WEBVTT 00:00:00.790 --> 00:00:03.130 The following content is provided under a Creative 00:00:03.130 --> 00:00:04.550 Commons license. 00:00:04.550 --> 00:00:06.760 Your support will help MIT OpenCourseWare 00:00:06.760 --> 00:00:10.850 continue to offer high quality educational resources for free. 00:00:10.850 --> 00:00:13.390 To make a donation or to view additional materials 00:00:13.390 --> 00:00:17.320 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:17.320 --> 00:00:18.570 at ocw.mit.edu. 00:00:28.762 --> 00:00:31.140 JOHN GUTTAG: So today, we're going 00:00:31.140 --> 00:00:34.560 to move on to a fairly different world than the world 00:00:34.560 --> 00:00:36.090 we've been living in. 00:00:36.090 --> 00:00:37.950 And this will be a world we'll be living in 00:00:37.950 --> 00:00:40.580 for quite a few lectures. 00:00:40.580 --> 00:00:42.990 But before I do that, I want to get back 00:00:42.990 --> 00:00:47.170 to just finish up something that Professor Grimson started. 00:00:47.170 --> 00:00:50.520 You may recall he talked about family trees 00:00:50.520 --> 00:00:52.650 and raised the question, was it actually 00:00:52.650 --> 00:00:55.890 possible to represent all ancestral relationships 00:00:55.890 --> 00:00:57.560 as a tree? 00:00:57.560 --> 00:00:59.890 Well, as a counterexample, I'm sure some of you 00:00:59.890 --> 00:01:03.770 are familiar with Oedipus Rex. 00:01:03.770 --> 00:01:05.269 For those of you who are not, I'm 00:01:05.269 --> 00:01:07.940 happy give you a plot summary at the end of the lecture. 00:01:07.940 --> 00:01:10.880 It's a rather bizarre plot. 00:01:10.880 --> 00:01:16.110 But it was captured in a wonderful song by Tom Lehrer. 00:01:16.110 --> 00:01:19.160 The short story is Oedipus ended up marrying his mother 00:01:19.160 --> 00:01:22.430 and having four children. 00:01:22.430 --> 00:01:25.100 And Tom Lehrer, if you've never heard of Tom Lehrer, 00:01:25.100 --> 00:01:29.660 you're missing one of the world's funniest songwriters. 00:01:29.660 --> 00:01:32.300 And he had a wonderful song called "Oedipus Rex," 00:01:32.300 --> 00:01:38.510 and I recommend this YouTube as a way to go and listen to it. 00:01:38.510 --> 00:01:44.870 And you can gather from the quote what the story is about. 00:01:44.870 --> 00:01:46.970 I also recommend the play, by the way. 00:01:46.970 --> 00:01:50.330 It's really kind of appalling what goes on, 00:01:50.330 --> 00:01:53.090 but it's beautiful. 00:01:53.090 --> 00:01:57.050 Back to the main topic, here's the relevant reading-- 00:01:59.570 --> 00:02:05.250 a small bit from later in the book and then chapter 14. 00:02:05.250 --> 00:02:07.260 You may notice that we're not actually going 00:02:07.260 --> 00:02:09.264 through the book in order. 00:02:09.264 --> 00:02:11.430 And the reason we're not doing that is because we're 00:02:11.430 --> 00:02:13.440 trying to get you information you need in time 00:02:13.440 --> 00:02:14.395 to do problem sets. 00:02:18.810 --> 00:02:24.480 So the topic of today is really uncertainty and the fact 00:02:24.480 --> 00:02:29.550 that the world is really annoyingly hard to understand. 00:02:32.520 --> 00:02:36.480 This is a signpost related to 6.0002, 00:02:36.480 --> 00:02:41.170 but we won't go into too much detail about it. 00:02:41.170 --> 00:02:43.050 We'd rather things were certain. 00:02:43.050 --> 00:02:47.330 But in fact, they usually are not. 00:02:47.330 --> 00:02:51.710 And this is a place where 6.0002 diverges 00:02:51.710 --> 00:02:53.930 from the typical introductory computer science 00:02:53.930 --> 00:02:58.250 course, which focuses on things that are functional-- 00:02:58.250 --> 00:03:02.030 given an input, you always get the same output. 00:03:02.030 --> 00:03:03.950 It's predictable. 00:03:03.950 --> 00:03:07.760 And we like to do that, because that's easier to teach. 00:03:07.760 --> 00:03:11.300 But in fact, for reasons we'll be talking about, 00:03:11.300 --> 00:03:14.210 it's not nearly as useful if you're 00:03:14.210 --> 00:03:16.580 trying to actually write computations that 00:03:16.580 --> 00:03:18.860 help you understand the world. 00:03:18.860 --> 00:03:21.110 You have to face uncertainty head on. 00:03:25.030 --> 00:03:27.860 An analogy is for many years people, believed 00:03:27.860 --> 00:03:31.490 in Newtonian mechanics-- 00:03:31.490 --> 00:03:34.520 I guess they still do in 8.01 maybe-- 00:03:34.520 --> 00:03:38.030 that every effect has a cause. 00:03:38.030 --> 00:03:40.430 An apple falls from the tree because of gravity, 00:03:40.430 --> 00:03:42.390 and you know where it's going to land. 00:03:42.390 --> 00:03:45.080 And the world can be understood causally. 00:03:45.080 --> 00:03:50.090 And people believed this really for quite a long time, 00:03:50.090 --> 00:03:54.500 most of history, until the early part 00:03:54.500 --> 00:03:58.250 of the 20th century, when the so-called Copenhagen 00:03:58.250 --> 00:04:00.770 doctrine was put forth. 00:04:03.880 --> 00:04:06.670 The doctrine there from Bohr and Heisenberg, 00:04:06.670 --> 00:04:09.910 two very famous physicists, was one 00:04:09.910 --> 00:04:13.930 of what they called causal nondeterminism. 00:04:13.930 --> 00:04:17.709 And their assertion was that the world at its very most 00:04:17.709 --> 00:04:24.610 fundamental level behaves in a way that you cannot predict. 00:04:24.610 --> 00:04:28.990 It's OK to make a statement that x is highly likely to occur, 00:04:28.990 --> 00:04:33.430 almost certain to occur, but for no case can 00:04:33.430 --> 00:04:36.310 you make a statement x will occur. 00:04:36.310 --> 00:04:40.360 Nothing has a probability of one. 00:04:40.360 --> 00:04:43.720 This was hard for us to imagine today, when we all 00:04:43.720 --> 00:04:45.580 know quantum mechanics. 00:04:45.580 --> 00:04:50.320 But at the turn of the century, this was a shocking statement. 00:04:50.320 --> 00:04:53.230 And two other very well-known physicists, 00:04:53.230 --> 00:04:55.900 Albert Einstein and Schrodinger, basically 00:04:55.900 --> 00:04:57.460 said, no, this is wrong. 00:04:57.460 --> 00:05:00.130 Bohr, Heisenberg, you guys are idiots. 00:05:00.130 --> 00:05:01.570 It's just not true. 00:05:01.570 --> 00:05:03.670 They probably didn't call them idiots. 00:05:03.670 --> 00:05:06.730 And this is most exemplified by Einstein's famous quote 00:05:06.730 --> 00:05:11.230 that "God does not play dice," which is indicative of the fact 00:05:11.230 --> 00:05:13.990 that this was actually a discussion that permeated 00:05:13.990 --> 00:05:19.570 not just the world of physics, but society in general people 00:05:19.570 --> 00:05:22.150 really turned it into literally a religious issue, 00:05:22.150 --> 00:05:24.900 as did Einstein. 00:05:24.900 --> 00:05:26.940 Well, so now we should ask the question, 00:05:26.940 --> 00:05:28.830 does it really matter? 00:05:28.830 --> 00:05:31.260 And to illustrate that, I need two coins. 00:05:31.260 --> 00:05:33.900 I forgot to bring any coins with me. 00:05:33.900 --> 00:05:35.840 Does anyone got a coin they can lend me? 00:05:35.840 --> 00:05:37.301 AUDIENCE: I have some coins. 