:31.950 --> 00:21:35.490 align:middle line:84% like a garbage can or a potato or something. 00:21:35.490 --> 00:21:37.190 align:middle line:84% But, yeah, they were just called zonks. 00:21:37.190 --> 00:21:44.886 align:middle line:90% 00:21:44.886 --> 00:21:47.118 align:middle line:90% So now, are we done? 00:21:47.118 --> 00:21:49.535 align:middle line:84% Is there any other randomness that we haven't modeled yet? 00:21:49.535 --> 00:21:55.380 align:middle line:90% 00:21:55.380 --> 00:21:58.650 align:middle line:90% Who thinks yes? 00:21:58.650 --> 00:21:59.530 align:middle line:90% Oh, OK. 00:21:59.530 --> 00:22:01.730 align:middle line:90% So what are we missing, then? 00:22:01.730 --> 00:22:03.730 align:middle line:84% Everybody seems to think we're missing somebody. 00:22:03.730 --> 00:22:05.190 align:middle line:90% Or are we just asleep? 00:22:05.190 --> 00:22:09.780 align:middle line:84% It is very early in the morning, I understand. 00:22:09.780 --> 00:22:12.330 align:middle line:90% Are we missing anything? 00:22:12.330 --> 00:22:15.630 align:middle line:84% Who thinks we're not missing anything? 00:22:15.630 --> 00:22:18.420 align:middle line:90% Well, a few hands, OK. 00:22:18.420 --> 00:22:23.490 align:middle line:84% Well, yeah, it turns out that this is all of the randomness. 00:22:23.490 --> 00:22:27.870 align:middle line:84% You could think about a different problem, where 00:22:27.870 --> 00:22:30.370 align:middle line:90% Marilyn has a random strategy. 00:22:30.370 --> 00:22:32.700 align:middle line:84% But for now, we're just-- our question 00:22:32.700 --> 00:22:35.080 align:middle line:84% is, Marilyn is always going to switch. 00:22:35.080 --> 00:22:37.145 align:middle line:84% What's the probability that she wins? 00:22:37.145 --> 00:22:41.010 align:middle line:84% So this is all the randomness that we've got. 00:22:41.010 --> 00:22:47.380 align:middle line:84% So now that we have our tree structure, 00:22:47.380 --> 00:22:50.050 align:middle line:90% we can label our outcomes. 00:22:50.050 --> 00:22:52.560 align:middle line:84% And that'll tell us what the set S is. 00:22:52.560 --> 00:22:58.800 align:middle line:84% So the set S is just going to be the set of leaves of our tree. 00:22:58.800 --> 00:23:04.980 align:middle line:84% Each path corresponds to some way 00:23:04.980 --> 00:23:06.835 align:middle line:84% that the random process could turn out. 00:23:06.835 --> 00:23:08.293 align:middle line:84% Those are going to be our outcomes. 00:23:08.293 --> 00:23:10.970 align:middle line:90% 00:23:10.970 --> 00:23:14.620 align:middle line:84% So formally, we could label these as tuples. 00:23:14.620 --> 00:23:19.890 align:middle line:84% So let's say that this one is PPP. 00:23:19.890 --> 00:23:24.050 align:middle line:84% So that represents Monty hides the prize behind door P, 00:23:24.050 --> 00:23:26.360 align:middle line:84% Marilyn picks door P, and then Monty 00:23:26.360 --> 00:23:30.980 align:middle line:90% reveals a zonk behind door G. 00:23:30.980 --> 00:23:35.180 align:middle line:90% This one's going to be PPS. 00:23:35.180 --> 00:23:39.260 align:middle line:90% This one's going to be PG. 00:23:39.260 --> 00:23:42.590 align:middle line:84% And now, I haven't actually labeled this edge. 00:23:42.590 --> 00:23:47.000 align:middle line:84% But if Monty hides the prize behind door P, 00:23:47.000 --> 00:23:51.713 align:middle line:84% and then Marilyn picks door G, where 00:23:51.713 --> 00:23:52.880 align:middle line:90% is Monty revealing the zonk? 00:23:52.880 --> 00:23:57.530 align:middle line:90% 00:23:57.530 --> 00:24:00.290 align:middle line:90% S, yeah. 00:24:00.290 --> 00:24:05.900 align:middle line:84% So let's label that, and we can continue all the way down. 00:24:05.900 --> 00:24:09.510 align:middle line:90% 00:24:09.510 --> 00:24:15.770 align:middle line:84% So S-- oh, I've just realized that I've overloaded it. 00:24:15.770 --> 00:24:18.700 align:middle line:84% It's a script S in my notes, and so it looks different. 00:24:18.700 --> 00:24:20.110 align:middle line:90% I'm sorry. 00:24:20.110 --> 00:24:26.190 align:middle line:84% [LAUGHS] OK, so our squiggly S is going to be the set of all 00:24:26.190 --> 00:24:35.415 align:middle line:90% tuples, so x, y, z, in PGS-- 00:24:35.415 --> 00:24:38.520 align:middle line:90% 00:24:38.520 --> 00:24:41.890 align:middle line:90% oops-- cubed. 00:24:41.890 --> 00:24:45.900 align:middle line:84% So it's going to be triples of P, G, and S, 00:24:45.900 --> 00:24:55.490 align:middle line:84% such that z is not equal to x, and z is not equal to y. 00:24:55.490 --> 00:25:01.947 align:middle line:90% 00:25:01.947 --> 00:25:04.200 align:middle line:84% So that's how we're mathematically going 00:25:04.200 --> 00:25:07.150 align:middle line:90% to represent our sample space. 00:25:07.150 --> 00:25:11.190 align:middle line:84% So x is the door that Monty hides the prize behind. 00:25:11.190 --> 00:25:14.100 align:middle line:84% y is the door that Marilyn initially chooses. 00:25:14.100 --> 00:25:17.020 align:middle line:84% And then z is going to be the door that Monty reveals, 00:25:17.020 --> 00:25:17.990 align:middle line:90% which has a goat. 00:25:17.990 --> 00:25:21.580 align:middle line:84% And by our rule, that door cannot be either of the first 00:25:21.580 --> 00:25:25.875 align:middle line:84% two, but anything else is a valid outcome. 00:25:25.875 --> 00:25:32.050 align:middle line:90% 00:25:32.050 --> 00:25:34.510 align:middle line:84% OK, do we have any questions about the sample space 00:25:34.510 --> 00:25:36.270 align:middle line:90% and how we compute it? 00:25:36.270 --> 00:25:45.382 align:middle line:90% 00:25:45.382 --> 00:25:53.680 align:middle line:84% So our next step, now that we've got our sample space, 00:25:53.680 --> 00:25:55.680 align:middle line:84% we want to figure out our probability function. 00:25:55.680 --> 00:26:10.420 align:middle line:90% 00:26:10.420 --> 00:26:13.360 align:middle line:84% So we'd like to figure out what the probability of each 00:26:13.360 --> 00:26:16.600 align:middle line:90% of these outcomes is. 00:26:16.600 --> 00:26:20.490 align:middle line:84% So to do this, we're going to assign a probability 00:26:20.490 --> 00:26:22.785 align:middle line:84% to every edge that we've got in our tree diagram. 00:26:22.785 --> 00:26:25.680 align:middle line:90% 00:26:25.680 --> 00:26:29.400 align:middle line:84% And so that's just going to be the probability that if we 00:26:29.400 --> 00:26:33.300 align:middle line:84% get to the source of that edge, what's the probability that we 00:26:33.300 --> 00:26:36.390 align:middle line:90% proceed along that edge? 00:26:36.390 --> 00:26:42.900 align:middle line:84% So at the very beginning, if we're at the root, 00:26:42.900 --> 00:26:45.300 align:middle line:84% what's the probability that, given 00:26:45.300 --> 00:26:49.865 align:middle line:84% that we're at the start, Monty hides the prize behind door P? 00:26:49.865 --> 00:26:52.480 align:middle line:90% 00:26:52.480 --> 00:26:52.980 align:middle line:90% Yeah? 00:26:52.980 --> 00:26:53.760 align:middle line:90% AUDIENCE: A third. 00:26:53.760 --> 00:26:56.040 align:middle line:84% BRYNMOR CHAPMAN: One-third, yeah. 00:26:56.040 --> 00:26:58.470 align:middle line:84% So we're going to label this with one-third. 00:26:58.470 --> 00:27:01.050 align:middle line:90% 00:27:01.050 --> 00:27:02.135 align:middle line:90% What about G? 00:27:02.135 --> 00:27:03.510 align:middle line:84% What's the probability that Monty 00:27:03.510 --> 00:27:05.270 align:middle line:90% hides the prize behind door G? 00:27:05.270 --> 00:27:10.530 align:middle line:90% 00:27:10.530 --> 00:27:11.520 align:middle line:90% Anybody? 