WEBVTT 00:00:00.000 --> 00:00:02.400 align:middle line:90% [SQUEAKING] 00:00:02.400 --> 00:00:03.820 align:middle line:90% [RUSTLING] 00:00:03.820 --> 00:00:04.320 align:middle line:90% 00:00:04.320 --> 00:00:05.760 align:middle line:90% [CLICKING] 00:00:05.760 --> 00:00:10.560 align:middle line:90% 00:00:10.560 --> 00:00:11.960 align:middle line:90% BRYNMOR CHAPMAN: Hello. 00:00:11.960 --> 00:00:14.030 align:middle line:90% Can people hear me in the back? 00:00:14.030 --> 00:00:15.300 align:middle line:90% Yeah, thumbs up. 00:00:15.300 --> 00:00:17.116 align:middle line:90% OK. 00:00:17.116 --> 00:00:19.130 align:middle line:84% So today, we're going to be continuing 00:00:19.130 --> 00:00:21.830 align:middle line:90% our discussion of probability. 00:00:21.830 --> 00:00:25.040 align:middle line:84% In the last lecture, we saw how to compute probabilities-- 00:00:25.040 --> 00:00:25.820 align:middle line:90% [BANG] 00:00:25.820 --> 00:00:27.890 align:middle line:90% --using the-- oh, dear-- 00:00:27.890 --> 00:00:29.700 align:middle line:90% using the tree method. 00:00:29.700 --> 00:00:31.280 align:middle line:84% And in this lecture, we are going 00:00:31.280 --> 00:00:33.830 align:middle line:84% to see how some of the tools that we learned 00:00:33.830 --> 00:00:37.880 align:middle line:84% in the last unit for counting, for reasoning 00:00:37.880 --> 00:00:40.110 align:middle line:84% about the sizes of sets and things like that, 00:00:40.110 --> 00:00:46.715 align:middle line:84% can also carry over naturally into the realm of probability. 00:00:46.715 --> 00:00:48.530 align:middle line:84% And we'll also learn how to express 00:00:48.530 --> 00:00:52.410 align:middle line:84% mathematically statements like, if you recall, 00:00:52.410 --> 00:00:54.830 align:middle line:84% I was saying things last time like if the prize is 00:00:54.830 --> 00:00:58.640 align:middle line:84% behind door A, what is the probability that Marilyn selects 00:00:58.640 --> 00:01:00.830 align:middle line:90% door B or something like that? 00:01:00.830 --> 00:01:05.050 align:middle line:84% So we're going to see more formally 00:01:05.050 --> 00:01:07.090 align:middle line:84% what these statements are trying to express 00:01:07.090 --> 00:01:09.941 align:middle line:90% and how to reason about them. 00:01:09.941 --> 00:01:14.990 align:middle line:84% So recall how we defined the probability of an event A. 00:01:14.990 --> 00:01:19.000 align:middle line:84% So an event is just a set of outcomes. 00:01:19.000 --> 00:01:22.990 align:middle line:84% Our probability function, Pr, is a function 00:01:22.990 --> 00:01:28.420 align:middle line:84% from the set of outcomes, the sample space, to the interval 0 00:01:28.420 --> 00:01:29.120 align:middle line:90% to 1. 00:01:29.120 --> 00:01:31.690 align:middle line:84% And we can extend this naturally to events 00:01:31.690 --> 00:01:33.700 align:middle line:84% by saying that the probability of an event 00:01:33.700 --> 00:01:35.740 align:middle line:84% is just the sum of the probabilities of all 00:01:35.740 --> 00:01:39.010 align:middle line:90% of the outcomes in that event. 00:01:39.010 --> 00:01:41.950 align:middle line:84% So an immediate consequence of this definition, 00:01:41.950 --> 00:01:44.450 align:middle line:84% or several immediate consequences I suppose, 00:01:44.450 --> 00:01:45.700 align:middle line:90% are the following. 00:01:45.700 --> 00:01:47.475 align:middle line:90% So first, we have the sum rule. 00:01:47.475 --> 00:01:52.330 align:middle line:90% 00:01:52.330 --> 00:01:56.380 align:middle line:84% So if we have two events, A and B, and they're disjoint-- 00:01:56.380 --> 00:02:00.395 align:middle line:90% so A and B are disjoint events-- 00:02:00.395 --> 00:02:04.800 align:middle line:90% 00:02:04.800 --> 00:02:08.955 align:middle line:84% this tells us that the probability of the union of A 00:02:08.955 --> 00:02:11.830 align:middle line:90% and B is, what? 00:02:11.830 --> 00:02:14.280 align:middle line:90% What do we think that is? 00:02:14.280 --> 00:02:15.005 align:middle line:90% Any ideas? 00:02:15.005 --> 00:02:18.685 align:middle line:90% 00:02:18.685 --> 00:02:19.185 align:middle line:90% Yeah? 00:02:19.185 --> 00:02:20.940 align:middle line:84% AUDIENCE: Sum of the probabilities. 00:02:20.940 --> 00:02:23.140 align:middle line:84% BRYNMOR CHAPMAN: Yeah, the sum of the probabilities. 00:02:23.140 --> 00:02:27.300 align:middle line:84% So probability of A plus the probability 00:02:27.300 --> 00:02:34.360 align:middle line:84% of B. So very similar to what we saw with sets. 00:02:34.360 --> 00:02:37.050 align:middle line:84% If you have two disjoint sets, the size of their union 00:02:37.050 --> 00:02:40.510 align:middle line:84% is the same as the sum of the individual sizes. 00:02:40.510 --> 00:02:44.550 align:middle line:84% The same thing happens with probability. 00:02:44.550 --> 00:02:46.265 align:middle line:84% We also have the complement rule. 00:02:46.265 --> 00:02:51.360 align:middle line:90% 00:02:51.360 --> 00:02:54.150 align:middle line:84% So if we have some event A, what do we 00:02:54.150 --> 00:02:57.890 align:middle line:84% think the probability of the complement of A 00:02:57.890 --> 00:03:00.595 align:middle line:84% is, so the probability that A doesn't occur? 00:03:00.595 --> 00:03:03.380 align:middle line:90% 00:03:03.380 --> 00:03:05.925 align:middle line:90% So I should probably-- 00:03:05.925 --> 00:03:06.425 align:middle line:90% yeah. 00:03:06.425 --> 00:03:11.570 align:middle line:90% 00:03:11.570 --> 00:03:15.170 align:middle line:90% What's the probability of not A? 00:03:15.170 --> 00:03:15.860 align:middle line:90% Anybody? 00:03:15.860 --> 00:03:16.900 align:middle line:90% Yeah? 00:03:16.900 --> 00:03:18.980 align:middle line:84% AUDIENCE: 1 minus the probability of A. 00:03:18.980 --> 00:03:25.820 align:middle line:84% BRYNMOR CHAPMAN: Yeah, 1 minus the probability of A because A 00:03:25.820 --> 00:03:28.170 align:middle line:90% and not A are disjoint. 00:03:28.170 --> 00:03:30.120 align:middle line:90% And one of them must happen. 00:03:30.120 --> 00:03:33.780 align:middle line:84% So with certainty, one of A or not A happens. 00:03:33.780 --> 00:03:35.820 align:middle line:84% So the probability of their union is 1. 