00:05:37.301 --> 00:05:39.900 JOHN GUTTAG: All right. 00:05:39.900 --> 00:05:42.300 Now, this is where I see how much the students trust me. 00:05:42.300 --> 00:05:44.190 Do I get a penny? 00:05:44.190 --> 00:05:46.440 Do I get a silver dollar? 00:05:46.440 --> 00:05:47.460 So what do we got here? 00:05:50.500 --> 00:05:54.600 This is someone who's entrusting me with quarters, not so bad. 00:05:57.500 --> 00:06:00.149 So we'll take these quarters, and we'll shake them up, 00:06:00.149 --> 00:06:01.690 and we'll put them down on the table. 00:06:04.240 --> 00:06:07.000 And now, we'll ask a question-- 00:06:07.000 --> 00:06:13.140 do we have two heads, two tails, or one head and one tail? 00:06:13.140 --> 00:06:17.220 So who thinks we have two heads? 00:06:17.220 --> 00:06:20.370 Who thinks we have two tails? 00:06:20.370 --> 00:06:23.230 Who thinks we have one of each? 00:06:23.230 --> 00:06:26.580 Well, clearly, everyone except a few people-- for example, 00:06:26.580 --> 00:06:29.730 the Indians fan, who clearly believe in the counterfactual-- 00:06:33.030 --> 00:06:37.080 made the most probabilistic decision. 00:06:37.080 --> 00:06:40.550 But in fact, there is no nondeterminism here. 00:06:40.550 --> 00:06:43.040 I know the answer. 00:06:43.040 --> 00:06:47.600 And so in some sense, it doesn't matter 00:06:47.600 --> 00:06:49.820 whether it's deterministic, because in fact, it's 00:06:49.820 --> 00:06:52.070 not causally nondeterministic. 00:06:52.070 --> 00:06:58.120 The answer is quite clear, but you don't know the answer. 00:06:58.120 --> 00:07:03.870 And so whether or not the world is inherently unpredictable, 00:07:03.870 --> 00:07:08.760 the fact that we never have complete knowledge of the world 00:07:08.760 --> 00:07:10.770 suggests that we might as well treat 00:07:10.770 --> 00:07:15.130 it as inherently unpredictable. 00:07:15.130 --> 00:07:19.060 And so this is called predictive nondeterminism. 00:07:19.060 --> 00:07:21.365 And this really is what's going to underline 00:07:21.365 --> 00:07:23.740 pretty much everything else we're going to be doing here. 00:07:30.370 --> 00:07:34.000 No comments about that? 00:07:34.000 --> 00:07:37.150 I wouldn't do that to you. 00:07:37.150 --> 00:07:39.700 Thank you. 00:07:39.700 --> 00:07:42.260 I know you are wishing to get interest on the money, 00:07:42.260 --> 00:07:44.140 but you don't get any. 00:07:44.140 --> 00:07:46.060 AUDIENCE: Was it heads or tails? 00:07:51.376 --> 00:07:52.500 JOHN GUTTAG: What was that? 00:07:56.160 --> 00:08:00.660 So when we think about nondeterminism in computation, 00:08:00.660 --> 00:08:04.150 we use the word stochastic process. 00:08:04.150 --> 00:08:07.020 And that's any process that's ongoing 00:08:07.020 --> 00:08:12.180 in which the next state depends upon the previous states 00:08:12.180 --> 00:08:14.800 in some random element. 00:08:14.800 --> 00:08:18.450 So typically up till now when we've written code, 00:08:18.450 --> 00:08:20.890 one line of code did depended only 00:08:20.890 --> 00:08:23.260 on what the previous lines of code did. 00:08:23.260 --> 00:08:25.810 There was no randomness. 00:08:25.810 --> 00:08:28.282 Here, we're going to have randomness. 00:08:28.282 --> 00:08:29.740 And we can see the difference if we 00:08:29.740 --> 00:08:34.450 look at these two specifications of rolling a die. 00:08:34.450 --> 00:08:38.320 The first one, returns an int between 1 and 6, 00:08:38.320 --> 00:08:41.890 is what I'll call underdetermined. 00:08:41.890 --> 00:08:45.940 By that I mean you can't tell what it's going to return. 00:08:45.940 --> 00:08:49.540 Maybe it will return a different number each time you call it, 00:08:49.540 --> 00:08:51.700 but it's not required to. 00:08:51.700 --> 00:08:55.120 Maybe it will return three every time you call it. 00:08:55.120 --> 00:08:58.690 The second specification requires randomness. 00:08:58.690 --> 00:09:01.360 It says, it returns are randomly chosen int. 00:09:01.360 --> 00:09:06.710 So it requires a stochastic implementation. 00:09:06.710 --> 00:09:11.090 Let's look at how we implement a random process in Python. 00:09:11.090 --> 00:09:15.890 We start by importing the library random. 00:09:15.890 --> 00:09:17.520 This is not to say you can import 00:09:17.520 --> 00:09:19.770 any random library you want. 00:09:19.770 --> 00:09:22.530 It's to say you import the library called random. 00:09:22.530 --> 00:09:23.810 Let me get my pen out of here. 00:09:27.310 --> 00:09:29.230 And we'll use that a lot. 00:09:29.230 --> 00:09:32.590 And then we're going to use the function in random called 00:09:32.590 --> 00:09:34.940 random.choice. 00:09:34.940 --> 00:09:39.530 It takes as an argument a sequence, in this case a list, 00:09:39.530 --> 00:09:43.850 and randomly chooses one member of the list. 00:09:43.850 --> 00:09:46.160 And it chooses it uniformly. 00:09:49.010 --> 00:09:52.930 It's a uniform distribution. 00:09:52.930 --> 00:09:56.860 And what that means is that it's equally probable 00:09:56.860 --> 00:09:59.650 that it will choose any number in that list each time 00:09:59.650 --> 00:10:01.690 you call it. 00:10:01.690 --> 00:10:03.820 We'll later look at distributions 00:10:03.820 --> 00:10:06.700 that are not uniform, not equally probable, 00:10:06.700 --> 00:10:08.320 where things are weighted. 00:10:08.320 --> 00:10:10.375 But here, it's quite simple, it's just uniform. 00:10:13.470 --> 00:10:16.980 And then we can test it using testRoll-- 00:10:16.980 --> 00:10:21.930 take some number of n and rolls the die that many times 00:10:21.930 --> 00:10:24.970 and creates a string telling us what we got. 00:10:29.750 --> 00:10:36.300 So let's consider running this on, say, testRoll of five. 00:10:36.300 --> 00:10:38.680 And we'll ask the question, if we run it, 00:10:38.680 --> 00:10:43.180 how probable is it that it's going to return a string 00:10:43.180 --> 00:10:43.960 of five 1's? 00:10:50.100 --> 00:10:51.120 How do we do that? 00:10:51.120 --> 00:10:54.420 Now, how many people here are either in 6.041 00:10:54.420 --> 00:10:56.670 or would have taken 6.041? 00:10:56.670 --> 00:10:59.280 Raise your hand. 00:10:59.280 --> 00:10:59.850 Oh, good. 00:10:59.850 --> 00:11:02.830 So very few of you know probability. 00:11:02.830 --> 00:11:03.330 That helps. 00:11:06.450 --> 00:11:09.170 So how do we think about that question? 00:11:09.170 --> 00:11:14.480 Well, probability, to me at least, is all about counting, 00:11:14.480 --> 00:11:16.740 especially discrete probability, which 00:11:16.740 --> 00:11:19.900 is what we're looking at here. 00:11:19.900 --> 00:11:23.830 What you do is you start by counting the number of events 00:11:23.830 --> 00:11:29.710 that have the property of interest 00:11:29.710 --> 00:11:31.480 and the number of possible events 00:11:31.480 --> 00:11:32.940 and divide one by the other. 00:11:35.580 --> 00:11:41.430 So if we think about rolling a die five times, 00:11:41.430 --> 00:11:44.070 we can enumerate all of the possible outcomes 00:11:44.070 --> 00:11:44.885 of five rolls. 