00:27:11.520 --> 00:27:17.010 align:middle line:90% 00:27:17.010 --> 00:27:19.220 align:middle line:84% I mean, yeah, arithmetic is pretty difficult. 00:27:19.220 --> 00:27:21.900 align:middle line:90% But I'll just do it for you. 00:27:21.900 --> 00:27:25.028 align:middle line:90% It's also one-third. 00:27:25.028 --> 00:27:27.830 align:middle line:84% And the probability that Monty hides it behind door 00:27:27.830 --> 00:27:30.290 align:middle line:90% S, also a third. 00:27:30.290 --> 00:27:31.285 align:middle line:90% Yeah, question? 00:27:31.285 --> 00:27:34.127 align:middle line:84% AUDIENCE: So they have to add up to 1-- 00:27:34.127 --> 00:27:36.210 align:middle line:84% BRYNMOR CHAPMAN: Yeah, that's a great observation. 00:27:36.210 --> 00:27:40.100 align:middle line:84% So all of these three probabilities 00:27:40.100 --> 00:27:45.470 align:middle line:84% should add up to 1, because if we're at this vertex here, 00:27:45.470 --> 00:27:47.280 align:middle line:90% we should proceed somewhere. 00:27:47.280 --> 00:27:50.633 align:middle line:90% 00:27:50.633 --> 00:27:54.380 align:middle line:84% So it wouldn't make a whole lot of sense to have probability 00:27:54.380 --> 00:27:57.620 align:middle line:84% 1/3, probability 2/3, and probability 1, 00:27:57.620 --> 00:28:00.895 align:middle line:84% because then you have to proceed down this path, 00:28:00.895 --> 00:28:03.270 align:middle line:84% but then you also have to proceed down one of the others. 00:28:03.270 --> 00:28:06.170 align:middle line:90% It's a bit sticky. 00:28:06.170 --> 00:28:09.660 align:middle line:84% So given that you're at this vertex, 00:28:09.660 --> 00:28:11.250 align:middle line:84% you've got to reach a leaf somehow. 00:28:11.250 --> 00:28:13.320 align:middle line:84% You've got to get to some outcome. 00:28:13.320 --> 00:28:15.790 align:middle line:90% So you've got to proceed. 00:28:15.790 --> 00:28:21.710 align:middle line:84% So all of these edges should sum up to 1. 00:28:21.710 --> 00:28:24.020 align:middle line:90% Great observation. 00:28:24.020 --> 00:28:25.970 align:middle line:84% It's not quite true, though, that that 00:28:25.970 --> 00:28:29.030 align:middle line:90% holds for the entire column. 00:28:29.030 --> 00:28:32.510 align:middle line:84% So what happens if we look at the next column? 00:28:32.510 --> 00:28:36.290 align:middle line:84% So given that Monty hides the prize behind door P, 00:28:36.290 --> 00:28:39.010 align:middle line:84% what's the probability that Marilyn then picks door P? 00:28:39.010 --> 00:28:43.880 align:middle line:90% 00:28:43.880 --> 00:28:46.370 align:middle line:90% Any ideas? 00:28:46.370 --> 00:28:47.137 align:middle line:90% Yeah? 00:28:47.137 --> 00:28:47.970 align:middle line:90% AUDIENCE: One-third. 00:28:47.970 --> 00:28:50.090 align:middle line:84% BRYNMOR CHAPMAN: One-third, yeah. 00:28:50.090 --> 00:28:53.430 align:middle line:84% So Marilyn picks this with probability one-third. 00:28:53.430 --> 00:28:55.880 align:middle line:84% Marilyn doesn't know or care what's 00:28:55.880 --> 00:28:59.180 align:middle line:84% going on with where the prize actually is. 00:28:59.180 --> 00:29:02.150 align:middle line:84% She's just picking each door with probability one-third, 00:29:02.150 --> 00:29:04.140 align:middle line:90% regardless. 00:29:04.140 --> 00:29:08.050 align:middle line:84% So we've kind of got the same thing going on here. 00:29:08.050 --> 00:29:11.830 align:middle line:84% Each of these three edges has probability one-third. 00:29:11.830 --> 00:29:16.120 align:middle line:84% But now the same holds true for this branch, right? 00:29:16.120 --> 00:29:18.760 align:middle line:84% If Monty has hidden the door-- or hidden the prize behind door 00:29:18.760 --> 00:29:21.710 align:middle line:90% G, now the same logic applies. 00:29:21.710 --> 00:29:25.450 align:middle line:84% Marilyn is still picking each door with probability one-third. 00:29:25.450 --> 00:29:29.650 align:middle line:84% So we've also got one-third here, one-third here, one-third 00:29:29.650 --> 00:29:30.690 align:middle line:90% here. 00:29:30.690 --> 00:29:34.330 align:middle line:84% So it's not quite that all of the edges in a column 00:29:34.330 --> 00:29:36.730 align:middle line:84% add up to 1, but all of the edges coming out 00:29:36.730 --> 00:29:40.060 align:middle line:90% of a single vertex add up to 1. 00:29:40.060 --> 00:29:41.700 align:middle line:84% And same thing for this bottom branch. 00:29:41.700 --> 00:29:49.282 align:middle line:90% 00:29:49.282 --> 00:29:50.745 align:middle line:90% Now, what about the last column? 00:29:50.745 --> 00:29:53.440 align:middle line:90% 00:29:53.440 --> 00:29:55.480 align:middle line:84% If Monty has hidden the prize behind door P, 00:29:55.480 --> 00:29:58.630 align:middle line:84% and Marilyn has also picked door P, 00:29:58.630 --> 00:30:01.680 align:middle line:84% what's the probability that we get to outcome PPG? 00:30:01.680 --> 00:30:04.750 align:middle line:90% 00:30:04.750 --> 00:30:05.913 align:middle line:90% Yeah? 00:30:05.913 --> 00:30:07.030 align:middle line:90% AUDIENCE: One-half? 00:30:07.030 --> 00:30:08.697 align:middle line:84% BRYNMOR CHAPMAN: One-half, that's right. 00:30:08.697 --> 00:30:11.924 align:middle line:90% 00:30:11.924 --> 00:30:15.450 align:middle line:84% And it's also one-half-- whoops, that's not a 2. 00:30:15.450 --> 00:30:16.440 align:middle line:90% That's a 3. 00:30:16.440 --> 00:30:18.372 align:middle line:90% There we go, one-half. 00:30:18.372 --> 00:30:21.560 align:middle line:84% So probability one-half for each of them. 00:30:21.560 --> 00:30:24.507 align:middle line:84% We've got probability one-half that the zonk behind G 00:30:24.507 --> 00:30:26.840 align:middle line:84% is revealed, probability one-half that the zonk behind S 00:30:26.840 --> 00:30:29.210 align:middle line:90% is revealed. 00:30:29.210 --> 00:30:33.890 align:middle line:84% And as we were saying before, if Monty hides the prize behind P, 00:30:33.890 --> 00:30:38.195 align:middle line:84% and Marilyn picks G, Monty now has no choice. 00:30:38.195 --> 00:30:43.865 align:middle line:84% So we're just going to say that this is probability 1. 00:30:43.865 --> 00:30:47.660 align:middle line:84% And same with this one-- this one also has one PSG. 00:30:47.660 --> 00:30:50.690 align:middle line:90% 00:30:50.690 --> 00:30:52.590 align:middle line:84% And similarly, for the other three branches, 00:30:52.590 --> 00:31:00.990 align:middle line:84% we've got 1, one-half, one-half, 1, 1, 1, one-half, and one-half. 00:31:00.990 --> 00:31:11.020 align:middle line:90% 00:31:11.020 --> 00:31:13.810 align:middle line:84% So do these edge probabilities make sense to everybody? 00:31:13.810 --> 00:31:25.206 align:middle line:90% 00:31:25.206 --> 00:31:29.640 align:middle line:84% So now, remember what the probability function was. 00:31:29.640 --> 00:31:37.240 align:middle line:84% We have to assign a probability to every outcome. 00:31:37.240 --> 00:31:40.925 align:middle line:84% The outcomes are the leaves of our tree, not the edges. 00:31:40.925 --> 00:31:43.510 align:middle line:90% 00:31:43.510 --> 00:31:49.890 align:middle line:84% So what is the probability that we end up with outcome PPG, 00:31:49.890 --> 00:31:50.600 align:middle line:90% for instance? 