00:03:35.820 --> 00:03:37.870 align:middle line:84% So this kind of just follows from the sum rule. 00:03:37.870 --> 00:03:43.730 align:middle line:90% 00:03:43.730 --> 00:03:44.920 align:middle line:90% What else do we have? 00:03:44.920 --> 00:03:47.930 align:middle line:90% 00:03:47.930 --> 00:03:48.850 align:middle line:90% The difference rule. 00:03:48.850 --> 00:03:54.520 align:middle line:90% 00:03:54.520 --> 00:03:59.560 align:middle line:84% So in this case, suppose we have, again, two events, A 00:03:59.560 --> 00:04:04.930 align:middle line:84% and B. And this time, we'd like to look at the difference 00:04:04.930 --> 00:04:10.827 align:middle line:84% A set minus B. What do we think that might 00:04:10.827 --> 00:04:12.660 align:middle line:84% be in terms of the probabilities of A and B? 00:04:12.660 --> 00:04:26.240 align:middle line:90% 00:04:26.240 --> 00:04:26.740 align:middle line:90% Yeah? 00:04:26.740 --> 00:04:29.410 align:middle line:84% AUDIENCE: The probability of A minus the intersection 00:04:29.410 --> 00:04:30.100 align:middle line:90% of A and B. 00:04:30.100 --> 00:04:31.210 align:middle line:90% BRYNMOR CHAPMAN: Exactly. 00:04:31.210 --> 00:04:39.280 align:middle line:84% So the probability of A minus the probability 00:04:39.280 --> 00:04:40.970 align:middle line:90% of the intersection. 00:04:40.970 --> 00:04:44.530 align:middle line:90% 00:04:44.530 --> 00:04:45.260 align:middle line:90% Why is that? 00:04:45.260 --> 00:04:48.730 align:middle line:84% Well, A is just the disjoint union of A minus B, 00:04:48.730 --> 00:04:53.370 align:middle line:90% and A intersects B. 00:04:53.370 --> 00:04:55.110 align:middle line:84% We also have an analog of principle 00:04:55.110 --> 00:04:57.010 align:middle line:90% of inclusion/exclusion. 00:04:57.010 --> 00:04:59.670 align:middle line:90% 00:04:59.670 --> 00:05:09.220 align:middle line:84% So the probability of A union B for more general A and B. 00:05:09.220 --> 00:05:13.830 align:middle line:84% So they're not necessarily disjoint anymore. 00:05:13.830 --> 00:05:15.570 align:middle line:84% Just like we had with sets, this is 00:05:15.570 --> 00:05:21.960 align:middle line:84% going to be the probability of A plus the probability of B 00:05:21.960 --> 00:05:25.840 align:middle line:84% minus the probability of the intersection. 00:05:25.840 --> 00:05:33.325 align:middle line:90% 00:05:33.325 --> 00:05:38.580 align:middle line:84% And we have a couple of new rules 00:05:38.580 --> 00:05:45.060 align:middle line:84% that don't really have analogs in the case of counting sets. 00:05:45.060 --> 00:05:48.355 align:middle line:84% So first, we have the union bound. 00:05:48.355 --> 00:05:48.855 align:middle line:90% Oops. 00:05:48.855 --> 00:05:55.870 align:middle line:90% 00:05:55.870 --> 00:06:01.660 align:middle line:84% So probability of A union B is less than 00:06:01.660 --> 00:06:07.427 align:middle line:84% or equal to probability of A plus probability of B. 00:06:07.427 --> 00:06:10.010 align:middle line:84% Or I suppose I shouldn't say it doesn't really have an analog, 00:06:10.010 --> 00:06:11.677 align:middle line:84% but we didn't really look at the analog. 00:06:11.677 --> 00:06:15.478 align:middle line:90% 00:06:15.478 --> 00:06:18.620 align:middle line:90% And why might that be the case? 00:06:18.620 --> 00:06:19.120 align:middle line:90% Yeah? 00:06:19.120 --> 00:06:19.960 align:middle line:90% AUDIENCE: I have a question. 00:06:19.960 --> 00:06:21.043 align:middle line:90% BRYNMOR CHAPMAN: Oh, yeah. 00:06:21.043 --> 00:06:23.755 align:middle line:84% AUDIENCE: What's the difference between the sum rule and pi? 00:06:23.755 --> 00:06:26.390 align:middle line:84% BRYNMOR CHAPMAN: So the question is, 00:06:26.390 --> 00:06:29.060 align:middle line:84% what's the difference between the sum rule and pi? 00:06:29.060 --> 00:06:34.400 align:middle line:84% So in the sum rule, we assume that A and B are disjoint. 00:06:34.400 --> 00:06:38.810 align:middle line:84% So if A and B are disjoint, then their intersection is empty. 00:06:38.810 --> 00:06:41.800 align:middle line:84% So the intersection has zero probability. 00:06:41.800 --> 00:06:45.160 align:middle line:84% So pi is just a way to generalize the sum 00:06:45.160 --> 00:06:47.380 align:middle line:84% rule to the case where they're not 00:06:47.380 --> 00:06:48.800 align:middle line:90% necessarily disjoint anymore. 00:06:48.800 --> 00:06:52.950 align:middle line:84% They may have a nontrivial intersection. 00:06:52.950 --> 00:06:53.860 align:middle line:90% Good question. 00:06:53.860 --> 00:06:57.420 align:middle line:90% Any other questions? 00:06:57.420 --> 00:06:58.200 align:middle line:90% No? 00:06:58.200 --> 00:06:59.460 align:middle line:90% OK. 00:06:59.460 --> 00:07:03.620 align:middle line:84% So can anybody give a reason why the union bound might be true? 00:07:03.620 --> 00:07:07.800 align:middle line:90% 00:07:07.800 --> 00:07:09.042 align:middle line:90% Yeah? 00:07:09.042 --> 00:07:11.250 align:middle line:84% AUDIENCE: Because the probability of the intersection 00:07:11.250 --> 00:07:11.917 align:middle line:90% is non-negative. 00:07:11.917 --> 00:07:13.980 align:middle line:84% BRYNMOR CHAPMAN: Yeah, so the answer 00:07:13.980 --> 00:07:18.110 align:middle line:84% was the probability of the intersection is non-negative. 00:07:18.110 --> 00:07:25.690 align:middle line:84% So if we just take pi, so this holds with equality. 00:07:25.690 --> 00:07:29.650 align:middle line:84% And if we just get rid of this negative term here, 00:07:29.650 --> 00:07:31.950 align:middle line:84% we can only increase the probability 00:07:31.950 --> 00:07:34.505 align:middle line:84% because all probabilities have to be between 0 and 1. 00:07:34.505 --> 00:07:38.110 align:middle line:90% 00:07:38.110 --> 00:07:40.030 align:middle line:90% We also have monotonicity. 00:07:40.030 --> 00:07:43.200 align:middle line:90% 00:07:43.200 --> 00:07:48.770 align:middle line:84% Again, I don't think we saw an analog of this with sets. 00:07:48.770 --> 00:07:54.980 align:middle line:84% But if A is a subset of B, what do you think that tells us 00:07:54.980 --> 00:07:56.486 align:middle line:90% about the probabilities? 00:07:56.486 --> 00:07:59.700 align:middle line:90% 00:07:59.700 --> 00:08:00.200 align:middle line:90% Any ideas? 00:08:00.200 --> 00:08:05.000 align:middle line:90% 00:08:05.000 --> 00:08:08.090 align:middle line:90% I'm seeing some hand motions. 00:08:08.090 --> 00:08:19.160 align:middle line:84% So as you might expect from the name, if is A subset of B, 00:08:19.160 --> 00:08:20.900 align:middle line:84% then its probability should be smaller 00:08:20.900 --> 00:08:29.520 align:middle line:84% than the probability of B. So do these make sense to everybody? 00:08:29.520 --> 00:08:36.380 align:middle line:90% 00:08:36.380 --> 00:08:42.830 align:middle line:84% So just as we saw with sets, both the sum rule, 00:08:42.830 --> 00:08:46.250 align:middle line:84% the union bound, and the principle of inclusion/exclusion 00:08:46.250 --> 00:08:51.590 align:middle line:84% can all generalize to arbitrary numbers of events 00:08:51.590 --> 00:08:55.610 align:middle line:90% or even countably many. 00:08:55.610 --> 00:08:58.050 align:middle line:84% I suppose principle of inclusion/exclusion, 00:08:58.050 --> 00:09:01.200 align:middle line:84% you should probably only think about finite cases. 