00:11:47.390 --> 00:11:50.870 So if we look at that, what are the outcomes? 00:11:50.870 --> 00:11:54.150 Well, I could get five 1's. 00:11:54.150 --> 00:12:00.720 I could get four 1's and a 2 or four 1's and 3, skip a few. 00:12:00.720 --> 00:12:05.040 The next one would be three 1's, a 2 and a 1, then a 2 and 2, 00:12:05.040 --> 00:12:08.850 and finally, at the end, all 6's. 00:12:08.850 --> 00:12:13.320 So remember, we looked before at when 00:12:13.320 --> 00:12:17.160 we're looking at optimization problems about binary numbers. 00:12:17.160 --> 00:12:20.670 And we said we can look at all the possible choices of items 00:12:20.670 --> 00:12:24.460 in the knapsack by a vector of 0's and 1's. 00:12:24.460 --> 00:12:27.340 We said, how many possible choices are there? 00:12:27.340 --> 00:12:30.200 Well, it depended on how many binary numbers you could 00:12:30.200 --> 00:12:32.910 get in that number of digits. 00:12:32.910 --> 00:12:36.410 Well, here we're doing the same thing, but instead of base 2, 00:12:36.410 --> 00:12:37.710 it's base 6. 00:12:40.590 --> 00:12:45.150 And so the number of possible outcomes of five rolls 00:12:45.150 --> 00:12:45.990 is quite high. 00:12:48.760 --> 00:12:50.860 How many of those are five 1's? 00:12:50.860 --> 00:12:54.180 Only one of them, right? 00:12:54.180 --> 00:12:58.300 So in order to get the probability of a five 1's, I 00:12:58.300 --> 00:13:00.370 divide 1 by 6 to the fifth. 00:13:03.232 --> 00:13:06.720 Does that makes sense to everybody? 00:13:06.720 --> 00:13:10.565 So in fact, we see it's highly unlikely. 00:13:10.565 --> 00:13:15.460 The probability of a five 1's is quite small. 00:13:15.460 --> 00:13:17.770 Now, suppose we were to ask about the probability 00:13:17.770 --> 00:13:19.870 of something else-- 00:13:19.870 --> 00:13:27.120 instead of five 1's, say 53421. 00:13:27.120 --> 00:13:31.230 It kind of looks more likely than that than five 1's 00:13:31.230 --> 00:13:33.630 in a row, but of course, it isn't, right? 00:13:33.630 --> 00:13:37.620 Any specific combination is equally probable. 00:13:37.620 --> 00:13:40.420 And there are a lot of them. 00:13:40.420 --> 00:13:44.920 So this is all the probability we're going to think about we 00:13:44.920 --> 00:13:48.550 could think about this way, as simply a matter of counting-- 00:13:48.550 --> 00:13:51.640 the number of possible events, the number of events that have 00:13:51.640 --> 00:13:54.970 the property of interest-- in this case being all 1's-- 00:13:54.970 --> 00:13:56.680 and then simple division. 00:13:59.530 --> 00:14:03.010 Given that framework, there were three basic facts 00:14:03.010 --> 00:14:07.870 about probability we're going to be using a lot of. 00:14:07.870 --> 00:14:15.980 So one, probabilities always range from 0 to 1. 00:14:15.980 --> 00:14:17.460 How do we know that? 00:14:17.460 --> 00:14:19.930 Well, we've got a fraction, right? 00:14:19.930 --> 00:14:25.190 And the denominator is all possible events. 00:14:25.190 --> 00:14:29.840 The numerator is the subset of that that's of interest. 00:14:29.840 --> 00:14:35.680 So it has to range from 0 to the denominator. 00:14:35.680 --> 00:14:37.330 And that tells us that the fraction 00:14:37.330 --> 00:14:40.250 has to range from 0 to 1. 00:14:40.250 --> 00:14:43.430 So 1 says it's always going to happen, 0 never. 00:14:46.870 --> 00:14:50.290 So if the probability of an event occurring is p, 00:14:50.290 --> 00:14:54.250 what's the probability of it not occurring? 00:14:54.250 --> 00:14:57.060 This follows from the first bullet. 00:14:57.060 --> 00:15:04.050 It's simply going to be 1 minus p. 00:15:04.050 --> 00:15:07.650 This is a trick that we'll find we'll use a lot. 00:15:07.650 --> 00:15:09.660 Because it's often the case when you 00:15:09.660 --> 00:15:13.080 want to compute the probability of something happening, 00:15:13.080 --> 00:15:16.680 it's easier to compute the probability of it not happening 00:15:16.680 --> 00:15:18.980 and subtract it from 1. 00:15:18.980 --> 00:15:21.560 And we'll see an example of that later today. 00:15:24.550 --> 00:15:27.940 Now, here's the biggie. 00:15:27.940 --> 00:15:31.650 When events are independent of each other, 00:15:31.650 --> 00:15:35.000 the probability of all of the events occurring 00:15:35.000 --> 00:15:39.380 is equal to the product of the probabilities of each 00:15:39.380 --> 00:15:40.885 of the events occurring. 00:15:44.280 --> 00:15:53.890 So if the probability of A is 0.5 and the probability of B 00:15:53.890 --> 00:16:01.150 is 0.4, the probability of A and B is what? 00:16:06.110 --> 00:16:07.670 0.5 times 0.4. 00:16:07.670 --> 00:16:10.680 You guys can figure that out. 00:16:10.680 --> 00:16:14.330 I think that's 0.2. 00:16:14.330 --> 00:16:16.100 So you'd expect that, that it should 00:16:16.100 --> 00:16:20.390 be much smaller than either of the first two probabilities. 00:16:20.390 --> 00:16:22.060 This is the most common rule, it's 00:16:22.060 --> 00:16:24.460 something we use all the time in probabilities, 00:16:24.460 --> 00:16:28.360 the so-called multiplicative law. 00:16:28.360 --> 00:16:33.120 We have to be careful about it, however, 00:16:33.120 --> 00:16:37.470 in that it only holds if the events are actually 00:16:37.470 --> 00:16:40.570 independent. 00:16:40.570 --> 00:16:44.920 Two events are independent if the outcome of one 00:16:44.920 --> 00:16:47.110 has no influence on the outcome of the other. 00:16:50.010 --> 00:16:52.370 So when we roll the die, we assume 00:16:52.370 --> 00:16:54.350 that the first roll, the outcome, 00:16:54.350 --> 00:16:55.870 was independent of the-- 00:16:55.870 --> 00:16:58.370 or the second roll was independent of the first roll, 00:16:58.370 --> 00:17:00.910 independent of the fourth roll. 00:17:00.910 --> 00:17:02.560 When we looked at the two coins, we 00:17:02.560 --> 00:17:05.410 assume that heads and tails of each coin 00:17:05.410 --> 00:17:08.460 was independent of the other coin. 00:17:08.460 --> 00:17:10.200 I didn't, for example, look at one coin 00:17:10.200 --> 00:17:12.304 and make sure that the other one was different. 00:17:15.700 --> 00:17:19.079 The danger here is that people often 00:17:19.079 --> 00:17:22.950 compute probabilities assuming independence when you don't 00:17:22.950 --> 00:17:26.099 actually have independence. 00:17:26.099 --> 00:17:29.470 So let's look at an example. 00:17:29.470 --> 00:17:32.980 For those of you familiar with American football, 00:17:32.980 --> 00:17:35.800 the New England Patriots and the Denver Broncos 00:17:35.800 --> 00:17:38.380 are two prominent teams. 00:17:38.380 --> 00:17:40.660 And let's look at computing the probability 00:17:40.660 --> 00:17:45.690 of whether one of them will lose on a given Sunday. 00:17:45.690 --> 00:17:48.840 So the Patriots have a winning percentage of 7 of 8-- 00:17:48.840 --> 00:17:51.420 they've won 7 of their 8 games so far-- 00:17:51.420 --> 00:17:54.590 and the Broncos 6 of 8. 