00:31:50.600 --> 00:32:00.120 align:middle line:90% 00:32:00.120 --> 00:32:00.810 align:middle line:90% Yeah? 00:32:00.810 --> 00:32:02.240 align:middle line:90% AUDIENCE: It would be 1/18. 00:32:02.240 --> 00:32:03.480 align:middle line:90% BRYNMOR CHAPMAN: 1/18. 00:32:03.480 --> 00:32:04.755 align:middle line:90% And how are you getting that? 00:32:04.755 --> 00:32:07.130 align:middle line:84% AUDIENCE: It's one-third times one-third times one-third. 00:32:07.130 --> 00:32:11.840 align:middle line:84% BRYNMOR CHAPMAN: Yeah, one-third times one-third times one-third. 00:32:11.840 --> 00:32:13.610 align:middle line:84% We're going to multiply together all 00:32:13.610 --> 00:32:17.905 align:middle line:84% of the probabilities along the path from the root to that leaf. 00:32:17.905 --> 00:32:21.011 align:middle line:90% 00:32:21.011 --> 00:32:23.180 align:middle line:84% So Monty hides the prize behind door P 00:32:23.180 --> 00:32:25.580 align:middle line:90% with probability one-third. 00:32:25.580 --> 00:32:28.190 align:middle line:84% And then, conditioned on that happening, 00:32:28.190 --> 00:32:31.580 align:middle line:84% Marilyn then picks door P with probability one-third. 00:32:31.580 --> 00:32:33.740 align:middle line:84% So then there's probability one-ninth 00:32:33.740 --> 00:32:37.100 align:middle line:84% that we make it to this vertex here. 00:32:37.100 --> 00:32:43.676 align:middle line:84% And now that one-ninth is split again among these two outcomes. 00:32:43.676 --> 00:32:47.090 align:middle line:84% Does that make sense to everybody? 00:32:47.090 --> 00:32:51.440 align:middle line:84% So if we do that for every outcome, we've got-- 00:32:51.440 --> 00:32:56.480 align:middle line:84% oops, 1/18, not 1/8, another 1/18 for PPS. 00:32:56.480 --> 00:32:57.483 align:middle line:90% Then what about PGS? 00:32:57.483 --> 00:33:01.470 align:middle line:90% 00:33:01.470 --> 00:33:02.430 align:middle line:90% Yeah? 00:33:02.430 --> 00:33:03.840 align:middle line:84% AUDIENCE: It'd just be one-ninth. 00:33:03.840 --> 00:33:05.632 align:middle line:84% BRYNMOR CHAPMAN: Yeah, it's just one-ninth. 00:33:05.632 --> 00:33:08.580 align:middle line:90% 00:33:08.580 --> 00:33:10.925 align:middle line:84% So same deal for the first two layers, 00:33:10.925 --> 00:33:12.300 align:middle line:84% but then in the last layer, we've 00:33:12.300 --> 00:33:13.540 align:middle line:90% just got a probability of 1. 00:33:13.540 --> 00:33:16.180 align:middle line:84% We don't have to split again, so it remains at one-ninth. 00:33:16.180 --> 00:33:19.140 align:middle line:90% 00:33:19.140 --> 00:33:21.160 align:middle line:90% And once again, we can continue. 00:33:21.160 --> 00:33:31.070 align:middle line:84% We've got 1/9 here, 1/18, 1/18, 1/9, and we can continue. 00:33:31.070 --> 00:33:37.578 align:middle line:90% 00:33:37.578 --> 00:33:40.960 align:middle line:84% So we're going to justify this more formally later. 00:33:40.960 --> 00:33:43.500 align:middle line:84% But for now, just take it on faith 00:33:43.500 --> 00:33:46.590 align:middle line:84% that in order to get the probability of an outcome, 00:33:46.590 --> 00:33:49.530 align:middle line:84% you multiply together the probabilities along the path 00:33:49.530 --> 00:33:52.230 align:middle line:90% to that outcome. 00:33:52.230 --> 00:33:54.870 align:middle line:84% Hopefully, it at least makes intuitive sense. 00:33:54.870 --> 00:33:57.710 align:middle line:84% But we'll reason about it more formally later. 00:33:57.710 --> 00:34:02.975 align:middle line:90% 00:34:02.975 --> 00:34:11.940 align:middle line:84% So now, we have our sample space, 00:34:11.940 --> 00:34:13.710 align:middle line:84% we have our probability function, 00:34:13.710 --> 00:34:15.952 align:middle line:84% so we've got the entire probability space that we're 00:34:15.952 --> 00:34:16.910 align:middle line:90% trying to reason about. 00:34:16.910 --> 00:34:19.760 align:middle line:90% 00:34:19.760 --> 00:34:28.460 align:middle line:84% The next thing that we want, we want 00:34:28.460 --> 00:34:32.705 align:middle line:84% to figure out what we call events we care about. 00:34:32.705 --> 00:34:41.120 align:middle line:90% 00:34:41.120 --> 00:34:47.920 align:middle line:84% So an event is simply a subset of our sample space. 00:34:47.920 --> 00:34:58.944 align:middle line:90% 00:34:58.944 --> 00:35:00.333 align:middle line:90% So it's a set of outcomes. 00:35:00.333 --> 00:35:03.570 align:middle line:90% 00:35:03.570 --> 00:35:13.475 align:middle line:84% So up here, we could think about the event that Marilyn wins. 00:35:13.475 --> 00:35:16.400 align:middle line:90% 00:35:16.400 --> 00:35:19.835 align:middle line:84% So what is that formally as a set? 00:35:19.835 --> 00:35:35.070 align:middle line:90% 00:35:35.070 --> 00:35:35.640 align:middle line:90% Yeah? 00:35:35.640 --> 00:35:38.850 align:middle line:90% AUDIENCE: Subset of [INAUDIBLE]. 00:35:38.850 --> 00:35:40.660 align:middle line:90% BRYNMOR CHAPMAN: Yeah, exactly. 00:35:40.660 --> 00:35:44.520 align:middle line:84% So the answer was, it's the subset of events 00:35:44.520 --> 00:35:46.090 align:middle line:90% where x and y are different. 00:35:46.090 --> 00:35:53.610 align:middle line:90% 00:35:53.610 --> 00:35:59.640 align:middle line:84% So the notation that we usually use for events is the following. 00:35:59.640 --> 00:36:02.313 align:middle line:84% We put the description of the event in square brackets. 00:36:02.313 --> 00:36:05.300 align:middle line:90% 00:36:05.300 --> 00:36:22.811 align:middle line:84% So this is just going to be the set P, G, S, P, S, G, G, P, S, 00:36:22.811 --> 00:36:40.640 align:middle line:84% G, S, P, S, G, P, S, P, G. So the 6 outcomes 00:36:40.640 --> 00:36:44.270 align:middle line:84% where Marilyn and Monty originally 00:36:44.270 --> 00:36:46.430 align:middle line:84% picked different doors, and then Monty is 00:36:46.430 --> 00:36:48.205 align:middle line:90% forced to reveal the last one. 00:36:48.205 --> 00:36:53.024 align:middle line:90% 00:36:53.024 --> 00:37:04.440 align:middle line:84% Similarly, we could also define other events like Monty opens 00:37:04.440 --> 00:37:24.540 align:middle line:84% door P. So this is going to be the set G, G, P, S, S, P, G, S, 00:37:24.540 --> 00:37:36.870 align:middle line:84% P, and S, G, P. So does this make sense to everybody? 00:37:36.870 --> 00:37:39.510 align:middle line:90% Any subset of events-- 00:37:39.510 --> 00:37:42.900 align:middle line:84% or, sorry, any subset of outcomes is called an event. 00:37:42.900 --> 00:37:45.360 align:middle line:84% We can take our English description 00:37:45.360 --> 00:37:49.025 align:middle line:84% and then look at which actual outcomes this corresponds to. 00:37:49.025 --> 00:37:56.180 align:middle line:90% 00:37:56.180 --> 00:38:00.190 align:middle line:84% So what we want to do now is figure out 00:38:00.190 --> 00:38:05.980 align:middle line:84% what event we care about, or possibly what events. 00:38:05.980 --> 00:38:08.790 align:middle line:84% So what are the events that we care about for this problem? 00:38:08.790 --> 00:38:15.160 align:middle line:90% 00:38:15.160 --> 00:38:15.880 align:middle line:90% Anybody? 00:38:15.880 --> 00:38:28.400 align:middle line:90% 00:38:28.400 --> 00:38:30.650 align:middle line:84% What was the probability that we're trying to compute? 00:38:30.650 --> 00:38:31.622 align:middle line:90% Yeah? 00:38:31.622 --> 00:38:33.760 align:middle line:84% AUDIENCE: [INAUDIBLE] when Monty loses-- 00:38:33.760 --> 00:38:35.280 align:middle line:90% or Marilyn loses. 