00:09:01.200 --> 00:09:03.920 align:middle line:84% But the other two, they fairly, naturally, 00:09:03.920 --> 00:09:09.170 align:middle line:84% generalize to countable sets of events. 00:09:09.170 --> 00:09:11.180 align:middle line:84% So do these rules make sense to everybody? 00:09:11.180 --> 00:09:12.930 align:middle line:84% You'll probably be using them quite a lot. 00:09:12.930 --> 00:09:15.700 align:middle line:84% So make sure you're familiar with them. 00:09:15.700 --> 00:09:18.710 align:middle line:90% 00:09:18.710 --> 00:09:19.776 align:middle line:90% Any questions? 00:09:19.776 --> 00:09:24.410 align:middle line:90% 00:09:24.410 --> 00:09:27.650 align:middle line:84% So in that case, let us move on to the main topic 00:09:27.650 --> 00:09:29.690 align:middle line:90% of today's lecture-- 00:09:29.690 --> 00:09:31.470 align:middle line:90% conditional probability. 00:09:31.470 --> 00:09:44.645 align:middle line:90% 00:09:44.645 --> 00:09:47.850 align:middle line:84% So if you recall on Tuesday when we were talking about the Monty 00:09:47.850 --> 00:09:52.060 align:middle line:84% Hall problem, when we were drawing our tree diagram, 00:09:52.060 --> 00:09:55.140 align:middle line:84% we were making statements such as, 00:09:55.140 --> 00:09:57.960 align:middle line:84% if the car is behind door 1, what's the probability that we 00:09:57.960 --> 00:09:58.630 align:middle line:90% pick door 2? 00:09:58.630 --> 00:10:02.020 align:middle line:84% Or I guess it was purple, gold, and green-- or purple, gold, 00:10:02.020 --> 00:10:02.530 align:middle line:90% silver? 00:10:02.530 --> 00:10:05.250 align:middle line:90% Silver, yeah. 00:10:05.250 --> 00:10:09.450 align:middle line:84% So how do we express this mathematically? 00:10:09.450 --> 00:10:13.140 align:middle line:84% So that's where conditional probability comes in. 00:10:13.140 --> 00:10:15.990 align:middle line:84% We're going to make use of the following definition. 00:10:15.990 --> 00:10:25.110 align:middle line:84% So if we have two events, A and B, 00:10:25.110 --> 00:10:34.780 align:middle line:84% we define the conditional probability 00:10:34.780 --> 00:10:46.086 align:middle line:84% of A given B, which we denote as the following. 00:10:46.086 --> 00:10:50.180 align:middle line:90% 00:10:50.180 --> 00:10:56.840 align:middle line:84% So we say A conditional B. We denote that with a vertical bar. 00:10:56.840 --> 00:11:01.190 align:middle line:84% We define this to be the probability 00:11:01.190 --> 00:11:14.051 align:middle line:84% of A intersect B divided by the probability of B. 00:11:14.051 --> 00:11:16.460 align:middle line:84% So intuitively, you can think of it 00:11:16.460 --> 00:11:21.350 align:middle line:84% a little bit as we're trying to scale our probability space so 00:11:21.350 --> 00:11:24.740 align:middle line:84% that B becomes our entire sample space. 00:11:24.740 --> 00:11:29.360 align:middle line:84% So we kind of want to divide out this probability of B 00:11:29.360 --> 00:11:33.050 align:middle line:84% so that we still get a probability space, all 00:11:33.050 --> 00:11:34.880 align:middle line:90% of the probabilities sum to 1. 00:11:34.880 --> 00:11:38.120 align:middle line:84% Does that make sense to everybody? 00:11:38.120 --> 00:11:42.190 align:middle line:84% And then the analog of A, we're only 00:11:42.190 --> 00:11:44.210 align:middle line:84% looking at everything that's already within B. 00:11:44.210 --> 00:11:52.344 align:middle line:84% So we're looking at the intersection of A and B. 00:11:52.344 --> 00:11:56.130 align:middle line:84% Now, if we rewrite this, we can get a product rule. 00:11:56.130 --> 00:12:04.006 align:middle line:90% 00:12:04.006 --> 00:12:08.060 align:middle line:84% So if we want to know the probability of A intersect B, 00:12:08.060 --> 00:12:10.030 align:middle line:84% we can just move this probability of B 00:12:10.030 --> 00:12:11.980 align:middle line:90% over to the other side. 00:12:11.980 --> 00:12:18.370 align:middle line:84% And we end up with probability of A intersect B equals 00:12:18.370 --> 00:12:31.440 align:middle line:84% probability of A given B times the probability of B. 00:12:31.440 --> 00:12:36.610 align:middle line:84% And as you might expect, you can extend this to multiple events. 00:12:36.610 --> 00:12:38.690 align:middle line:84% Or I guess this is already multiple events. 00:12:38.690 --> 00:12:42.580 align:middle line:84% You can extend it to more than two events. 00:12:42.580 --> 00:12:49.970 align:middle line:84% So you can also have, e.g., the probability of A intersect B 00:12:49.970 --> 00:13:00.670 align:middle line:84% intersect C equals the probability of A given B and C 00:13:00.670 --> 00:13:04.990 align:middle line:90% times the probability of B-- 00:13:04.990 --> 00:13:08.740 align:middle line:84% or maybe I should write this out more explicitly-- 00:13:08.740 --> 00:13:11.570 align:middle line:90% B intersect C. 00:13:11.570 --> 00:13:15.280 align:middle line:84% And then if we apply our binary product rule again, 00:13:15.280 --> 00:13:23.710 align:middle line:84% we can expand out this term as the probability of A given 00:13:23.710 --> 00:13:38.945 align:middle line:84% B and C, probability of B given C, probability of C. 00:13:38.945 --> 00:13:42.530 align:middle line:84% So that's basically what we were doing on Tuesday written out 00:13:42.530 --> 00:13:45.092 align:middle line:90% more formally. 00:13:45.092 --> 00:13:51.560 align:middle line:84% So probability of C corresponds to what 00:13:51.560 --> 00:13:54.860 align:middle line:84% is the first edge we chose, the first edge going away 00:13:54.860 --> 00:13:56.060 align:middle line:90% from the root. 00:13:56.060 --> 00:13:59.570 align:middle line:84% And then conditioned on reaching a particular vertex, 00:13:59.570 --> 00:14:02.360 align:middle line:84% like conditioned on Monty hiding the prize 00:14:02.360 --> 00:14:07.670 align:middle line:84% in the gold box, what's the probability that Marilyn chooses 00:14:07.670 --> 00:14:10.020 align:middle line:90% the silver one? 00:14:10.020 --> 00:14:11.960 align:middle line:84% So that's the conditional probability 00:14:11.960 --> 00:14:15.140 align:middle line:84% of choosing the silver box, conditioned on the prize being 00:14:15.140 --> 00:14:17.786 align:middle line:90% in the gold box. 00:14:17.786 --> 00:14:23.983 align:middle line:84% And then, as you continue along the path to a leaf, 00:14:23.983 --> 00:14:26.150 align:middle line:84% all of the edge probabilities that you're looking at 00:14:26.150 --> 00:14:28.250 align:middle line:84% are secret-- they were secretly just 00:14:28.250 --> 00:14:30.620 align:middle line:90% basic conditional probabilities. 