00:17:54.590 --> 00:17:57.560 The probability of both winning next Sunday, 00:17:57.560 --> 00:18:00.860 assuming that this is indicative of how good they are, 00:18:00.860 --> 00:18:03.470 we can get with the multiplicative rule. 00:18:03.470 --> 00:18:08.750 So it's 7/8 times 6/8, or 42/64. 00:18:08.750 --> 00:18:12.060 We could simplify that fraction, I suppose. 00:18:12.060 --> 00:18:14.370 Does that makes sense? 00:18:14.370 --> 00:18:17.840 So this is probably a pretty good estimate of both of them 00:18:17.840 --> 00:18:20.600 winning next Sunday. 00:18:20.600 --> 00:18:24.380 So the probability of at least one of them losing 00:18:24.380 --> 00:18:27.740 is 1 minus that. 00:18:27.740 --> 00:18:30.430 So here's an example of why we often use 00:18:30.430 --> 00:18:34.120 the 1 minus rule, because we could 00:18:34.120 --> 00:18:38.020 compute the probability of both of them 00:18:38.020 --> 00:18:41.440 winning by simply multiplying. 00:18:41.440 --> 00:18:44.130 And we subtract that from 1. 00:18:44.130 --> 00:18:47.220 However, what about Sunday, December 18? 00:18:47.220 --> 00:18:50.440 What's the probability? 00:18:50.440 --> 00:18:53.920 Well, as it happens, that day the Patriots 00:18:53.920 --> 00:18:55.025 are playing the Broncos. 00:18:58.380 --> 00:19:02.230 So now suddenly, the outcomes are not independent. 00:19:02.230 --> 00:19:05.550 The probability of one of them losing 00:19:05.550 --> 00:19:10.470 is influenced by the probability of the other winning. 00:19:10.470 --> 00:19:13.540 So you would expect the probability of one 00:19:13.540 --> 00:19:17.989 of them losing is much closer to 1 than 22/64, 00:19:17.989 --> 00:19:18.780 which is about 1/3. 00:19:21.780 --> 00:19:25.490 So in this case, it's easy. 00:19:25.490 --> 00:19:28.430 But as we'll see, as we get through the term, 00:19:28.430 --> 00:19:30.560 there are lots of cases where you 00:19:30.560 --> 00:19:33.950 have to work pretty hard to understand whether or not two 00:19:33.950 --> 00:19:36.350 events really are independent. 00:19:36.350 --> 00:19:40.410 And if you get it wrong, you get a totally bogus answer. 00:19:40.410 --> 00:19:45.530 1/3 versus 1 is a pretty big difference. 00:19:45.530 --> 00:19:49.010 By the way, as it happens, the probability of the Broncos 00:19:49.010 --> 00:19:50.070 losing is about 1. 00:19:56.190 --> 00:19:58.400 Let's go look at some code. 00:20:01.040 --> 00:20:03.260 And we'll go back to our dice, because it's 00:20:03.260 --> 00:20:05.420 much easier to simulate dice games 00:20:05.420 --> 00:20:08.300 than it is to simulate football games. 00:20:11.510 --> 00:20:13.340 So here it is. 00:20:13.340 --> 00:20:17.030 And we're going to talk a lot about simulations. 00:20:17.030 --> 00:20:18.980 So here, rather than rolling the die, 00:20:18.980 --> 00:20:20.480 I've written a program to do it. 00:20:23.980 --> 00:20:27.660 We've already seen the code for rolling a die. 00:20:27.660 --> 00:20:32.990 And so to run this simulation, typically what we're doing here 00:20:32.990 --> 00:20:35.480 is I'm giving you the goal-- 00:20:35.480 --> 00:20:38.510 for example, are we going to get five 1's-- 00:20:38.510 --> 00:20:41.800 the number of trials-- 00:20:41.800 --> 00:20:47.060 each trial, in this case, will be say of length 5-- 00:20:47.060 --> 00:20:48.770 so I'm going to roll the same die 00:20:48.770 --> 00:20:55.130 five times say 1,000 different times, and then just some text 00:20:55.130 --> 00:20:57.910 as to what I'm going to print. 00:20:57.910 --> 00:21:01.090 Almost all the simulations we look at 00:21:01.090 --> 00:21:05.630 are going to start with lines that look a lot like that. 00:21:05.630 --> 00:21:08.650 We're going to initialize some variable. 00:21:08.650 --> 00:21:11.755 And then we're going to run some number of trials. 00:21:16.160 --> 00:21:19.860 So in this case, we're going to get 00:21:19.860 --> 00:21:21.340 from the length of the goal-- 00:21:21.340 --> 00:21:23.790 so if the goal is five 1's, then we're 00:21:23.790 --> 00:21:26.490 going to roll the dice five times; if it's 10 runs, 00:21:26.490 --> 00:21:29.830 we'll roll it 10 times. 00:21:29.830 --> 00:21:35.310 So this is essentially one trial, one attempt. 00:21:38.850 --> 00:21:41.850 And then we'll check the result. And if it 00:21:41.850 --> 00:21:43.720 has the property we want-- 00:21:43.720 --> 00:21:47.460 in this case, it's equal to the goal-- 00:21:47.460 --> 00:21:50.040 then we're going to increment the total, which 00:21:50.040 --> 00:21:54.380 we initialized up here by 1. 00:21:54.380 --> 00:21:57.170 So we'll keep track with just the counting-- 00:21:57.170 --> 00:22:01.610 the number of trials that actually meet the goal. 00:22:01.610 --> 00:22:04.990 And then when we're done, what we're going to do 00:22:04.990 --> 00:22:08.560 is divide the number that met the goal 00:22:08.560 --> 00:22:10.870 by the number of trials-- 00:22:10.870 --> 00:22:14.170 exactly the counting argument we just looked at. 00:22:14.170 --> 00:22:19.700 And then we'll print the result. 00:22:19.700 --> 00:22:22.220 Almost every simulation we look at 00:22:22.220 --> 00:22:24.360 is going to have this structure. 00:22:24.360 --> 00:22:27.680 There'll be an outer loop, which is the number of trials. 00:22:27.680 --> 00:22:29.870 And then inside-- maybe it'll have a loop, 00:22:29.870 --> 00:22:32.600 or maybe it won't-- will be a single trial. 00:22:32.600 --> 00:22:33.770 We'll sum up the results. 00:22:33.770 --> 00:22:36.920 And then we'll divide by the number of trials. 00:22:36.920 --> 00:22:37.490 Let's run it. 00:22:45.300 --> 00:22:49.650 So a couple of things are going to go on here. 00:22:49.650 --> 00:22:59.570 If you look at the code as we've looked at it before, 00:22:59.570 --> 00:23:02.780 what you're seeing is I'm computing the estimated 00:23:02.780 --> 00:23:05.180 probability by the simulation. 00:23:05.180 --> 00:23:08.270 And I'm comparing it to the actual probability, which we've 00:23:08.270 --> 00:23:09.590 already seen how to compute. 00:23:12.117 --> 00:23:14.700 So if you look at it, there are a couple of things to look at. 00:23:17.370 --> 00:23:19.260 The estimated probability is pretty 00:23:19.260 --> 00:23:24.704 close to the actual probability but not the same. 00:23:24.704 --> 00:23:26.590 So let's go back to the PowerPoint. 00:23:31.860 --> 00:23:34.240 Here are the results. 00:23:34.240 --> 00:23:37.680 And there are at least two questions raised 00:23:37.680 --> 00:23:40.050 by this result. First of all, how 00:23:40.050 --> 00:23:43.290 did I know that this is what would get printed? 00:23:43.290 --> 00:23:45.610 Remember, this is random. 00:23:45.610 --> 00:23:48.520 How did I know that the estimate-- well, there's 00:23:48.520 --> 00:23:51.790 nothing random about the actual probability. 00:23:51.790 --> 00:23:55.390 But how did I know that the estimated probability 00:23:55.390 --> 00:23:57.180 would be 0? 00:23:57.180 --> 00:23:58.470 And why did it print it twice? 00:23:58.470 --> 00:24:00.330 Because I messed up the PowerPoint. 