00:38:35.280 --> 00:38:36.280 align:middle line:90% BRYNMOR CHAPMAN: Pardon? 00:38:36.280 --> 00:38:37.630 align:middle line:90% AUDIENCE: When Marilyn loses. 00:38:37.630 --> 00:38:38.547 align:middle line:90% BRYNMOR CHAPMAN: Yeah. 00:38:38.547 --> 00:38:41.265 align:middle line:84% So the events Marilyn wins, and Marilyn loses. 00:38:41.265 --> 00:38:44.584 align:middle line:84% These are the events we care about. 00:38:44.584 --> 00:38:58.580 align:middle line:84% So we could say Marilyn loses is the complement 00:38:58.580 --> 00:39:01.465 align:middle line:84% of Marilyn wins, which we denote with an overbar. 00:39:01.465 --> 00:39:05.810 align:middle line:90% 00:39:05.810 --> 00:39:07.900 align:middle line:84% So these are the two events we care about. 00:39:07.900 --> 00:39:20.620 align:middle line:90% 00:39:20.620 --> 00:39:35.730 align:middle line:84% So the final step is to compute the answer. 00:39:35.730 --> 00:39:39.380 align:middle line:84% We've identified the event that we care about. 00:39:39.380 --> 00:39:41.285 align:middle line:84% So now we want to figure out the probability 00:39:41.285 --> 00:39:42.475 align:middle line:90% that that event occurs. 00:39:42.475 --> 00:39:46.616 align:middle line:90% 00:39:46.616 --> 00:39:50.223 align:middle line:84% So we can extend the probability function two events. 00:39:50.223 --> 00:39:53.770 align:middle line:90% 00:39:53.770 --> 00:39:57.495 align:middle line:84% So if E is an event, we can think 00:39:57.495 --> 00:39:58.870 align:middle line:84% about the probability of an event 00:39:58.870 --> 00:40:02.380 align:middle line:84% just as we think about the probability of outcomes. 00:40:02.380 --> 00:40:05.890 align:middle line:84% And we're going to define this to be 00:40:05.890 --> 00:40:12.850 align:middle line:84% the sum over all outcomes omega in E 00:40:12.850 --> 00:40:14.535 align:middle line:90% of the probability of omega. 00:40:14.535 --> 00:40:23.404 align:middle line:90% 00:40:23.404 --> 00:40:26.830 align:middle line:84% So hopefully, intuitively, that makes sense. 00:40:26.830 --> 00:40:32.650 align:middle line:84% All of these outcomes, exactly one outcome is going to happen. 00:40:32.650 --> 00:40:35.770 align:middle line:84% So if we're trying to figure out the probability that one 00:40:35.770 --> 00:40:38.380 align:middle line:84% of a set of outcomes happens, it's 00:40:38.380 --> 00:40:40.630 align:middle line:84% just going to be the sum of the probabilities 00:40:40.630 --> 00:40:42.420 align:middle line:90% that each individual one occurs. 00:40:42.420 --> 00:40:45.040 align:middle line:90% 00:40:45.040 --> 00:40:47.760 align:middle line:84% So we know the probabilities of each of the outcomes. 00:40:47.760 --> 00:40:51.055 align:middle line:84% That's what we just computed here. 00:40:51.055 --> 00:40:53.720 align:middle line:90% 00:40:53.720 --> 00:40:55.640 align:middle line:84% So now we can use that to compute 00:40:55.640 --> 00:41:00.380 align:middle line:84% the probability of the events we just talked about. 00:41:00.380 --> 00:41:04.585 align:middle line:84% So what is the probability that Marilyn wins? 00:41:04.585 --> 00:41:20.540 align:middle line:90% 00:41:20.540 --> 00:41:24.830 align:middle line:84% So we have our set of outcomes here. 00:41:24.830 --> 00:41:26.290 align:middle line:84% What's the probability of each one? 00:41:26.290 --> 00:41:42.980 align:middle line:90% 00:41:42.980 --> 00:41:46.440 align:middle line:84% Now, what's the probability of outcome PGS? 00:41:46.440 --> 00:41:46.980 align:middle line:90% Yeah? 00:41:46.980 --> 00:41:47.580 align:middle line:90% AUDIENCE: One-ninth. 00:41:47.580 --> 00:41:48.955 align:middle line:84% BRYNMOR CHAPMAN: One-ninth, yeah. 00:41:48.955 --> 00:41:50.796 align:middle line:84% That one's written up there explicitly. 00:41:50.796 --> 00:41:55.860 align:middle line:90% 00:41:55.860 --> 00:41:57.365 align:middle line:84% What about the probability of PSG? 00:41:57.365 --> 00:42:01.170 align:middle line:90% 00:42:01.170 --> 00:42:02.725 align:middle line:90% Same, one-ninth. 00:42:02.725 --> 00:42:06.270 align:middle line:90% 00:42:06.270 --> 00:42:08.143 align:middle line:84% What about the probability of GPS? 00:42:08.143 --> 00:42:10.800 align:middle line:90% 00:42:10.800 --> 00:42:12.562 align:middle line:90% Also a ninth. 00:42:12.562 --> 00:42:14.320 align:middle line:90% All of these are symmetric. 00:42:14.320 --> 00:42:23.670 align:middle line:90% 00:42:23.670 --> 00:42:24.625 align:middle line:90% What's that, 2/3? 00:42:24.625 --> 00:42:27.830 align:middle line:90% 00:42:27.830 --> 00:42:31.590 align:middle line:84% So we've got six events-- or six outcomes, each of which 00:42:31.590 --> 00:42:34.230 align:middle line:84% occurs with probability one-ninth. 00:42:34.230 --> 00:42:38.040 align:middle line:84% So if you add those all together, you get 2/3. 00:42:38.040 --> 00:42:42.660 align:middle line:84% So this seems to support what we found empirically. 00:42:42.660 --> 00:42:46.490 align:middle line:84% Marilyn wins with better than probability one-half. 00:42:46.490 --> 00:42:48.227 align:middle line:84% The probability that she loses is going 00:42:48.227 --> 00:42:49.435 align:middle line:90% to be the complement of this. 00:42:49.435 --> 00:43:01.298 align:middle line:90% 00:43:01.298 --> 00:43:04.340 align:middle line:84% So it's going to be the sum of the probabilities of all 00:43:04.340 --> 00:43:06.070 align:middle line:90% of the other outcomes. 00:43:06.070 --> 00:43:09.060 align:middle line:84% And we said that all probabilities fit out to 1, 00:43:09.060 --> 00:43:12.050 align:middle line:84% so this is going to be the complement with respect to 1. 00:43:12.050 --> 00:43:15.020 align:middle line:90% 00:43:15.020 --> 00:43:18.095 align:middle line:84% So as it turns out, Marilyn was correct. 00:43:18.095 --> 00:43:20.150 align:middle line:84% We didn't have to rely on intuition or anything 00:43:20.150 --> 00:43:23.520 align:middle line:84% like that, no ingenious analogies, et cetera. 00:43:23.520 --> 00:43:27.060 align:middle line:90% It's just arithmetic over Q. 00:43:27.060 --> 00:43:29.930 align:middle line:84% And I think the hardest part of this for many people 00:43:29.930 --> 00:43:35.450 align:middle line:84% is resisting the temptation to jump to some obvious conclusion. 00:43:35.450 --> 00:43:39.620 align:middle line:84% It might be tempting to get to the end of the tree diagram 00:43:39.620 --> 00:43:42.210 align:middle line:84% and be like, oh, we've got six outcomes where Marilyn wins, 00:43:42.210 --> 00:43:47.550 align:middle line:84% six where she loses, So probability 6/12 is 1/2. 00:43:47.550 --> 00:43:50.970 align:middle line:84% So if we actually set all of that intuition 00:43:50.970 --> 00:43:53.380 align:middle line:84% aside, go through everything formally, 00:43:53.380 --> 00:43:58.050 align:middle line:84% it's all quite simple and straightforward. 00:43:58.050 --> 00:43:59.620 align:middle line:84% Does that make sense to everybody? 00:43:59.620 --> 00:44:06.610 align:middle line:90% 00:44:06.610 --> 00:44:09.540 align:middle line:84% So let's move on to a different example. 00:44:09.540 --> 00:44:15.300 align:middle line:84% Suppose I have three strange dice. 00:44:15.300 --> 00:44:17.213 align:middle line:90% So excuse my drawing-- 00:44:17.213 --> 00:44:19.830 align:middle line:90% 00:44:19.830 --> 00:44:22.410 align:middle line:90% whoops-- there. 