00:14:30.620 --> 00:14:32.970 align:middle line:84% Does that make sense to everybody? 00:14:32.970 --> 00:14:37.300 align:middle line:90% 00:14:37.300 --> 00:14:38.346 align:middle line:90% Any questions? 00:14:38.346 --> 00:14:46.264 align:middle line:90% 00:14:46.264 --> 00:14:53.110 align:middle line:84% So another extension of this that you may find useful 00:14:53.110 --> 00:14:54.590 align:middle line:90% is the following-- 00:14:54.590 --> 00:14:57.670 align:middle line:90% 00:14:57.670 --> 00:15:09.130 align:middle line:84% A intersect B given C is going to be probability of A given B 00:15:09.130 --> 00:15:24.088 align:middle line:84% intersect C probability of B given C. So can anybody 00:15:24.088 --> 00:15:25.380 align:middle line:90% see why that might be the case? 00:15:25.380 --> 00:15:30.880 align:middle line:90% 00:15:30.880 --> 00:15:34.920 align:middle line:84% Well, first maybe we should look at, what exactly is it saying? 00:15:34.920 --> 00:15:44.520 align:middle line:84% So if you squint a little bit and get rid of the C's, it 00:15:44.520 --> 00:15:49.050 align:middle line:84% kind of looks a little bit like the product rule. 00:15:49.050 --> 00:15:56.310 align:middle line:84% A intersect B is probability of A given B times 00:15:56.310 --> 00:16:02.050 align:middle line:84% probability of B. It looks kind of like that. 00:16:02.050 --> 00:16:04.890 align:middle line:84% But we've added a bunch of C's everywhere. 00:16:04.890 --> 00:16:07.170 align:middle line:84% So basically what we're saying here 00:16:07.170 --> 00:16:11.460 align:middle line:84% is this is the product rule in the probability space 00:16:11.460 --> 00:16:15.090 align:middle line:84% where we're assuming C. It's the product rule conditioned 00:16:15.090 --> 00:16:19.810 align:middle line:84% on C. Does that make sense to everybody? 00:16:19.810 --> 00:16:22.590 align:middle line:90% 00:16:22.590 --> 00:16:26.430 align:middle line:84% And we can obtain it from the previous line 00:16:26.430 --> 00:16:39.200 align:middle line:84% just by dividing out by the probability of C. 00:16:39.200 --> 00:16:40.560 align:middle line:90% Do people have questions? 00:16:40.560 --> 00:16:42.510 align:middle line:84% Are we happy to move on to some examples? 00:16:42.510 --> 00:16:43.700 align:middle line:90% Yeah? 00:16:43.700 --> 00:16:44.485 align:middle line:90% AUDIENCE: Sorry. 00:16:44.485 --> 00:16:45.443 align:middle line:90% BRYNMOR CHAPMAN: Sorry. 00:16:45.443 --> 00:16:47.960 align:middle line:84% AUDIENCE: Is it sets on the left board? 00:16:47.960 --> 00:16:50.510 align:middle line:84% Or is it still events [INAUDIBLE] on the board? 00:16:50.510 --> 00:16:51.510 align:middle line:90% BRYNMOR CHAPMAN: Oh, OK. 00:16:51.510 --> 00:16:54.840 align:middle line:84% So the question was, are we talking about sets or events? 00:16:54.840 --> 00:16:56.340 align:middle line:90% Yes. 00:16:56.340 --> 00:17:02.735 align:middle line:84% So remember that events are defined to be sets of outcomes. 00:17:02.735 --> 00:17:05.880 align:middle line:90% So we're talking about events. 00:17:05.880 --> 00:17:08.940 align:middle line:84% But events are sets, and we can treat them as such. 00:17:08.940 --> 00:17:11.878 align:middle line:84% So that's why we're using lots of notation 00:17:11.878 --> 00:17:14.420 align:middle line:84% here that you would typically associate with sets, like union 00:17:14.420 --> 00:17:15.420 align:middle line:90% intersection, et cetera. 00:17:15.420 --> 00:17:18.450 align:middle line:90% 00:17:18.450 --> 00:17:19.560 align:middle line:90% Any other questions? 00:17:19.560 --> 00:17:20.075 align:middle line:90% Oh, yeah? 00:17:20.075 --> 00:17:27.140 align:middle line:84% AUDIENCE: So if you have A and B and C [INAUDIBLE] 00:17:27.140 --> 00:17:32.472 align:middle line:84% or is it more like a single node of [INAUDIBLE]? 00:17:32.472 --> 00:17:37.370 align:middle line:84% BRYNMOR CHAPMAN: So these are not nodes in the tree. 00:17:37.370 --> 00:17:37.870 align:middle line:90% Oh, sorry. 00:17:37.870 --> 00:17:40.350 align:middle line:84% Question was, if we have an event that's like A union 00:17:40.350 --> 00:17:43.740 align:middle line:84% B union C or something, are these general sets? 00:17:43.740 --> 00:17:46.230 align:middle line:90% Or are they nodes in the tree? 00:17:46.230 --> 00:17:47.650 align:middle line:90% That is a good observation. 00:17:47.650 --> 00:17:52.330 align:middle line:84% Some of the nodes in the tree naturally correspond to events. 00:17:52.330 --> 00:17:56.820 align:middle line:84% Like if we draw out our tree diagram, 00:17:56.820 --> 00:17:59.640 align:middle line:84% this node here fairly naturally corresponds 00:17:59.640 --> 00:18:02.430 align:middle line:90% to this set of events-- 00:18:02.430 --> 00:18:07.710 align:middle line:84% or sorry, this set of outcomes, those two leaves in its subtree. 00:18:07.710 --> 00:18:11.670 align:middle line:84% But not every event necessarily corresponds to an internal node 00:18:11.670 --> 00:18:13.290 align:middle line:90% of the tree like that. 00:18:13.290 --> 00:18:20.220 align:middle line:84% For instance, you could have these two outcomes. 00:18:20.220 --> 00:18:22.720 align:middle line:84% If you consider the set of those two outcomes, 00:18:22.720 --> 00:18:24.700 align:middle line:90% that's still a valid event. 00:18:24.700 --> 00:18:26.520 align:middle line:84% But it doesn't really naturally correspond 00:18:26.520 --> 00:18:29.390 align:middle line:90% to a tree node in the same way. 00:18:29.390 --> 00:18:31.622 align:middle line:90% Does that make sense? 00:18:31.622 --> 00:18:33.655 align:middle line:84% AUDIENCE: I think you misunderstood my question. 00:18:33.655 --> 00:18:35.545 align:middle line:84% BRYNMOR CHAPMAN: Oh, that is quite possible. 00:18:35.545 --> 00:18:38.540 align:middle line:84% AUDIENCE: I was meaning the probability 00:18:38.540 --> 00:18:43.040 align:middle line:84% of A given B or A given C, [INAUDIBLE] 00:18:43.040 --> 00:18:46.160 align:middle line:90% is that still a similar set? 00:18:46.160 --> 00:18:47.488 align:middle line:90% Or how that would be different? 00:18:47.488 --> 00:18:49.280 align:middle line:84% But I think it would just be helpful to see 00:18:49.280 --> 00:18:51.085 align:middle line:90% what happened on the tree. 00:18:51.085 --> 00:18:53.318 align:middle line:90% BRYNMOR CHAPMAN: OK. 00:18:53.318 --> 00:18:55.928 align:middle line:84% So asking about the conditional probabilities as they 00:18:55.928 --> 00:18:56.720 align:middle line:90% appear in the tree. 00:18:56.720 --> 00:19:01.080 align:middle line:90% 00:19:01.080 --> 00:19:02.487 align:middle line:90% TA: Brynmor? 