00:24:00.330 --> 00:24:04.140 Any rate, so how do I know what would get printed? 00:24:04.140 --> 00:24:12.610 Well a confession-- random.choice 00:24:12.610 --> 00:24:14.920 is not actually random. 00:24:14.920 --> 00:24:20.140 In fact, nothing we can do in a computer is actually random. 00:24:20.140 --> 00:24:23.650 You can prove that it's impossible to build 00:24:23.650 --> 00:24:28.950 a computer that actually generates truly random numbers. 00:24:28.950 --> 00:24:32.520 What they do instead is generate numbers 00:24:32.520 --> 00:24:34.050 that called pseudorandom. 00:24:42.120 --> 00:24:44.740 How do they do that? 00:24:44.740 --> 00:24:48.930 They have an algorithm that given one number generates 00:24:48.930 --> 00:24:52.700 the next number in a sequence. 00:24:52.700 --> 00:24:56.375 And they start that algorithm with a seed. 00:25:00.050 --> 00:25:02.630 Now, typically, they get that seed 00:25:02.630 --> 00:25:05.930 by reading the clock of the computer. 00:25:05.930 --> 00:25:08.090 So most computers have a clock that, say, 00:25:08.090 --> 00:25:12.080 keeps track of the number of microseconds since January 1, 00:25:12.080 --> 00:25:14.174 1978. 00:25:14.174 --> 00:25:15.590 I don't know if that's still true. 00:25:15.590 --> 00:25:18.590 That's what Unix used to do. 00:25:18.590 --> 00:25:22.070 So the notion is, you start your program, 00:25:22.070 --> 00:25:26.420 there's no way of knowing how many microseconds have elapsed. 00:25:26.420 --> 00:25:29.395 And so you're getting a random number to start the process. 00:25:32.040 --> 00:25:33.660 Since you don't know where it starts, 00:25:33.660 --> 00:25:34.800 you don't know what the second number 00:25:34.800 --> 00:25:37.050 is, you don't know what the third number is, you don't 00:25:37.050 --> 00:25:38.580 know what the fourth number is. 00:25:38.580 --> 00:25:42.570 And so it's predictably nondeterministic, 00:25:42.570 --> 00:25:46.600 because you don't know what the seed is going to be. 00:25:46.600 --> 00:25:49.180 Now, you can imagine that this makes 00:25:49.180 --> 00:25:52.460 programs really hard to debug. 00:25:52.460 --> 00:25:55.850 Every time you run it, something different could happen. 00:25:55.850 --> 00:25:59.220 Now, we'll see often you want them to be unpredictable. 00:25:59.220 --> 00:26:02.300 But for now, we want them to be predictable, makes it easier 00:26:02.300 --> 00:26:04.130 prepare PowerPoint. 00:26:04.130 --> 00:26:08.635 So what you have is a command. 00:26:13.040 --> 00:26:19.190 You can call random.seed and give it a value 00:26:19.190 --> 00:26:21.800 and say, I don't want you to just choose some random seed, 00:26:21.800 --> 00:26:24.890 I want you to use 0 as the seed. 00:26:24.890 --> 00:26:27.530 For the same seed, you always get the same sequence 00:26:27.530 --> 00:26:30.120 of random values. 00:26:30.120 --> 00:26:33.410 And so what I've done is I set the seed to be, I think, 0 00:26:33.410 --> 00:26:36.620 in this case, not because there's anything magic about 0, 00:26:36.620 --> 00:26:38.780 it's just sort of habit. 00:26:38.780 --> 00:26:41.540 But it made it predictable. 00:26:41.540 --> 00:26:43.640 As you write programs with randomness 00:26:43.640 --> 00:26:45.980 in and when you're debugging it, you will almost surely 00:26:45.980 --> 00:26:49.550 want to start by setting random.seed to a value 00:26:49.550 --> 00:26:51.590 so you get the same answer. 00:26:51.590 --> 00:26:54.950 But make sure you debug it with more than one value of this, 00:26:54.950 --> 00:26:58.320 so you didn't just get lucky with your seed. 00:26:58.320 --> 00:27:01.460 So that's how I knew what would get printed. 00:27:01.460 --> 00:27:06.480 The next question is, why did the simulation 00:27:06.480 --> 00:27:09.670 give me the wrong answer? 00:27:09.670 --> 00:27:14.530 The actual probability is three 0's and 1286. 00:27:14.530 --> 00:27:16.630 But it's estimated a probability of 0. 00:27:19.150 --> 00:27:20.140 Why is it wrong? 00:27:24.200 --> 00:27:27.100 Well, let's think about this. 00:27:27.100 --> 00:27:30.020 I ran 1,000 trials. 00:27:30.020 --> 00:27:32.430 What does it mean to say the probability is zero? 00:27:32.430 --> 00:27:36.670 It means that I tried it 1,000 times and didn't ever get 00:27:36.670 --> 00:27:39.380 a sequence of five 1's. 00:27:39.380 --> 00:27:44.500 So the numerator of the division at the bottom was 0. 00:27:44.500 --> 00:27:46.150 Hence, the answer is 0. 00:27:46.150 --> 00:27:47.890 Is this surprising? 00:27:47.890 --> 00:27:49.440 Well, no. 00:27:49.440 --> 00:27:54.200 Because if that's the actual probability of getting five 00:27:54.200 --> 00:27:58.075 1's, it's not very shocking that in 1,000 trials 00:27:58.075 --> 00:27:58.825 it never happened. 00:28:02.260 --> 00:28:06.140 It's not a surprising result. And so we 00:28:06.140 --> 00:28:09.230 have to be careful when we run these things to understand 00:28:09.230 --> 00:28:14.250 the difference between what's in this case an actual probability 00:28:14.250 --> 00:28:17.510 and what statisticians call a sample probability. 00:28:25.530 --> 00:28:28.970 So what we got with the sample was 0. 00:28:28.970 --> 00:28:32.740 So what's the obvious thing to do? 00:28:32.740 --> 00:28:35.590 If you're doing a simulation of an event 00:28:35.590 --> 00:28:39.020 and the event is pretty rare, you 00:28:39.020 --> 00:28:43.520 want to try it on a very large number of trials. 00:28:43.520 --> 00:28:45.050 So let's go back to our code. 00:28:51.350 --> 00:28:58.720 And we'll change it to instead of 1,000, 1,000,000. 00:28:58.720 --> 00:29:01.572 You can see up here, by the way, where I set the seed. 00:29:01.572 --> 00:29:02.840 And now, let's run it. 00:29:17.760 --> 00:29:19.650 We did a lot better. 00:29:19.650 --> 00:29:22.470 If we look at here our estimated probability, 00:29:22.470 --> 00:29:25.980 it's three 0's 128, still not quite 00:29:25.980 --> 00:29:30.142 the actual probability but darn close. 00:29:30.142 --> 00:29:31.600 And maybe if I had done 10 million, 00:29:31.600 --> 00:29:32.891 it would have been even closer. 00:29:35.610 --> 00:29:38.040 So if you're writing a simulation 00:29:38.040 --> 00:29:41.130 to compute the probability of an event 00:29:41.130 --> 00:29:44.040 and the event is moderately rare, 00:29:44.040 --> 00:29:47.310 then you better run a lot of trials 00:29:47.310 --> 00:29:51.750 before you believe your estimated probability. 00:29:51.750 --> 00:29:55.440 In a week or so, we'll actually look at that more 00:29:55.440 --> 00:29:57.810 mathematically and say, what is a lot, 00:29:57.810 --> 00:29:59.130 how do we know what is enough. 00:30:12.110 --> 00:30:13.550 What are the morals here? 00:30:13.550 --> 00:30:15.430 Moral one, I've just told you-- 00:30:15.430 --> 00:30:18.950 takes a lot of trials to get a good estimate of the frequency 00:30:18.950 --> 00:30:21.510 of a rare event. 00:30:21.510 --> 00:30:26.470 Moral two, we should always, if we're getting an estimated 00:30:26.470 --> 00:30:29.290 probability, know that, and probably 00:30:29.290 --> 00:30:33.570 say that, and not confuse it with the actual probability. 