00:44:22.410 --> 00:44:26.820 align:middle line:84% OK, so I've got one die that has 2 written 00:44:26.820 --> 00:44:32.340 align:middle line:84% on two of the faces, so top and bottom, 6 written on these two 00:44:32.340 --> 00:44:37.680 align:middle line:84% faces, and 7 written on those two faces. 00:44:37.680 --> 00:44:42.770 align:middle line:84% Hopefully, that looks vaguely cubical. 00:44:42.770 --> 00:44:44.715 align:middle line:84% I've got a second die, which is similar. 00:44:44.715 --> 00:44:47.380 align:middle line:90% 00:44:47.380 --> 00:44:52.540 align:middle line:84% But instead of 2, 6, and 7, I've got 1, 5, and 9. 00:44:52.540 --> 00:45:00.870 align:middle line:84% Then I've got a third one, which has 3, 4, and 8. 00:45:00.870 --> 00:45:04.360 align:middle line:90% 00:45:04.360 --> 00:45:09.580 align:middle line:90% So consider the following game. 00:45:09.580 --> 00:45:12.310 align:middle line:90% I pick a die. 00:45:12.310 --> 00:45:16.720 align:middle line:84% Now you pick one of the other dies-- 00:45:16.720 --> 00:45:19.420 align:middle line:84% or dice-- after seeing what I pick. 00:45:19.420 --> 00:45:23.090 align:middle line:90% And we each roll our dice. 00:45:23.090 --> 00:45:25.035 align:middle line:84% And whoever gets the higher number wins. 00:45:25.035 --> 00:45:27.700 align:middle line:90% 00:45:27.700 --> 00:45:28.832 align:middle line:90% Who should win? 00:45:28.832 --> 00:45:30.165 align:middle line:90% Should I win, or should you win? 00:45:30.165 --> 00:45:32.830 align:middle line:90% 00:45:32.830 --> 00:45:33.660 align:middle line:90% Who thinks I win? 00:45:33.660 --> 00:45:37.330 align:middle line:90% 00:45:37.330 --> 00:45:39.110 align:middle line:90% OK, who thinks you win? 00:45:39.110 --> 00:45:43.770 align:middle line:90% 00:45:43.770 --> 00:45:47.850 align:middle line:84% Who thinks we should just do the tree diagram and figure it out? 00:45:47.850 --> 00:45:49.180 align:middle line:90% Yeah, good call. 00:45:49.180 --> 00:45:50.190 align:middle line:90% OK. 00:45:50.190 --> 00:45:58.862 align:middle line:84% So let's call these dice R, G, and B. Yeah, let's 00:45:58.862 --> 00:45:59.945 align:middle line:90% draw out our tree diagram. 00:45:59.945 --> 00:46:03.920 align:middle line:90% 00:46:03.920 --> 00:46:09.330 align:middle line:84% So what happens if, say, I pick the red die, and you 00:46:09.330 --> 00:46:12.630 align:middle line:90% pick the green die? 00:46:12.630 --> 00:46:13.420 align:middle line:90% Who should win? 00:46:13.420 --> 00:46:15.360 align:middle line:90% Let's see. 00:46:15.360 --> 00:46:20.650 align:middle line:84% So we've got-- when I roll my die, 00:46:20.650 --> 00:46:22.270 align:middle line:90% there are three possibilities. 00:46:22.270 --> 00:46:28.920 align:middle line:84% I could either get a 2, a 6, or a 7. 00:46:28.920 --> 00:46:32.700 align:middle line:84% And now you roll the green die, and you could have either 00:46:32.700 --> 00:46:36.790 align:middle line:90% a 1, a 5, or a 9-- 00:46:36.790 --> 00:46:38.510 align:middle line:90% not minus 5, 5-- 00:46:38.510 --> 00:46:46.174 align:middle line:90% 1, 5, 9, 1, 5, 9. 00:46:46.174 --> 00:46:50.704 align:middle line:90% 00:46:50.704 --> 00:46:53.230 align:middle line:84% And now what is the probability-- 00:46:53.230 --> 00:46:57.490 align:middle line:84% or, sorry, what are the outcomes? 00:46:57.490 --> 00:47:00.942 align:middle line:90% We have 9 leaves, so 9 outcomes. 00:47:00.942 --> 00:47:02.400 align:middle line:84% How are we going to represent them? 00:47:02.400 --> 00:47:15.670 align:middle line:90% 00:47:15.670 --> 00:47:18.610 align:middle line:84% Well, we can do it basically the same as how we represented them 00:47:18.610 --> 00:47:20.120 align:middle line:90% up here. 00:47:20.120 --> 00:47:26.060 align:middle line:84% We can just look at a tuple of edge labels. 00:47:26.060 --> 00:47:29.440 align:middle line:84% So our sample space is going to be 00:47:29.440 --> 00:47:39.755 align:middle line:84% the set 2, 6, 7, a Cartesian product with 1, 5, 9. 00:47:39.755 --> 00:47:42.551 align:middle line:90% 00:47:42.551 --> 00:47:44.402 align:middle line:84% So this is going to be ordered pairs 00:47:44.402 --> 00:47:46.610 align:middle line:84% where the first element is one of these three things, 00:47:46.610 --> 00:47:49.410 align:middle line:84% and the second is one of these three things. 00:47:49.410 --> 00:47:53.040 align:middle line:84% So this one, for instance, is 2, comma 1. 00:47:53.040 --> 00:47:55.350 align:middle line:84% This one is 2, comma, 5, et cetera. 00:47:55.350 --> 00:48:00.650 align:middle line:90% 00:48:00.650 --> 00:48:02.575 align:middle line:84% Now what are the probabilities on each edge? 00:48:02.575 --> 00:48:07.520 align:middle line:90% 00:48:07.520 --> 00:48:08.900 align:middle line:90% Yeah? 00:48:08.900 --> 00:48:11.720 align:middle line:90% One-ninth. 00:48:11.720 --> 00:48:14.937 align:middle line:84% So those are the probabilities on the outcomes. 00:48:14.937 --> 00:48:16.895 align:middle line:90% Yeah, one-third on each edge. 00:48:16.895 --> 00:48:23.636 align:middle line:90% 00:48:23.636 --> 00:48:26.420 align:middle line:84% So when I roll my die, I've got a one-third chance each 00:48:26.420 --> 00:48:28.040 align:middle line:90% of getting 2, 6, or 7. 00:48:28.040 --> 00:48:30.350 align:middle line:84% When you roll yours, you've got a one-third chance each 00:48:30.350 --> 00:48:32.400 align:middle line:90% of getting 1, 5, or 9. 00:48:32.400 --> 00:48:36.220 align:middle line:90% 00:48:36.220 --> 00:48:39.010 align:middle line:84% And it's going to be the same for the other two branches. 00:48:39.010 --> 00:48:41.710 align:middle line:84% So when we multiply together the probabilities 00:48:41.710 --> 00:48:47.730 align:middle line:84% along paths to leaves, we get one-ninth for every outcome. 00:48:47.730 --> 00:48:52.354 align:middle line:90% 00:48:52.354 --> 00:48:54.400 align:middle line:84% So when you have a probability space 00:48:54.400 --> 00:48:57.580 align:middle line:84% like this that's particularly nice, 00:48:57.580 --> 00:48:59.940 align:middle line:84% this is what we call a uniform probability space. 00:48:59.940 --> 00:49:04.460 align:middle line:90% 00:49:04.460 --> 00:49:08.740 align:middle line:90% So let's say S, comma, Pr-- 00:49:08.740 --> 00:49:11.770 align:middle line:84% so this is our probability space-- 00:49:11.770 --> 00:49:19.905 align:middle line:90% is uniform if Pr is constant. 00:49:19.905 --> 00:49:26.270 align:middle line:90% 00:49:26.270 --> 00:49:28.060 align:middle line:84% So in this case, it's a constant function 00:49:28.060 --> 00:49:31.190 align:middle line:84% that maps every outcome to a probability of one-ninth. 00:49:31.190 --> 00:49:32.975 align:middle line:84% So it's a uniform probability space. 00:49:32.975 --> 00:49:35.810 align:middle line:90% 00:49:35.810 --> 00:49:38.812 align:middle line:84% And uniform spaces are particularly 00:49:38.812 --> 00:49:41.270 align:middle line:84% nice because we don't even have to worry about the addition 00:49:41.270 --> 00:49:42.950 align:middle line:90% that we were doing before. 00:49:42.950 --> 00:49:46.250 align:middle line:84% In order to figure out probabilities of events, 00:49:46.250 --> 00:49:48.830 align:middle line:84% you just count up how many outcomes are in the event 00:49:48.830 --> 00:49:52.740 align:middle line:84% and then divide by the total number of outcomes. 00:49:52.740 --> 00:49:54.880 align:middle line:84% So it just becomes a counting problem. 