00:19:02.487 --> 00:19:04.055 align:middle line:90% BRYNMOR CHAPMAN: Yes? 00:19:04.055 --> 00:19:06.030 align:middle line:90% TA: What is A given B? 00:19:06.030 --> 00:19:07.050 align:middle line:90% Is that an event? 00:19:07.050 --> 00:19:08.150 align:middle line:90% Is it a set? 00:19:08.150 --> 00:19:10.290 align:middle line:84% How can we describe it, not as a probability but-- 00:19:10.290 --> 00:19:12.120 align:middle line:90% BRYNMOR CHAPMAN: Oh, I see. 00:19:12.120 --> 00:19:13.620 align:middle line:90% OK. 00:19:13.620 --> 00:19:19.070 align:middle line:84% So I guess the way that I like to think about A given B, 00:19:19.070 --> 00:19:22.670 align:middle line:84% it is an event in a different probability space. 00:19:22.670 --> 00:19:25.630 align:middle line:90% So you can think of-- 00:19:25.630 --> 00:19:28.490 align:middle line:90% if you limit the scope-- 00:19:28.490 --> 00:19:31.630 align:middle line:84% rather than talking about your entire sample space S, 00:19:31.630 --> 00:19:34.600 align:middle line:84% if you limit the scope to just C, 00:19:34.600 --> 00:19:37.120 align:middle line:84% so you scale all of the probabilities down 00:19:37.120 --> 00:19:40.090 align:middle line:84% by the probability of C, you end up with a different probability 00:19:40.090 --> 00:19:41.030 align:middle line:90% space. 00:19:41.030 --> 00:19:45.220 align:middle line:84% So now the probability of-- or sorry, the event B given C 00:19:45.220 --> 00:19:47.920 align:middle line:84% is an event in that probability space. 00:19:47.920 --> 00:19:49.430 align:middle line:90% Does that make sense? 00:19:49.430 --> 00:19:50.157 align:middle line:90% AUDIENCE: Yeah. 00:19:50.157 --> 00:19:50.990 align:middle line:90% BRYNMOR CHAPMAN: OK. 00:19:50.990 --> 00:19:51.855 align:middle line:90% Yeah, sorry. 00:19:51.855 --> 00:19:54.800 align:middle line:84% [CHUCKLES] Thank you for the clarification. 00:19:54.800 --> 00:19:59.400 align:middle line:90% 00:19:59.400 --> 00:20:01.270 align:middle line:84% So why don't we move on to some examples 00:20:01.270 --> 00:20:05.692 align:middle line:84% then if this makes reasonable amount of sense. 00:20:05.692 --> 00:20:07.650 align:middle line:84% Hopefully it'll make more sense after examples. 00:20:07.650 --> 00:20:10.274 align:middle line:90% 00:20:10.274 --> 00:20:12.670 align:middle line:84% So suppose Ash and Gary are having 00:20:12.670 --> 00:20:16.090 align:middle line:84% a bunch of Pokemon battles against each other. 00:20:16.090 --> 00:20:17.660 align:middle line:90% And they're doing a tournament. 00:20:17.660 --> 00:20:21.010 align:middle line:84% The first to win two battles wins the-- 00:20:21.010 --> 00:20:22.070 align:middle line:90% or I guess the series. 00:20:22.070 --> 00:20:25.564 align:middle line:84% The first to win two battles wins the series. 00:20:25.564 --> 00:20:31.330 align:middle line:84% And the probabilities of victory are not entirely uniform. 00:20:31.330 --> 00:20:34.282 align:middle line:84% So for the first match, maybe it's a toss-up. 00:20:34.282 --> 00:20:35.740 align:middle line:84% Ash wins with probability one half. 00:20:35.740 --> 00:20:39.920 align:middle line:84% Garry wins with probability one half. 00:20:39.920 --> 00:20:42.630 align:middle line:84% But after the first battle, there's 00:20:42.630 --> 00:20:46.060 align:middle line:84% an advantage to whoever won the last battle. 00:20:46.060 --> 00:20:48.160 align:middle line:84% Their morale is high or something. 00:20:48.160 --> 00:20:49.270 align:middle line:90% They do better. 00:20:49.270 --> 00:20:53.075 align:middle line:84% So they now have probability of 2/3 of winning the next battle. 00:20:53.075 --> 00:20:55.782 align:middle line:90% 00:20:55.782 --> 00:20:57.520 align:middle line:90% And there aren't any draws. 00:20:57.520 --> 00:20:58.737 align:middle line:90% They're bitter rivals. 00:20:58.737 --> 00:20:59.945 align:middle line:90% They will fight to the death. 00:20:59.945 --> 00:21:05.586 align:middle line:90% 00:21:05.586 --> 00:21:08.550 align:middle line:84% So let's define a couple of events. 00:21:08.550 --> 00:21:19.085 align:middle line:84% We'll say that A is the event Ash wins the series. 00:21:19.085 --> 00:21:22.800 align:middle line:90% 00:21:22.800 --> 00:21:31.330 align:middle line:84% And B is the event Ash wins the first battle. 00:21:31.330 --> 00:21:39.230 align:middle line:90% 00:21:39.230 --> 00:21:52.865 align:middle line:84% So now we can ask for the probability of A given B. 00:21:52.865 --> 00:21:56.780 align:middle line:84% So does the problem setup make sense? 00:21:56.780 --> 00:21:57.320 align:middle line:90% Yeah? 00:21:57.320 --> 00:21:58.280 align:middle line:90% OK. 00:21:58.280 --> 00:22:00.580 align:middle line:90% So how could we compute this? 00:22:00.580 --> 00:22:05.630 align:middle line:90% 00:22:05.630 --> 00:22:06.472 align:middle line:90% Any ideas? 00:22:06.472 --> 00:22:07.555 align:middle line:90% Where do we want to start? 00:22:07.555 --> 00:22:18.117 align:middle line:90% 00:22:18.117 --> 00:22:19.950 align:middle line:84% Should we just throw some numbers out there? 00:22:19.950 --> 00:22:20.450 align:middle line:90% Oh, yeah? 00:22:20.450 --> 00:22:21.828 align:middle line:90% AUDIENCE: Draw a tree diagram. 00:22:21.828 --> 00:22:23.870 align:middle line:84% BRYNMOR CHAPMAN: Yeah, let's draw a tree diagram. 00:22:23.870 --> 00:22:24.740 align:middle line:90% Nice one. 00:22:24.740 --> 00:22:30.220 align:middle line:84% OK, so drawing a tree diagram, what's 00:22:30.220 --> 00:22:33.590 align:middle line:84% the first random thing that happens? 00:22:33.590 --> 00:22:35.636 align:middle line:84% What's the first source of randomness? 00:22:35.636 --> 00:22:39.460 align:middle line:90% 00:22:39.460 --> 00:22:41.650 align:middle line:90% Yeah? 00:22:41.650 --> 00:22:43.450 align:middle line:90% AUDIENCE: [INAUDIBLE]. 00:22:43.450 --> 00:22:44.900 align:middle line:90% BRYNMOR CHAPMAN: Yeah. 00:22:44.900 --> 00:22:47.860 align:middle line:84% So Ash could win or lose the first battle. 00:22:47.860 --> 00:22:57.667 align:middle line:84% So we can actually label these as B and not B. 00:22:57.667 --> 00:22:59.625 align:middle line:84% What's the next random thing that could happen? 00:22:59.625 --> 00:23:06.790 align:middle line:90% 00:23:06.790 --> 00:23:08.400 align:middle line:90% Yeah? 00:23:08.400 --> 00:23:10.660 align:middle line:90% AUDIENCE: [INAUDIBLE]. 00:23:10.660 --> 00:23:12.430 align:middle line:84% BRYNMOR CHAPMAN: Yeah, that's right. 00:23:12.430 --> 00:23:16.840 align:middle line:84% So actually, maybe it's better to call these just W and L 00:23:16.840 --> 00:23:18.710 align:middle line:84% so we can use the same thing everywhere. 