00:30:33.570 --> 00:30:36.400 The third moral here is, it was kind of 00:30:36.400 --> 00:30:38.830 stupid to do a simulation. 00:30:38.830 --> 00:30:42.430 Since it was a very simple closed-form answer 00:30:42.430 --> 00:30:45.550 that we could compute that would really tell us 00:30:45.550 --> 00:30:48.220 what the actual probability is, why even 00:30:48.220 --> 00:30:51.550 bother with the simulation? 00:30:51.550 --> 00:30:53.880 Well, we're going to see why now, 00:30:53.880 --> 00:30:57.340 because simulations can be very useful. 00:30:57.340 --> 00:31:00.390 Let's look at another problem. 00:31:00.390 --> 00:31:02.070 This is the famous birthday problem. 00:31:02.070 --> 00:31:03.660 Some of you have seen it. 00:31:03.660 --> 00:31:06.240 What's the probability of at least two people in a group 00:31:06.240 --> 00:31:08.770 having the same birthday? 00:31:08.770 --> 00:31:10.600 There's a URL at the bottom. 00:31:10.600 --> 00:31:12.760 That's pointing to a Google form. 00:31:12.760 --> 00:31:15.940 I'd like please all of you who have a computing device 00:31:15.940 --> 00:31:20.100 to go to it and fill out your birthday. 00:31:20.100 --> 00:31:22.942 It's anonymous, so we won't know how old you are, don't worry. 00:31:22.942 --> 00:31:24.150 Actually, it's only the date. 00:31:24.150 --> 00:31:25.290 It's not the year. 00:31:27.880 --> 00:31:33.870 So suppose there were 367 people in the group, roughly 00:31:33.870 --> 00:31:40.680 the number of people who took the 6.0001 600 midterm. 00:31:40.680 --> 00:31:44.070 If they are 367 people, what's the probability of at least two 00:31:44.070 --> 00:31:45.230 of them sharing a birthday? 00:31:49.790 --> 00:31:54.110 One, by something called the pigeonhole principle. 00:31:54.110 --> 00:31:56.000 You got some number of holes. 00:31:56.000 --> 00:31:57.800 And if you have more pigeons than holes, 00:31:57.800 --> 00:32:01.430 two pigeons have to share a whole. 00:32:01.430 --> 00:32:04.040 What about smaller numbers? 00:32:04.040 --> 00:32:07.430 Well, if we make a simplifying assumption 00:32:07.430 --> 00:32:10.650 that each birthdate is equally likely, 00:32:10.650 --> 00:32:13.970 then there's actually a nice closed-form solution for it. 00:32:17.760 --> 00:32:20.730 Again, this is a question where it's easier 00:32:20.730 --> 00:32:24.210 to compute the opposite of what you're trying 00:32:24.210 --> 00:32:26.670 to do and subtract it from 1. 00:32:26.670 --> 00:32:32.160 And so this fraction is giving the probability of two people 00:32:32.160 --> 00:32:35.190 not sharing a birthday. 00:32:35.190 --> 00:32:38.560 The proof that this is right, it's a little bit elaborate. 00:32:38.560 --> 00:32:42.450 But you can trust me, it's accurate. 00:32:42.450 --> 00:32:46.150 But it's a formula, and it's not that complicated a formula. 00:32:46.150 --> 00:32:49.800 So numbers like 366 factorial are big. 00:32:55.240 --> 00:32:57.460 So let's approximate a solution. 00:32:57.460 --> 00:33:00.940 We'll right a simulation and see if we get the same answer 00:33:00.940 --> 00:33:03.920 that that formula gave us. 00:33:03.920 --> 00:33:05.200 So here's the code for that-- 00:33:07.810 --> 00:33:09.550 two arguments-- the number of people 00:33:09.550 --> 00:33:14.780 in the group and the number that we asking do 00:33:14.780 --> 00:33:17.520 they have the same birthday. 00:33:17.520 --> 00:33:21.120 So since I'm assuming for now that every birthday is equally 00:33:21.120 --> 00:33:26.100 likely, the possible dates range from 1 to 366, 00:33:26.100 --> 00:33:28.005 because some years have a February 29. 00:33:31.200 --> 00:33:35.490 I'll keep track of the number of people born in each date 00:33:35.490 --> 00:33:38.640 by starting with none. 00:33:38.640 --> 00:33:41.470 And then for p in the range of number of people, 00:33:41.470 --> 00:33:45.240 I'll make a random choice of the possible dates 00:33:45.240 --> 00:33:49.999 and increment that element of the list by 1. 00:33:49.999 --> 00:33:51.540 And then at the end, we can say, look 00:33:51.540 --> 00:33:54.330 at the maximum number of birthdays 00:33:54.330 --> 00:33:59.560 and see if it's greater than or equal to the number of same. 00:33:59.560 --> 00:34:01.240 So that tells us that. 00:34:04.490 --> 00:34:07.220 And then we can actually look at the birthday problem-- 00:34:07.220 --> 00:34:09.640 number of people, the number of same, and, as usual, 00:34:09.640 --> 00:34:10.514 the number of trials. 00:34:13.750 --> 00:34:17.840 So the number of hits is 0 for t in range number of trials. 00:34:17.840 --> 00:34:21.940 If sameDate is true, then we'll increment the number 00:34:21.940 --> 00:34:28.590 of hits by 1 and then as usual divide by the number of trials. 00:34:28.590 --> 00:34:34.739 And we'll try it for 10, 20, 40, and 100 people. 00:34:37.310 --> 00:34:41.480 And then just, we'll print the estimated probability 00:34:41.480 --> 00:34:46.429 and the actual probability computed using 00:34:46.429 --> 00:34:48.320 that formula I showed you. 00:34:48.320 --> 00:34:50.600 I have not shown you, but I've imported 00:34:50.600 --> 00:34:53.480 a library called math, because it 00:34:53.480 --> 00:34:55.040 is a factorial implementation. 00:34:55.040 --> 00:34:56.900 It's way faster than the recursive one 00:34:56.900 --> 00:35:00.270 that we've seen before. 00:35:00.270 --> 00:35:00.880 Let's run it. 00:35:23.920 --> 00:35:25.040 And we'll see what we get. 00:35:25.040 --> 00:35:30.580 So for 10, the estimated probability is 0.11 now. 00:35:30.580 --> 00:35:36.720 So you can see, the estimates are really pretty good. 00:35:36.720 --> 00:35:39.450 Once again, we have this business that for 100, 00:35:39.450 --> 00:35:43.450 we're estimating 1, when the real answer is point many, 00:35:43.450 --> 00:35:45.150 many 9's. 00:35:45.150 --> 00:35:47.400 But again, this is sample probability. 00:35:47.400 --> 00:35:53.250 It just means in the number of trials we did, every 1 00:35:53.250 --> 00:35:56.190 for 100 people, there was a shared birthday. 00:35:56.190 --> 00:35:59.010 This is a number that usually surprises people, 00:35:59.010 --> 00:36:03.690 as to why with 100 people the probability is so high. 00:36:03.690 --> 00:36:06.990 But we could work out the formula and see it. 00:36:06.990 --> 00:36:08.460 And as you can see, the estimates 00:36:08.460 --> 00:36:10.930 are pretty good from my simulation. 00:36:20.252 --> 00:36:22.210 Now, we're going to see why we did a simulation 00:36:22.210 --> 00:36:23.720 in the first place. 00:36:23.720 --> 00:36:27.970 Suppose we want the probability of three people sharing 00:36:27.970 --> 00:36:29.260 a birthday instead of two. 00:36:34.030 --> 00:36:37.240 It's pretty easy to see how we changed the simulation. 00:36:37.240 --> 00:36:38.980 I even made a parameter. 00:36:38.980 --> 00:36:42.190 I just changed the number 2 to number 3. 00:36:42.190 --> 00:36:45.200 The math, on the other hand, is ugly. 00:36:48.030 --> 00:36:52.190 Why is the math so much uglier for 3 than for 2? 