00:49:54.880 --> 00:50:00.270 align:middle line:90% 00:50:00.270 --> 00:50:03.925 align:middle line:84% So using that, what's the probability that red wins? 00:50:03.925 --> 00:50:11.540 align:middle line:90% 00:50:11.540 --> 00:50:12.170 align:middle line:90% Yeah? 00:50:12.170 --> 00:50:13.340 align:middle line:90% AUDIENCE: 5/9. 00:50:13.340 --> 00:50:15.155 align:middle line:90% BRYNMOR CHAPMAN: 5/9, yeah. 00:50:15.155 --> 00:50:23.570 align:middle line:90% 00:50:23.570 --> 00:50:25.700 align:middle line:90% So let's break this down a bit. 00:50:25.700 --> 00:50:30.580 align:middle line:84% So we're looking at the probability of the set, 00:50:30.580 --> 00:50:35.080 align:middle line:84% 2, comma, 1, red wins; 6, 1, 6, 5, red wins; 00:50:35.080 --> 00:50:37.950 align:middle line:90% and 7, 1, 7, 5, red wins. 00:50:37.950 --> 00:50:41.950 align:middle line:90% 00:50:41.950 --> 00:50:56.440 align:middle line:90% So 2, 1, 6, 1, 6, 5, 7, 1, 7, 5. 00:50:56.440 --> 00:50:58.780 align:middle line:90% We have five of these. 00:50:58.780 --> 00:51:00.985 align:middle line:90% So that's 5/9. 00:51:00.985 --> 00:51:17.991 align:middle line:90% 00:51:17.991 --> 00:51:21.550 align:middle line:84% Now, what if instead we had red versus blue 00:51:21.550 --> 00:51:22.785 align:middle line:90% instead of red versus green? 00:51:22.785 --> 00:51:31.040 align:middle line:90% 00:51:31.040 --> 00:51:34.140 align:middle line:84% So if instead of red versus green, we have red versus blue, 00:51:34.140 --> 00:51:40.380 align:middle line:84% this column now gets replaced with 3, 4, 8 instead of 1, 5, 9. 00:51:40.380 --> 00:51:45.428 align:middle line:90% 00:51:45.428 --> 00:51:49.490 align:middle line:84% So our tree diagram is going to look like the following. 00:51:49.490 --> 00:51:53.690 align:middle line:90% So we've still got 2, 6, 7 here. 00:51:53.690 --> 00:51:57.840 align:middle line:84% But this time, this is going to be 3, 4, 8-- 00:51:57.840 --> 00:52:01.460 align:middle line:90% 00:52:01.460 --> 00:52:06.590 align:middle line:90% 3, 4, 8, 3, 4, 8. 00:52:06.590 --> 00:52:10.310 align:middle line:90% 00:52:10.310 --> 00:52:13.190 align:middle line:84% And I'm going to omit the probabilities because, as we 00:52:13.190 --> 00:52:14.670 align:middle line:90% just established, it's uniform. 00:52:14.670 --> 00:52:16.003 align:middle line:90% So it's just a counting problem. 00:52:16.003 --> 00:52:24.446 align:middle line:90% 00:52:24.446 --> 00:52:25.863 align:middle line:84% But let's write down the outcomes. 00:52:25.863 --> 00:52:41.994 align:middle line:90% 00:52:41.994 --> 00:52:44.525 align:middle line:84% So in how many of these outcomes does red win? 00:52:44.525 --> 00:52:48.600 align:middle line:90% 00:52:48.600 --> 00:52:54.930 align:middle line:84% Well, not this one, not this one, not this one. 00:52:54.930 --> 00:53:01.990 align:middle line:84% So if red rolls 2, all of blue's numbers are going to be higher, 00:53:01.990 --> 00:53:04.290 align:middle line:90% so blue is always going to win. 00:53:04.290 --> 00:53:09.810 align:middle line:84% If red rolls 6, red wins if blue rolls 3 or 4; 00:53:09.810 --> 00:53:13.925 align:middle line:84% loses if blue rolls 8, and same for 7. 00:53:13.925 --> 00:53:19.662 align:middle line:90% 00:53:19.662 --> 00:53:26.820 align:middle line:84% So we have four outcomes where red wins. 00:53:26.820 --> 00:53:28.353 align:middle line:84% So red wins with probability 4/9. 00:53:28.353 --> 00:53:35.012 align:middle line:90% 00:53:35.012 --> 00:53:45.967 align:middle line:84% So we just saw that red beats green but loses to blue. 00:53:45.967 --> 00:53:47.800 align:middle line:84% What does that tell us about blue and green? 00:53:47.800 --> 00:53:50.630 align:middle line:90% 00:53:50.630 --> 00:53:52.720 align:middle line:84% If I pick blue, and you pick green, what happens? 00:53:52.720 --> 00:53:55.460 align:middle line:90% 00:53:55.460 --> 00:53:56.200 align:middle line:90% Who wins? 00:53:56.200 --> 00:54:00.080 align:middle line:90% 00:54:00.080 --> 00:54:00.770 align:middle line:90% Yeah? 00:54:00.770 --> 00:54:01.850 align:middle line:90% AUDIENCE: Green. 00:54:01.850 --> 00:54:02.820 align:middle line:90% BRYNMOR CHAPMAN: Green. 00:54:02.820 --> 00:54:04.180 align:middle line:90% Why? 00:54:04.180 --> 00:54:06.680 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:54:06.680 --> 00:54:08.575 align:middle line:84% BRYNMOR CHAPMAN: OK, so green beats red. 00:54:08.575 --> 00:54:11.660 align:middle line:90% 00:54:11.660 --> 00:54:15.025 align:middle line:84% So I think we just established that red beats green, right? 00:54:15.025 --> 00:54:20.720 align:middle line:90% 00:54:20.720 --> 00:54:26.580 align:middle line:84% So here, we've got red beats green. 00:54:26.580 --> 00:54:29.585 align:middle line:84% Here, we've got red loses to blue. 00:54:29.585 --> 00:54:33.570 align:middle line:90% 00:54:33.570 --> 00:54:38.460 align:middle line:84% So if we have green and blue, who 00:54:38.460 --> 00:54:41.910 align:middle line:90% thinks that green should win? 00:54:41.910 --> 00:54:44.040 align:middle line:90% Who thinks that blue should win? 00:54:44.040 --> 00:54:46.320 align:middle line:90% Who has absolutely no idea? 00:54:46.320 --> 00:54:49.710 align:middle line:84% Who wants to do the tree diagram? 00:54:49.710 --> 00:54:51.448 align:middle line:84% Oh, still pretty enthusiastic about that, 00:54:51.448 --> 00:54:52.990 align:middle line:84% even though we've got some intuition. 00:54:52.990 --> 00:54:54.660 align:middle line:90% That's great. 00:54:54.660 --> 00:54:56.170 align:middle line:90% Intuition is bad. 00:54:56.170 --> 00:54:57.580 align:middle line:90% Discard the intuition. 00:54:57.580 --> 00:55:00.570 align:middle line:90% 00:55:00.570 --> 00:55:05.910 align:middle line:84% OK, so if we've got green versus blue, 00:55:05.910 --> 00:55:08.590 align:middle line:84% so this can be the green roll here. 00:55:08.590 --> 00:55:12.150 align:middle line:90% We've got 1, 5, 9. 00:55:12.150 --> 00:55:23.320 align:middle line:84% Blue roll can be 3, 4, 8, 3, 4, 8, 3, 4, 8. 00:55:23.320 --> 00:55:26.920 align:middle line:90% 00:55:26.920 --> 00:55:29.650 align:middle line:84% So in which of these does green win? 00:55:29.650 --> 00:55:33.520 align:middle line:84% Well, if green rolls 1, then green is screwed. 00:55:33.520 --> 00:55:37.990 align:middle line:84% So all of these are bad for green. 00:55:37.990 --> 00:55:43.210 align:middle line:84% If green rolls 5, well, 5 will beat 3 and 4, but not 8. 00:55:43.210 --> 00:55:47.740 align:middle line:90% So green wins these two. 00:55:47.740 --> 00:55:51.855 align:middle line:84% And if green rolls 9, then green beats whatever blue rolls. 00:55:51.855 --> 00:55:56.870 align:middle line:90% 00:55:56.870 --> 00:55:59.025 align:middle line:84% So it looks like green is winning in five of these. 00:55:59.025 --> 00:56:14.435 align:middle line:90% 00:56:14.435 --> 00:56:18.090 align:middle line:90% So this should be like that. 00:56:18.090 --> 00:56:25.050 align:middle line:90% 00:56:25.050 --> 00:56:31.050 align:middle line:84% So red beats green, green beats blue, and blue beats red. 00:56:31.050 --> 00:56:32.000 align:middle line:90% What's gone wrong? 00:56:32.000 --> 00:56:40.800 align:middle line:90% 00:56:40.800 --> 00:56:41.696 align:middle line:90% Yeah? 00:56:41.696 --> 00:56:44.480 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:56:44.480 --> 00:56:45.480 align:middle line:90% BRYNMOR CHAPMAN: Pardon? 00:56:45.480 --> 00:56:48.570 align:middle line:84% AUDIENCE: There is only a hierarchy [INAUDIBLE]. 