00:23:18.710 --> 00:23:22.080 align:middle line:90% 00:23:22.080 --> 00:23:25.830 align:middle line:90% So win, lose, win, lose. 00:23:25.830 --> 00:23:29.920 align:middle line:84% Regardless of whether he wins or loses the first battle, 00:23:29.920 --> 00:23:33.330 align:middle line:84% he can still either win or lose the second one. 00:23:33.330 --> 00:23:35.225 align:middle line:84% Now, what's the third layer going to be? 00:23:35.225 --> 00:23:42.488 align:middle line:90% 00:23:42.488 --> 00:23:45.910 align:middle line:84% AUDIENCE: If he's won two, it's over. 00:23:45.910 --> 00:23:48.310 align:middle line:84% If he's won one and he's lost one, 00:23:48.310 --> 00:23:50.580 align:middle line:84% you go again and split it into two separate outcomes. 00:23:50.580 --> 00:23:51.300 align:middle line:90% BRYNMOR CHAPMAN: Exactly. 00:23:51.300 --> 00:23:53.640 align:middle line:84% AUDIENCE: Lost two, that means that he's lost two. 00:23:53.640 --> 00:23:55.450 align:middle line:90% BRYNMOR CHAPMAN: Yeah, exactly. 00:23:55.450 --> 00:23:59.430 align:middle line:84% So if he's won two or lost two, those are just leaves now. 00:23:59.430 --> 00:24:00.520 align:middle line:90% We don't do anything more. 00:24:00.520 --> 00:24:02.470 align:middle line:90% The series is over. 00:24:02.470 --> 00:24:05.910 align:middle line:84% But if he's won one and lost one in either order, 00:24:05.910 --> 00:24:09.090 align:middle line:90% we can now split again. 00:24:09.090 --> 00:24:11.820 align:middle line:84% So that's a little bit long, but, oh, well. 00:24:11.820 --> 00:24:17.430 align:middle line:90% W, L, W, L. 00:24:17.430 --> 00:24:19.650 align:middle line:90% So now what are our outcomes? 00:24:19.650 --> 00:24:23.630 align:middle line:84% Again, let's look at the paths from root 00:24:23.630 --> 00:24:28.530 align:middle line:84% to leaves and just label with a string of edge labels. 00:24:28.530 --> 00:24:42.140 align:middle line:84% So this is going to be W comma W; W, L, W; W, L, L; L, W, W; L, 00:24:42.140 --> 00:24:50.530 align:middle line:90% W, L; and L, L. 00:24:50.530 --> 00:24:53.905 align:middle line:84% Now, what are the probabilities on each of the edges? 00:24:53.905 --> 00:24:59.330 align:middle line:90% 00:24:59.330 --> 00:25:01.790 align:middle line:84% So in the first layer, what's the probability 00:25:01.790 --> 00:25:04.430 align:middle line:90% that Ash wins the first game? 00:25:04.430 --> 00:25:05.150 align:middle line:90% First match. 00:25:05.150 --> 00:25:05.650 align:middle line:90% Yeah? 00:25:05.650 --> 00:25:06.590 align:middle line:90% AUDIENCE: One half. 00:25:06.590 --> 00:25:07.710 align:middle line:90% BRYNMOR CHAPMAN: One half. 00:25:07.710 --> 00:25:10.530 align:middle line:84% So each of these is probability one half. 00:25:10.530 --> 00:25:14.280 align:middle line:90% 00:25:14.280 --> 00:25:17.500 align:middle line:84% Now, the second layer, we said, are now 00:25:17.500 --> 00:25:19.160 align:middle line:90% conditional probabilities. 00:25:19.160 --> 00:25:22.790 align:middle line:84% So conditioned on winning the first match, 00:25:22.790 --> 00:25:26.192 align:middle line:84% what's the probability of winning the second match? 00:25:26.192 --> 00:25:27.900 align:middle line:84% What's the probability on this edge here? 00:25:27.900 --> 00:25:33.220 align:middle line:90% 00:25:33.220 --> 00:25:34.520 align:middle line:90% Do we remember? 00:25:34.520 --> 00:25:35.332 align:middle line:90% AUDIENCE: 2/3. 00:25:35.332 --> 00:25:36.945 align:middle line:90% BRYNMOR CHAPMAN: 2/3, yeah. 00:25:36.945 --> 00:25:39.580 align:middle line:90% 00:25:39.580 --> 00:25:42.290 align:middle line:84% And so the other branch is going to be the complement of that, 00:25:42.290 --> 00:25:44.560 align:middle line:90% so one third. 00:25:44.560 --> 00:25:47.830 align:middle line:84% And then for the bottom branch, it's 00:25:47.830 --> 00:25:49.940 align:middle line:84% going to be exactly the opposite. 00:25:49.940 --> 00:25:51.940 align:middle line:84% It's going to be one third chance of winning, 00:25:51.940 --> 00:25:53.220 align:middle line:90% 2/3 chance of losing. 00:25:53.220 --> 00:25:58.885 align:middle line:90% 00:25:58.885 --> 00:26:02.740 align:middle line:84% Now, conditioned on winning the first match 00:26:02.740 --> 00:26:05.140 align:middle line:84% and losing the second match, what's 00:26:05.140 --> 00:26:06.690 align:middle line:84% the probability of winning the third? 00:26:06.690 --> 00:26:14.140 align:middle line:90% 00:26:14.140 --> 00:26:14.650 align:middle line:90% Yeah? 00:26:14.650 --> 00:26:15.700 align:middle line:90% AUDIENCE: One third. 00:26:15.700 --> 00:26:19.020 align:middle line:84% BRYNMOR CHAPMAN: One third, yeah. 00:26:19.020 --> 00:26:21.990 align:middle line:90% So one third win, 2/3 loss. 00:26:21.990 --> 00:26:27.000 align:middle line:84% And this will be exactly the opposite, so 2/3 win, 00:26:27.000 --> 00:26:29.910 align:middle line:90% one third loss. 00:26:29.910 --> 00:26:32.010 align:middle line:84% So now we can assign probabilities 00:26:32.010 --> 00:26:35.310 align:middle line:90% to all of the outcomes. 00:26:35.310 --> 00:26:37.145 align:middle line:84% What's the probability of winning twice? 00:26:37.145 --> 00:26:41.460 align:middle line:90% 00:26:41.460 --> 00:26:43.510 align:middle line:84% How do we compute these probabilities? 00:26:43.510 --> 00:26:48.360 align:middle line:90% 00:26:48.360 --> 00:26:49.440 align:middle line:90% Anyone? 00:26:49.440 --> 00:26:49.940 align:middle line:90% Yeah? 00:26:49.940 --> 00:26:54.110 align:middle line:84% AUDIENCE: Why is the probability of winning [INAUDIBLE]? 00:26:54.110 --> 00:26:57.000 align:middle line:84% BRYNMOR CHAPMAN: Oh, so this was just what 00:26:57.000 --> 00:26:59.320 align:middle line:90% we assumed in the problem. 00:26:59.320 --> 00:27:04.020 align:middle line:84% If it's not the first game, you win with probability 2/3 if you 00:27:04.020 --> 00:27:06.450 align:middle line:84% won the last game or with probability one third if you 00:27:06.450 --> 00:27:07.415 align:middle line:90% lost the last game. 00:27:07.415 --> 00:27:11.880 align:middle line:90% 00:27:11.880 --> 00:27:14.610 align:middle line:90% So the first branch is 50-50. 00:27:14.610 --> 00:27:17.210 align:middle line:84% But the subsequent ones are going to be 2/3, one third, 00:27:17.210 --> 00:27:19.502 align:middle line:84% in one direction or the other, depending on the outcome 00:27:19.502 --> 00:27:20.455 align:middle line:90% of the previous game-- 00:27:20.455 --> 00:27:21.830 align:middle line:84% maybe I shouldn't use "outcome"-- 00:27:21.830 --> 00:27:23.705 align:middle line:84% depending on the result of the previous game. 00:27:23.705 --> 00:27:28.160 align:middle line:90% 00:27:28.160 --> 00:27:31.043 align:middle line:84% OK, so how do we compute probabilities of outcomes? 