00:36:52.190 --> 00:36:55.400 Because for 2, the complementary problem-- 00:36:55.400 --> 00:36:58.040 the number we're subtracting from 1-- 00:36:58.040 --> 00:37:03.640 is simply the question of, are all birthdays different? 00:37:03.640 --> 00:37:08.170 So did two people share a birthday is 1 minus or all 00:37:08.170 --> 00:37:11.570 does everybody have a different birthday. 00:37:11.570 --> 00:37:16.250 On the other hand, for 3 people, the complementary problem is 00:37:16.250 --> 00:37:19.490 a complicated disjunct-- a bunch of ors-- 00:37:19.490 --> 00:37:22.190 either all birthdays are distinct, 00:37:22.190 --> 00:37:26.240 or two people share a birthday and the rest are distinct, 00:37:26.240 --> 00:37:30.140 or there are two groups of two people sharing a birthday 00:37:30.140 --> 00:37:31.970 and everything is distinct. 00:37:31.970 --> 00:37:36.450 So you can see here, there's a lot of possibilities. 00:37:36.450 --> 00:37:40.800 And so it's 1 minus now a very complicated formula. 00:37:40.800 --> 00:37:42.840 And in fact, if you try and look how to do this, 00:37:42.840 --> 00:37:45.450 most people will tell you don't bother. 00:37:45.450 --> 00:37:48.490 Here's kind of a good approximation. 00:37:48.490 --> 00:37:50.320 But the math gets very hairy. 00:37:53.040 --> 00:37:57.160 In contrast, changing the simulation is dead easy. 00:37:57.160 --> 00:37:57.880 We can do that. 00:38:03.808 --> 00:38:06.280 Whoops. 00:38:06.280 --> 00:38:13.650 So if we come over here for the code, all I have to do 00:38:13.650 --> 00:38:15.075 is change this to 2 or 3. 00:38:25.090 --> 00:38:27.190 And I'm going to leave in this code, which 00:38:27.190 --> 00:38:31.180 is the wrong code, computing the actual probability now 00:38:31.180 --> 00:38:35.110 for 2 people sharing rather than 3, because I want 00:38:35.110 --> 00:38:37.660 to make it easy for you to see the difference between what 00:38:37.660 --> 00:38:41.260 happens when we look at 3 shared rather than 2 shared. 00:38:53.140 --> 00:38:55.980 And I get invalid syntax. 00:38:55.980 --> 00:38:58.766 That's not good. 00:38:58.766 --> 00:39:00.640 That's what happens when I type in real time. 00:39:07.970 --> 00:39:10.010 Why do I have invalid syntax? 00:39:10.010 --> 00:39:11.337 AUDIENCE: Line 56. 00:39:11.337 --> 00:39:12.170 JOHN GUTTAG: Pardon. 00:39:12.170 --> 00:39:13.631 AUDIENCE: Line 56. 00:39:13.631 --> 00:39:15.170 JOHN GUTTAG: One person, Anna. 00:39:15.170 --> 00:39:17.660 AUDIENCE: Line 56, there's a comma. 00:39:17.660 --> 00:39:20.532 JOHN GUTTAG: Oh. 00:39:20.532 --> 00:39:21.490 That's not a good line. 00:39:32.960 --> 00:39:40.410 So now, we see that if we get, say, to n equals 100, for 2, 00:39:40.410 --> 00:39:42.530 you'll remember, it was 0.99. 00:39:42.530 --> 00:39:46.000 But for 3, it's only 0.63. 00:39:46.000 --> 00:39:49.590 So we see going from two sharing to three sharing 00:39:49.590 --> 00:39:54.930 gets us a radically different answer, not surprisingly. 00:39:54.930 --> 00:39:57.240 But we also-- and the real thing I wanted you to see-- 00:39:57.240 --> 00:39:59.310 is how easy it was to answer this question 00:39:59.310 --> 00:40:01.810 with the simulation. 00:40:01.810 --> 00:40:05.940 And that's a primary reason we use simulations 00:40:05.940 --> 00:40:09.000 to get probabilistic questions rather 00:40:09.000 --> 00:40:11.190 than sitting down and the pencil and paper 00:40:11.190 --> 00:40:14.460 and doing fancy probability calculations, 00:40:14.460 --> 00:40:19.300 because it's often way easier to do a simulation. 00:40:19.300 --> 00:40:22.220 We can see that in spades if we look at the next question. 00:40:26.680 --> 00:40:28.210 Let's think about this assumption 00:40:28.210 --> 00:40:31.270 that all birthdays are equally likely. 00:40:31.270 --> 00:40:33.370 Well, as you can see, this is a chart 00:40:33.370 --> 00:40:38.440 of how common birthdates are in the US, a heat map. 00:40:38.440 --> 00:40:44.820 And you'll see, for example, that February 29 00:40:44.820 --> 00:40:47.930 is quite an uncommon birthday. 00:40:47.930 --> 00:40:52.010 So we should probably treat that differently. 00:40:52.010 --> 00:40:53.480 Somewhat surprisingly, you'll see 00:40:53.480 --> 00:40:57.160 that July 4 is a very uncommon birthday as well. 00:40:57.160 --> 00:41:00.410 It's easy to understand why February 29. 00:41:00.410 --> 00:41:02.570 The only thing I can figure out for July 4 00:41:02.570 --> 00:41:06.230 is obstetricians don't like working on holidays. 00:41:06.230 --> 00:41:08.300 And so they induce labor sometime 00:41:08.300 --> 00:41:10.790 around the 2nd or the 3rd, so they 00:41:10.790 --> 00:41:14.420 don't have to come to work on the 4th or the 5th. 00:41:14.420 --> 00:41:15.680 Sounds a horrible thought. 00:41:15.680 --> 00:41:19.952 But I can't think of any other explanation for this anomaly. 00:41:19.952 --> 00:41:21.410 You'll probably, if you look at it, 00:41:21.410 --> 00:41:25.580 see Christmas day is not so common either. 00:41:25.580 --> 00:41:27.170 So now, the question, which we can 00:41:27.170 --> 00:41:29.120 answer, since you've all fill out this form, 00:41:29.120 --> 00:41:32.810 is how exceptional are MIT students? 00:41:32.810 --> 00:41:35.930 We like to think that you're different in every respect. 00:41:35.930 --> 00:41:38.960 So are your birthdays distributed differently 00:41:38.960 --> 00:41:40.830 than other dates? 00:41:40.830 --> 00:41:43.220 Have we got that data? 00:41:43.220 --> 00:41:44.890 So now we'll go look at that. 00:41:49.180 --> 00:41:50.920 We should have a heat map for you guys. 00:41:53.900 --> 00:41:54.400 This one? 00:41:54.400 --> 00:41:56.850 AUDIENCE: Yep. 00:41:56.850 --> 00:41:59.300 I removed all the February 31. 00:41:59.300 --> 00:42:02.240 Thank you for those submissions. 00:42:02.240 --> 00:42:05.910 [LAUGHTER] 00:42:06.525 --> 00:42:08.750 JOHN GUTTAG: So here it is. 00:42:08.750 --> 00:42:13.310 And we can see that, well, they don't 00:42:13.310 --> 00:42:17.790 seem to be banded quite as much in the summer months, 00:42:17.790 --> 00:42:20.370 probably says more about your parents than it does about you. 00:42:23.030 --> 00:42:26.090 But you can see that, indeed, we do have-- 00:42:26.090 --> 00:42:28.280 wow, we have a day where there are 00:42:28.280 --> 00:42:30.110 five birthdays, that look like? 00:42:30.110 --> 00:42:30.620 Or no? 00:42:30.620 --> 00:42:32.556 AUDIENCE: February 12. 00:42:32.556 --> 00:42:33.355 JOHN GUTTAG: Wow. 00:42:36.654 --> 00:42:39.070 You want to raise your hand if you're born on February 12? 00:42:42.388 --> 00:42:45.800 [LAUGHTER] 00:42:46.670 --> 00:42:51.902 So you are exceptional in that you lie about when you're born. 00:42:51.902 --> 00:42:57.470 But if you hadn't lied, I think we would have still seen 00:42:57.470 --> 00:42:59.450 the probabilities would hold. 00:42:59.450 --> 00:43:03.155 How many people were there, do we know? 00:43:03.155 --> 00:43:07.865 AUDIENCE: 146 with 112 unique birthdays. 