00:56:48.570 --> 00:56:49.650 align:middle line:90% BRYNMOR CHAPMAN: Yeah. 00:56:49.650 --> 00:56:51.520 align:middle line:90% So nothing's gone wrong. 00:56:51.520 --> 00:56:54.420 align:middle line:84% All that's gone wrong is I kind of expected 00:56:54.420 --> 00:56:57.570 align:middle line:84% that there would be a linear order on them. 00:56:57.570 --> 00:56:59.920 align:middle line:84% If red beats green and green beats blue, 00:56:59.920 --> 00:57:01.440 align:middle line:84% it kind of intuitively makes sense 00:57:01.440 --> 00:57:02.980 align:middle line:84% that red should beat blue, right? 00:57:02.980 --> 00:57:05.550 align:middle line:90% No. 00:57:05.550 --> 00:57:07.210 align:middle line:90% No, no intuition. 00:57:07.210 --> 00:57:09.510 align:middle line:90% Just do the tree diagram. 00:57:09.510 --> 00:57:12.520 align:middle line:84% So yeah, this is perfectly fine, right? 00:57:12.520 --> 00:57:14.150 align:middle line:84% And, in fact, it gets even weirder. 00:57:14.150 --> 00:57:17.488 align:middle line:90% 00:57:17.488 --> 00:57:21.110 align:middle line:84% What do you think happens if we each choose two dice-- 00:57:21.110 --> 00:57:22.360 align:middle line:90% or, sorry, we each roll twice? 00:57:22.360 --> 00:57:27.250 align:middle line:90% 00:57:27.250 --> 00:57:33.200 align:middle line:84% So what if I pick the red die; I roll it twice; 00:57:33.200 --> 00:57:37.090 align:middle line:84% you pick the green die; you roll it twice? 00:57:37.090 --> 00:57:39.250 align:middle line:84% Red probably beats green, so is it true 00:57:39.250 --> 00:57:41.770 align:middle line:84% that two red rolls probably beat two green rolls 00:57:41.770 --> 00:57:42.600 align:middle line:90% if we sum them up? 00:57:42.600 --> 00:57:48.280 align:middle line:90% 00:57:48.280 --> 00:57:51.130 align:middle line:90% Who thinks yes? 00:57:51.130 --> 00:57:53.290 align:middle line:90% Who thinks no? 00:57:53.290 --> 00:57:55.600 align:middle line:84% Who wants to draw the tree diagram? 00:57:55.600 --> 00:57:57.560 align:middle line:90% Oh, wow, so eager. 00:57:57.560 --> 00:57:58.550 align:middle line:90% I don't. 00:57:58.550 --> 00:58:02.941 align:middle line:90% [LAUGHS] 00:58:02.941 --> 00:58:06.730 align:middle line:84% What is the tree diagram going to look like if we do that? 00:58:06.730 --> 00:58:08.420 align:middle line:90% I'm rolling twice. 00:58:08.420 --> 00:58:09.890 align:middle line:90% You're rolling twice. 00:58:09.890 --> 00:58:15.240 align:middle line:84% Each roll can have three possibilities. 00:58:15.240 --> 00:58:20.240 align:middle line:84% So we're going to end up with a pretty massive tree. 00:58:20.240 --> 00:58:23.750 align:middle line:84% I don't want to draw that, at least not on the board. 00:58:23.750 --> 00:58:26.600 align:middle line:84% Does somebody else want to come up and do it? 00:58:26.600 --> 00:58:28.010 align:middle line:90% Please? 00:58:28.010 --> 00:58:29.450 align:middle line:90% No, nobody? 00:58:29.450 --> 00:58:33.750 align:middle line:84% OK, well, instead of drawing out the entire tree diagram, 00:58:33.750 --> 00:58:36.775 align:middle line:84% let's start drawing it and see if we can find a pattern. 00:58:36.775 --> 00:58:41.795 align:middle line:90% 00:58:41.795 --> 00:58:47.233 align:middle line:90% So this is R2 versus G2. 00:58:47.233 --> 00:58:51.746 align:middle line:90% 00:58:51.746 --> 00:58:58.950 align:middle line:84% So let's start by drawing as we did before. 00:58:58.950 --> 00:59:02.780 align:middle line:90% So my first roll, that was red. 00:59:02.780 --> 00:59:05.960 align:middle line:90% It could be 2, 6, or 7. 00:59:05.960 --> 00:59:08.790 align:middle line:84% Second roll could also be 2, 6, or 7. 00:59:08.790 --> 00:59:21.406 align:middle line:90% 00:59:21.406 --> 00:59:28.510 align:middle line:84% Now, for green, we're basically just going 00:59:28.510 --> 00:59:32.980 align:middle line:84% to be drawing the same tree nine times, so let's not. 00:59:32.980 --> 00:59:40.900 align:middle line:84% So green could roll 1, 5, 9, and then can also roll 00:59:40.900 --> 00:59:43.965 align:middle line:90% 1, 5, 9 for the second roll. 00:59:43.965 --> 00:59:52.132 align:middle line:90% 00:59:52.132 --> 00:59:55.120 align:middle line:84% And rather than duplicating this tree nine times, 00:59:55.120 --> 00:59:57.860 align:middle line:84% I'm just going to leave this as it is. 00:59:57.860 --> 00:59:59.850 align:middle line:84% And we can see if we can find a pattern. 00:59:59.850 --> 01:00:04.012 align:middle line:90% 01:00:04.012 --> 01:00:05.990 align:middle line:84% Does that make sense to everybody? 01:00:05.990 --> 01:00:12.710 align:middle line:90% 01:00:12.710 --> 01:00:19.490 align:middle line:84% So an outcome now is going to be a 4-tuple, instead of a pair. 01:00:19.490 --> 01:00:23.240 align:middle line:84% And in some sense, we can think of it 01:00:23.240 --> 01:00:29.960 align:middle line:84% as combining a leaf of this tree with a leaf of this tree. 01:00:29.960 --> 01:00:32.300 align:middle line:84% We're going to take a red leaf and a green leaf. 01:00:32.300 --> 01:00:32.810 align:middle line:90% Question? 01:00:32.810 --> 01:00:34.790 align:middle line:90% Or no? 01:00:34.790 --> 01:00:35.960 align:middle line:90% No, OK. 01:00:35.960 --> 01:00:42.230 align:middle line:90% 01:00:42.230 --> 01:00:49.391 align:middle line:84% So we can then abbreviate it even further. 01:00:49.391 --> 01:00:52.920 align:middle line:84% Instead of looking at the pair of numbers that we've rolled, 01:00:52.920 --> 01:00:57.020 align:middle line:84% let's instead look at just the sum. 01:00:57.020 --> 01:01:02.420 align:middle line:84% So if red rolls two 2's, we've got 4 here. 01:01:02.420 --> 01:01:04.220 align:middle line:90% 2 and 6 is 8. 01:01:04.220 --> 01:01:06.830 align:middle line:90% 2 and 7 is 9. 01:01:06.830 --> 01:01:09.855 align:middle line:90% 6, 2 is 8. 01:01:09.855 --> 01:01:10.990 align:middle line:90% 6, 6 is not 9. 01:01:10.990 --> 01:01:12.480 align:middle line:90% That's 12. 01:01:12.480 --> 01:01:14.550 align:middle line:90% Never trust me to do arithmetic. 01:01:14.550 --> 01:01:16.170 align:middle line:90% 6, 7 is 13. 01:01:16.170 --> 01:01:17.880 align:middle line:90% 7 and 2 is 9. 01:01:17.880 --> 01:01:19.470 align:middle line:90% 7 and 6 is 13. 01:01:19.470 --> 01:01:20.630 align:middle line:90% 7 and 7 is 14. 01:01:20.630 --> 01:01:25.278 align:middle line:90% 01:01:25.278 --> 01:01:27.850 align:middle line:84% And if we do the same thing for green, 01:01:27.850 --> 01:01:38.005 align:middle line:84% we have 2, 6, 10, 6, 10, 14, 10, 14, 18. 01:01:38.005 --> 01:01:45.438 align:middle line:90% 01:01:45.438 --> 01:01:50.130 align:middle line:84% So now we can start counting outcomes. 01:01:50.130 --> 01:01:52.780 align:middle line:84% If we take this leaf of the red tree, 01:01:52.780 --> 01:01:56.040 align:middle line:84% how many outcomes are there where red wins? 01:01:56.040 --> 01:01:59.520 align:middle line:84% If red rolls four in total, the only way 01:01:59.520 --> 01:02:03.970 align:middle line:84% that red wins is if green gets snake eyes. 01:02:03.970 --> 01:02:05.015 align:middle line:90% So we've just got one. 01:02:05.015 --> 01:02:09.770 align:middle line:90% 01:02:09.770 --> 01:02:14.270 align:middle line:84% If red rolls 8, which could be either of these two, 01:02:14.270 --> 01:02:17.990 align:middle line:84% how many outcomes are there where red wins? 01:02:17.990 --> 01:02:21.890 align:middle line:84% Well, we've got the 2 here, a 6 here, and a 6 here. 01:02:21.890 --> 01:02:22.780 align:middle line:90% So we've got three. 