00:27:31.043 --> 00:27:31.960 align:middle line:90% Does anybody remember? 00:27:31.960 --> 00:27:34.845 align:middle line:90% 00:27:34.845 --> 00:27:35.345 align:middle line:90% Yeah? 00:27:35.345 --> 00:27:42.320 align:middle line:84% AUDIENCE: Do you just multiply to that each outcome-- 00:27:42.320 --> 00:27:44.780 align:middle line:84% or, yeah, you multiply the probabilities along the tree 00:27:44.780 --> 00:27:45.560 align:middle line:90% to that outcome. 00:27:45.560 --> 00:27:47.727 align:middle line:84% BRYNMOR CHAPMAN: Yeah, so you multiply probabilities 00:27:47.727 --> 00:27:49.930 align:middle line:90% along paths to outcomes. 00:27:49.930 --> 00:27:53.970 align:middle line:84% So for this top outcome, we've got one half times 2/3. 00:27:53.970 --> 00:27:56.570 align:middle line:84% So this is probability one third. 00:27:56.570 --> 00:27:59.450 align:middle line:84% Win, loss, win has probability one third by one 00:27:59.450 --> 00:28:00.990 align:middle line:90% half by one third. 00:28:00.990 --> 00:28:04.490 align:middle line:90% So that's 1/18. 00:28:04.490 --> 00:28:07.520 align:middle line:84% Win, loss, loss says probability one half times one third times 00:28:07.520 --> 00:28:08.250 align:middle line:90% 2/3. 00:28:08.250 --> 00:28:11.360 align:middle line:90% So that's going to be one ninth. 00:28:11.360 --> 00:28:14.620 align:middle line:90% And then these all reverse-- 00:28:14.620 --> 00:28:19.360 align:middle line:90% 1/18 and one third. 00:28:19.360 --> 00:28:21.460 align:middle line:90% Have I done that correctly? 00:28:21.460 --> 00:28:23.890 align:middle line:90% Yes, that looks fine. 00:28:23.890 --> 00:28:27.640 align:middle line:84% If I make an error, please shout at me. 00:28:27.640 --> 00:28:32.330 align:middle line:84% OK, so we've got our tree diagram. 00:28:32.330 --> 00:28:33.590 align:middle line:90% We've got our sample space. 00:28:33.590 --> 00:28:36.655 align:middle line:84% We've got our probability function. 00:28:36.655 --> 00:28:37.905 align:middle line:90% What's the next thing we need? 00:28:37.905 --> 00:28:42.280 align:middle line:90% 00:28:42.280 --> 00:28:42.930 align:middle line:90% Do we remember? 00:28:42.930 --> 00:29:01.630 align:middle line:90% 00:29:01.630 --> 00:29:14.000 align:middle line:84% So we've got, step 1, tree; step 2, the probabilities; 00:29:14.000 --> 00:29:16.360 align:middle line:90% 3 was the events of interest. 00:29:16.360 --> 00:29:21.774 align:middle line:90% 00:29:21.774 --> 00:29:23.785 align:middle line:84% So what are the events we care about here? 00:29:23.785 --> 00:29:32.270 align:middle line:90% 00:29:32.270 --> 00:29:33.790 align:middle line:90% Yeah? 00:29:33.790 --> 00:29:35.750 align:middle line:84% AUDIENCE: Winning the tournament [INAUDIBLE]. 00:29:35.750 --> 00:29:36.115 align:middle line:90% BRYNMOR CHAPMAN: Pardon? 00:29:36.115 --> 00:29:38.870 align:middle line:84% AUDIENCE: Winning the tournament and winning the first game. 00:29:38.870 --> 00:29:40.662 align:middle line:84% BRYNMOR CHAPMAN: OK, winning the tournament 00:29:40.662 --> 00:29:43.400 align:middle line:90% and winning the first game. 00:29:43.400 --> 00:29:49.910 align:middle line:84% So A was-- yep, A was winning the tournament. 00:29:49.910 --> 00:29:53.330 align:middle line:90% B was winning the first game. 00:29:53.330 --> 00:29:54.055 align:middle line:90% Is that all? 00:29:54.055 --> 00:29:56.413 align:middle line:90% 00:29:56.413 --> 00:29:58.330 align:middle line:84% Are there any other events that we care about? 00:29:58.330 --> 00:30:00.920 align:middle line:90% 00:30:00.920 --> 00:30:01.610 align:middle line:90% Yeah? 00:30:01.610 --> 00:30:03.440 align:middle line:90% AUDIENCE: A intersect B. 00:30:03.440 --> 00:30:04.760 align:middle line:90% BRYNMOR CHAPMAN: Yes, exactly. 00:30:04.760 --> 00:30:12.340 align:middle line:84% So we also want A intersect B. We also care about the event 00:30:12.340 --> 00:30:14.330 align:middle line:84% where you win the first battle and the series. 00:30:14.330 --> 00:30:15.000 align:middle line:90% Why is that? 00:30:15.000 --> 00:30:20.470 align:middle line:90% 00:30:20.470 --> 00:30:21.780 align:middle line:90% Would you like to explain? 00:30:21.780 --> 00:30:23.980 align:middle line:90% AUDIENCE: [INAUDIBLE]. 00:30:23.980 --> 00:30:25.640 align:middle line:90% BRYNMOR CHAPMAN: Yeah, exactly. 00:30:25.640 --> 00:30:28.370 align:middle line:84% So ultimately, we want this conditional probability, 00:30:28.370 --> 00:30:31.990 align:middle line:84% the probability of A given B. And if we go back 00:30:31.990 --> 00:30:36.220 align:middle line:84% to our definition, the probability of A given B 00:30:36.220 --> 00:30:39.130 align:middle line:84% is the probability of A intersect B divided 00:30:39.130 --> 00:30:41.620 align:middle line:90% by the probability of B. 00:30:41.620 --> 00:30:44.170 align:middle line:84% So as it turns out, we don't really 00:30:44.170 --> 00:30:49.120 align:middle line:84% need A. The two probabilities we really need are these two, 00:30:49.120 --> 00:30:50.475 align:middle line:90% are the two events I suppose. 00:30:50.475 --> 00:30:53.400 align:middle line:90% 00:30:53.400 --> 00:30:58.390 align:middle line:90% So finally-- 4, answer. 00:30:58.390 --> 00:31:00.610 align:middle line:90% What is our answer? 00:31:00.610 --> 00:31:05.540 align:middle line:84% Well, probability of A given B, as we just said, 00:31:05.540 --> 00:31:09.840 align:middle line:84% is the probability of A intersect B divided 00:31:09.840 --> 00:31:17.140 align:middle line:84% by probability of B. What are each of these probabilities? 00:31:17.140 --> 00:31:19.710 align:middle line:90% 00:31:19.710 --> 00:31:22.290 align:middle line:84% Well, which events-- or sorry, which outcomes 00:31:22.290 --> 00:31:23.780 align:middle line:90% are in the event A intersect B? 00:31:23.780 --> 00:31:32.520 align:middle line:90% 00:31:32.520 --> 00:31:37.200 align:middle line:84% It's W, W in A intersect B. Yeah? 00:31:37.200 --> 00:31:40.840 align:middle line:90% Win, loss, win-- also yes. 00:31:40.840 --> 00:31:43.920 align:middle line:90% Win, loss, loss-- nope. 00:31:43.920 --> 00:31:45.940 align:middle line:84% And all of these, he lost the first game. 00:31:45.940 --> 00:31:47.380 align:middle line:90% So these are also not in it. 00:31:47.380 --> 00:31:52.120 align:middle line:84% So we care about these two outcomes. 00:31:52.120 --> 00:32:00.570 align:middle line:84% So this is going to be, by the sum rule, one third plus 1/18. 00:32:00.570 --> 00:32:03.310 align:middle line:84% So that doesn't look much like an 8. 