00:43:07.865 --> 00:43:12.190 JOHN GUTTAG: 146 people, 112 unique birthdays. 00:43:12.190 --> 00:43:16.220 So indeed, the probability does work. 00:43:26.470 --> 00:43:28.990 So we know you're exceptional in a funny way. 00:43:28.990 --> 00:43:32.240 Well, you can imagine how hard it 00:43:32.240 --> 00:43:36.080 would be to adjust the analytic model to account 00:43:36.080 --> 00:43:40.370 for a weird distribution of birthdates. 00:43:40.370 --> 00:43:44.900 But again, adjusting the simulation model is easy. 00:43:44.900 --> 00:43:46.700 I could have gone back to that heat 00:43:46.700 --> 00:43:49.670 map I showed you of birthdays in the US 00:43:49.670 --> 00:43:52.550 and gotten a separate probability for each day, 00:43:52.550 --> 00:43:55.130 but I was too lazy. 00:43:55.130 --> 00:44:01.220 And instead, what I observed was that we had a few days, 00:44:01.220 --> 00:44:06.950 like February 29, highly unlikely, and this band 00:44:06.950 --> 00:44:10.040 in the middle of people who were conceived 00:44:10.040 --> 00:44:13.670 in the late fall and early winter. 00:44:13.670 --> 00:44:19.950 So what I did is I duplicated some dates. 00:44:19.950 --> 00:44:25.565 So the 58th day of the year, February 29, occurs only once. 00:44:28.750 --> 00:44:30.590 The dates before that, I said, let's 00:44:30.590 --> 00:44:32.730 pretend they occur four times. 00:44:32.730 --> 00:44:34.590 What only matters here is not how often 00:44:34.590 --> 00:44:36.450 they occur but the relative frequency. 00:44:40.700 --> 00:44:46.000 And then the dates after that occur four times 00:44:46.000 --> 00:44:49.480 except for the dates in that band, which is going 00:44:49.480 --> 00:44:52.180 to have occur yet more often. 00:44:52.180 --> 00:44:56.000 So now-- and don't worry about the exact details here-- 00:44:56.000 --> 00:44:58.840 but what I'm doing is simply adjusting the simulation 00:44:58.840 --> 00:45:02.140 to change the probability of each date getting 00:45:02.140 --> 00:45:04.378 chosen by same date. 00:45:07.190 --> 00:45:09.170 And then I can run the simulation model. 00:45:09.170 --> 00:45:13.450 And, again, with a very small change to code, 00:45:13.450 --> 00:45:15.900 I've modeled something that's mathematically 00:45:15.900 --> 00:45:18.360 enormously complex. 00:45:18.360 --> 00:45:22.050 I have no idea how to actually do this probability 00:45:22.050 --> 00:45:23.670 mathematically. 00:45:23.670 --> 00:45:27.046 But the code is, as you can see, quite straightforward. 00:45:33.850 --> 00:45:35.460 So let's go to that here. 00:45:39.090 --> 00:45:45.450 So what I'm going to do is comment this one out 00:45:45.450 --> 00:46:02.660 and uncomment this more complicated set of dates 00:46:02.660 --> 00:46:03.500 and see what we get. 00:46:14.020 --> 00:46:16.240 And again, it changes quite dramatically. 00:46:16.240 --> 00:46:18.240 You might remember, before it was around I think 00:46:18.240 --> 00:46:23.460 0.6-something for 100, and now, it's 0.75. 00:46:23.460 --> 00:46:26.460 So getting away from the notion that birthdays are uniformly 00:46:26.460 --> 00:46:28.710 distributed to saying some birthdays are 00:46:28.710 --> 00:46:32.010 more common than others, again, dramatically changes 00:46:32.010 --> 00:46:34.570 the answer. 00:46:34.570 --> 00:46:36.589 And we can easily look at that. 00:46:43.080 --> 00:46:49.730 So that gets us to the big topic of simulation models. 00:46:49.730 --> 00:46:52.820 It's a program that describes a computation that 00:46:52.820 --> 00:46:57.830 provides information about the possible behaviors of a system. 00:46:57.830 --> 00:47:00.050 I say possible behaviors, because I'm 00:47:00.050 --> 00:47:02.835 particularly interested in stochastic systems. 00:47:05.720 --> 00:47:10.350 They're descriptive not prescriptive in the sense 00:47:10.350 --> 00:47:13.740 that they describe the possible outcomes. 00:47:13.740 --> 00:47:18.800 They don't tell you how to achieve possible outcomes. 00:47:18.800 --> 00:47:20.720 This is different from what we've 00:47:20.720 --> 00:47:22.550 looked at earlier in the course, where we 00:47:22.550 --> 00:47:25.700 looked at optimization models. 00:47:25.700 --> 00:47:30.440 So an optimization model is prescriptive. 00:47:30.440 --> 00:47:33.800 It tells you how to achieve an effect, 00:47:33.800 --> 00:47:38.000 how to get the most value out of your knapsack, 00:47:38.000 --> 00:47:42.350 how to find the shortest path from A to B in a graph. 00:47:42.350 --> 00:47:44.750 In contrast, a simulation model says, 00:47:44.750 --> 00:47:48.170 if I do this, here's what happens. 00:47:48.170 --> 00:47:52.290 It doesn't tell you how to make something happened. 00:47:52.290 --> 00:47:53.970 So it's very different, and it's why 00:47:53.970 --> 00:47:57.390 we need both, why we need optimization models 00:47:57.390 --> 00:48:00.570 and we need simulation models. 00:48:00.570 --> 00:48:03.750 We have to remember that a simulation model is only 00:48:03.750 --> 00:48:06.570 an approximation to reality. 00:48:06.570 --> 00:48:10.110 I put in an approximation to the distribution of birthdates, 00:48:10.110 --> 00:48:12.910 but it wasn't quite right. 00:48:12.910 --> 00:48:16.770 And as the very famous statistician George Box said, 00:48:16.770 --> 00:48:22.320 "all models are wrong, but some are actually very useful." 00:48:22.320 --> 00:48:27.930 In the next lecture, we'll look at a useful class of models. 00:48:27.930 --> 00:48:30.610 When do we use simulations? 00:48:30.610 --> 00:48:33.310 Typically, as we've just shown, to model systems that 00:48:33.310 --> 00:48:37.180 are mathematically intractable, like the birthday problem 00:48:37.180 --> 00:48:39.740 we just looked at. 00:48:39.740 --> 00:48:43.130 In other situations, to extract intermediate results-- 00:48:43.130 --> 00:48:47.660 something happens along the way to the answer. 00:48:47.660 --> 00:48:50.410 And as I hope you've seen that simulations 00:48:50.410 --> 00:48:55.480 are used because we can play what if games by successively 00:48:55.480 --> 00:48:57.340 refining it. 00:48:57.340 --> 00:48:59.230 We started with a simple simulation 00:48:59.230 --> 00:49:01.960 that assumed that we only asked the question of, do 00:49:01.960 --> 00:49:04.540 two people share a birthday. 00:49:04.540 --> 00:49:08.080 We showed how we could change it to ask do three people share 00:49:08.080 --> 00:49:10.020 a birthday. 00:49:10.020 --> 00:49:11.910 We then saw that we could change it 00:49:11.910 --> 00:49:16.260 to assume a different distribution of birthdates 00:49:16.260 --> 00:49:18.620 in the group. 00:49:18.620 --> 00:49:20.520 And so we can start with something simple. 00:49:20.520 --> 00:49:23.310 And we get it ever more complexed 00:49:23.310 --> 00:49:25.415 to answer questions what if. 00:49:29.510 --> 00:49:32.030 We're going to start in the next lecture 00:49:32.030 --> 00:49:36.680 by producing a simulation of a random walk. 00:49:36.680 --> 00:49:38.120 And with that, I'll stop. 00:49:38.120 --> 00:49:40.840 And see you guys soon.