01:02:22.780 --> 01:02:25.880 align:middle line:90% 01:02:25.880 --> 01:02:30.500 align:middle line:84% And we've got two red leaves that have 8. 01:02:30.500 --> 01:02:31.320 align:middle line:90% What about 9? 01:02:31.320 --> 01:02:34.520 align:middle line:84% It looks like we've got two more that have 9. 01:02:34.520 --> 01:02:38.140 align:middle line:84% Same deal-- we don't have any 8's or 9's over on this side. 01:02:38.140 --> 01:02:43.940 align:middle line:90% 01:02:43.940 --> 01:02:45.330 align:middle line:90% What else do we have? 01:02:45.330 --> 01:02:47.540 align:middle line:90% 12. 01:02:47.540 --> 01:02:51.830 align:middle line:84% OK, well, if red rolls 12, then red 01:02:51.830 --> 01:02:55.620 align:middle line:84% is good if blue rolls 2, 6, 10, but not 14 or 18. 01:02:55.620 --> 01:02:59.143 align:middle line:84% So we've got 1, 2, 3, 4, 5, 6 outcomes. 01:02:59.143 --> 01:03:01.880 align:middle line:90% 01:03:01.880 --> 01:03:05.800 align:middle line:84% And we have one red leaf where red has 12. 01:03:05.800 --> 01:03:08.481 align:middle line:90% 01:03:08.481 --> 01:03:09.880 align:middle line:90% And then same thing for 13-- 01:03:09.880 --> 01:03:14.220 align:middle line:84% there are no 12's or 13's over here, and we have two 13's. 01:03:14.220 --> 01:03:19.560 align:middle line:90% 01:03:19.560 --> 01:03:23.100 align:middle line:90% Now, what about 14? 01:03:23.100 --> 01:03:28.170 align:middle line:84% Well, if red rolls 14, then same deal. 01:03:28.170 --> 01:03:29.915 align:middle line:84% But we've got these two ties here. 01:03:29.915 --> 01:03:35.490 align:middle line:90% 01:03:35.490 --> 01:03:36.943 align:middle line:90% So red wins. 01:03:36.943 --> 01:03:40.500 align:middle line:90% 01:03:40.500 --> 01:03:43.050 align:middle line:90% Draw, we've just got two. 01:03:43.050 --> 01:03:45.790 align:middle line:84% And then red loses is going to be the complement. 01:03:45.790 --> 01:03:59.800 align:middle line:90% 01:03:59.800 --> 01:04:02.640 align:middle line:84% So has anybody added all of these up yet? 01:04:02.640 --> 01:04:03.485 align:middle line:90% I haven't. 01:04:03.485 --> 01:04:07.012 align:middle line:90% 01:04:07.012 --> 01:04:11.965 align:middle line:90% 5, 30, 42, 43, is that right? 01:04:11.965 --> 01:04:15.277 align:middle line:90% 01:04:15.277 --> 01:04:15.860 align:middle line:90% AUDIENCE: Yes. 01:04:15.860 --> 01:04:22.850 align:middle line:90% 01:04:22.850 --> 01:04:26.845 align:middle line:84% BRYNMOR CHAPMAN: OK, so 43 outcomes for red wins. 01:04:26.845 --> 01:04:39.500 align:middle line:90% 01:04:39.500 --> 01:04:40.768 align:middle line:90% Is something wrong? 01:04:40.768 --> 01:04:41.935 align:middle line:90% This doesn't match my notes. 01:04:41.935 --> 01:04:46.130 align:middle line:90% 01:04:46.130 --> 01:04:47.840 align:middle line:90% Let's see. 01:04:47.840 --> 01:04:52.070 align:middle line:90% So we've got 1 for this subtree. 01:04:52.070 --> 01:04:55.700 align:middle line:90% For each of these, we have 3. 01:04:55.700 --> 01:04:59.200 align:middle line:84% For each of the 9's, we also have 3, 01:04:59.200 --> 01:05:03.453 align:middle line:90% so that's those, 12, 13, and 13. 01:05:03.453 --> 01:05:07.200 align:middle line:90% 01:05:07.200 --> 01:05:08.900 align:middle line:84% Oh, do I just have too many terms? 01:05:08.900 --> 01:05:10.025 align:middle line:90% I just have too many terms. 01:05:10.025 --> 01:05:13.200 align:middle line:90% 01:05:13.200 --> 01:05:14.256 align:middle line:90% OK, not 43. 01:05:14.256 --> 01:05:14.923 align:middle line:90% It should be 37. 01:05:14.923 --> 01:05:17.730 align:middle line:90% 01:05:17.730 --> 01:05:19.740 align:middle line:90% Is that right? 01:05:19.740 --> 01:05:25.470 align:middle line:84% Yeah, or 12, 13, 13, and 14 each have 6. 01:05:25.470 --> 01:05:26.460 align:middle line:90% AUDIENCE: Yes. 01:05:26.460 --> 01:05:28.020 align:middle line:90% BRYNMOR CHAPMAN: OK. 01:05:28.020 --> 01:05:31.090 align:middle line:84% OK, yeah, never trust me to do arithmetic. 01:05:31.090 --> 01:05:32.790 align:middle line:90% It's always a bad plan. 01:05:32.790 --> 01:05:36.850 align:middle line:84% OK, so red wins in 37 of the outcomes. 01:05:36.850 --> 01:05:38.620 align:middle line:90% There's a draw in 2 of them. 01:05:38.620 --> 01:05:40.430 align:middle line:84% And how many outcomes are there in total? 01:05:40.430 --> 01:05:46.770 align:middle line:90% 01:05:46.770 --> 01:05:47.880 align:middle line:90% Yeah? 01:05:47.880 --> 01:05:48.910 align:middle line:90% AUDIENCE: Is that 81? 01:05:48.910 --> 01:05:50.230 align:middle line:90% BRYNMOR CHAPMAN: Yeah, 81. 01:05:50.230 --> 01:05:59.130 align:middle line:84% So red loses in 81 minus 37 minus 2, which is that 42? 01:05:59.130 --> 01:06:00.020 align:middle line:90% I think that's 42. 01:06:00.020 --> 01:06:03.742 align:middle line:90% 01:06:03.742 --> 01:06:12.830 align:middle line:84% So because this is a uniform space, we have 37/81, 42/81, 01:06:12.830 --> 01:06:15.770 align:middle line:84% which is going to be-- well, actually, let's not reduce it. 01:06:15.770 --> 01:06:18.190 align:middle line:90% It's easier to see like that. 01:06:18.190 --> 01:06:19.845 align:middle line:90% And 2 out of 81 for a draw. 01:06:19.845 --> 01:06:22.630 align:middle line:90% 01:06:22.630 --> 01:06:27.500 align:middle line:84% So if we only had one roll, red had the advantage. 01:06:27.500 --> 01:06:29.680 align:middle line:84% Red wins with probability of 5/9. 01:06:29.680 --> 01:06:33.850 align:middle line:84% If we have two rolls, the probability that red wins 01:06:33.850 --> 01:06:36.728 align:middle line:84% is smaller than the probability that red loses. 01:06:36.728 --> 01:06:38.770 align:middle line:84% So now suddenly, green has the advantage instead. 01:06:38.770 --> 01:06:41.800 align:middle line:90% 01:06:41.800 --> 01:06:47.080 align:middle line:84% And as it turns out, the same thing 01:06:47.080 --> 01:06:50.650 align:middle line:84% happens with all three pairs of dice. 01:06:50.650 --> 01:06:55.430 align:middle line:84% So red beats green with one roll. 01:06:55.430 --> 01:06:57.530 align:middle line:90% Green beats blue with one roll. 01:06:57.530 --> 01:07:01.140 align:middle line:84% But blue beats green with two rolls. 01:07:01.140 --> 01:07:04.320 align:middle line:84% And blue beats red with one roll, but red 01:07:04.320 --> 01:07:07.410 align:middle line:90% beats blue with two rolls. 01:07:07.410 --> 01:07:13.320 align:middle line:84% And yeah, so the dice continue to not have 01:07:13.320 --> 01:07:15.130 align:middle line:90% this transitivity property. 01:07:15.130 --> 01:07:19.554 align:middle line:90% But every time, it switched. 01:07:19.554 --> 01:07:26.290 align:middle line:84% And as it turns out, you can even extend this. 01:07:26.290 --> 01:07:29.440 align:middle line:84% Like if you have more dice with different numbers, 01:07:29.440 --> 01:07:35.070 align:middle line:84% you can get all sorts of weird things that could happen. 01:07:35.070 --> 01:07:38.880 align:middle line:84% So there's a link to a rather interesting paper 01:07:38.880 --> 01:07:43.980 align:middle line:84% on arXiv if you want to see it in the lecture notes. 01:07:43.980 --> 01:07:47.200 align:middle line:84% But yeah, for now, I think that's all we have time for. 01:07:47.200 --> 01:07:49.660 align:middle line:84% Feel free to come up if you have questions. 01:07:49.660 --> 01:07:52.850 align:middle line:84% And otherwise, we will see you on Thursday. 01:07:52.850 --> 01:07:59.000 align:middle line:90%