00:32:03.310 --> 00:32:03.810 align:middle line:90% Oh, well. 00:32:03.810 --> 00:32:06.320 align:middle line:90% 00:32:06.320 --> 00:32:08.350 align:middle line:90% Now what's the probability of B? 00:32:08.350 --> 00:32:16.020 align:middle line:90% 00:32:16.020 --> 00:32:16.520 align:middle line:90% Yeah? 00:32:16.520 --> 00:32:17.312 align:middle line:90% AUDIENCE: One half. 00:32:17.312 --> 00:32:20.640 align:middle line:84% BRYNMOR CHAPMAN: Yeah, it's just one half, either by symmetry, 00:32:20.640 --> 00:32:22.860 align:middle line:84% or you can actually look at what those outcomes are. 00:32:22.860 --> 00:32:25.855 align:middle line:90% It's these top three. 00:32:25.855 --> 00:32:34.515 align:middle line:90% 00:32:34.515 --> 00:32:38.800 align:middle line:84% And if we do this arithmetic, we've got 7/18 divided by one 00:32:38.800 --> 00:32:40.163 align:middle line:90% half, which is 7/9. 00:32:40.163 --> 00:32:46.160 align:middle line:90% 00:32:46.160 --> 00:32:49.580 align:middle line:84% So the probability that Ash wins the entire series given that he 00:32:49.580 --> 00:32:51.813 align:middle line:84% won the first battle is going to be 7/9. 00:32:51.813 --> 00:32:54.668 align:middle line:90% 00:32:54.668 --> 00:32:58.250 align:middle line:84% So far this is pretty straightforward, fairly routine 00:32:58.250 --> 00:33:00.006 align:middle line:90% mathematical-- well, arithmetic. 00:33:00.006 --> 00:33:03.260 align:middle line:90% 00:33:03.260 --> 00:33:08.245 align:middle line:84% But what if, instead, we asked for the probability of B given 00:33:08.245 --> 00:33:08.745 align:middle line:90% A? 00:33:08.745 --> 00:33:31.720 align:middle line:90% 00:33:31.720 --> 00:33:33.176 align:middle line:90% What is that probability? 00:33:33.176 --> 00:33:44.860 align:middle line:90% 00:33:44.860 --> 00:33:46.840 align:middle line:84% Well, numerically, we can compute it 00:33:46.840 --> 00:33:48.000 align:middle line:90% in exactly the same way. 00:33:48.000 --> 00:33:50.500 align:middle line:84% It's going to be the probability of the intersection divided 00:33:50.500 --> 00:33:51.958 align:middle line:84% by, this time, the probability of A 00:33:51.958 --> 00:33:53.890 align:middle line:84% rather than the probability of B. 00:33:53.890 --> 00:33:56.230 align:middle line:84% But still, by symmetry, that probability 00:33:56.230 --> 00:33:58.160 align:middle line:90% is going to be one half. 00:33:58.160 --> 00:33:59.593 align:middle line:90% So this is also going to be 7/9. 00:33:59.593 --> 00:34:04.440 align:middle line:90% 00:34:04.440 --> 00:34:06.013 align:middle line:90% But what exactly does it mean? 00:34:06.013 --> 00:34:13.260 align:middle line:90% 00:34:13.260 --> 00:34:13.830 align:middle line:90% Yeah? 00:34:13.830 --> 00:34:16.199 align:middle line:90% AUDIENCE: The probability that-- 00:34:16.199 --> 00:34:17.620 align:middle line:90% he already won the series. 00:34:17.620 --> 00:34:19.305 align:middle line:84% What was the probability that he won the first battle? 00:34:19.305 --> 00:34:21.120 align:middle line:84% BRYNMOR CHAPMAN: Yeah, so the answer was, 00:34:21.120 --> 00:34:23.620 align:middle line:84% given that he already won the series, what's the probability 00:34:23.620 --> 00:34:26.790 align:middle line:90% that he won the first battle? 00:34:26.790 --> 00:34:32.380 align:middle line:84% Now, this might sound a little bit weird to some of you. 00:34:32.380 --> 00:34:35.139 align:middle line:84% Like, if we know that he's won the series, 00:34:35.139 --> 00:34:36.489 align:middle line:90% everything's already happened. 00:34:36.489 --> 00:34:39.989 align:middle line:84% He either won the battle or he didn't. 00:34:39.989 --> 00:34:44.040 align:middle line:84% But when we're modeling probability questions, 00:34:44.040 --> 00:34:48.389 align:middle line:84% we don't really care about any temporal relations or anything 00:34:48.389 --> 00:34:50.539 align:middle line:90% like that, or even causal ones. 00:34:50.539 --> 00:34:56.560 align:middle line:90% 00:34:56.560 --> 00:34:59.260 align:middle line:90% It's all just numerical. 00:34:59.260 --> 00:35:02.930 align:middle line:84% You can think of it a bit like an inference. 00:35:02.930 --> 00:35:05.612 align:middle line:90% You've got some uncertainty. 00:35:05.612 --> 00:35:08.600 align:middle line:84% Maybe you're told that he won the battle. 00:35:08.600 --> 00:35:10.590 align:middle line:84% Or, sorry, you're told that he won the series. 00:35:10.590 --> 00:35:14.330 align:middle line:84% You want to update your belief that maybe he won the first 00:35:14.330 --> 00:35:16.280 align:middle line:90% battle, maybe he didn't. 00:35:16.280 --> 00:35:19.070 align:middle line:84% And maybe you're not entirely confident in it. 00:35:19.070 --> 00:35:23.408 align:middle line:84% But you'd like to estimate how confident you should be. 00:35:23.408 --> 00:35:25.075 align:middle line:84% That might be one way to think about it. 00:35:25.075 --> 00:35:27.890 align:middle line:90% 00:35:27.890 --> 00:35:31.250 align:middle line:84% So in a sense, it's like looking at a backwards probability, 00:35:31.250 --> 00:35:34.100 align:middle line:84% that something in the past has happened given that we observed 00:35:34.100 --> 00:35:37.040 align:middle line:90% something in the future. 00:35:37.040 --> 00:35:43.190 align:middle line:84% So in general, they won't always be the same like this. 00:35:43.190 --> 00:35:45.500 align:middle line:84% You can relate the probability of B 00:35:45.500 --> 00:35:47.490 align:middle line:84% given A to the probability of A given B. 00:35:47.490 --> 00:35:50.360 align:middle line:90% They won't always be equal. 00:35:50.360 --> 00:35:51.715 align:middle line:90% Let's see how to relate them. 00:35:51.715 --> 00:35:55.550 align:middle line:90% 00:35:55.550 --> 00:35:59.963 align:middle line:84% It just so happens that in this case, these two things-- 00:35:59.963 --> 00:36:03.220 align:middle line:90% 00:36:03.220 --> 00:36:07.030 align:middle line:84% these two things had the same probability. 00:36:07.030 --> 00:36:10.510 align:middle line:84% So that's really the core reason why they were equal. 00:36:10.510 --> 00:36:13.580 align:middle line:84% But more generally, we have the following. 00:36:13.580 --> 00:36:19.780 align:middle line:90% 00:36:19.780 --> 00:36:20.905 align:middle line:90% This is called Bayes' rule. 00:36:20.905 --> 00:36:26.140 align:middle line:90% 00:36:26.140 --> 00:36:32.180 align:middle line:84% You can think of it as it's just the product rule in disguise. 00:36:32.180 --> 00:36:40.750 align:middle line:84% So the probability of B given A equals the probability-- 00:36:40.750 --> 00:36:44.920 align:middle line:84% oops, why don't I split this into two lines-- 00:36:44.920 --> 00:36:52.180 align:middle line:84% probability of A given B, probability of B 00:36:52.180 --> 00:36:57.140 align:middle line:90% divided by probability of A. 00:36:57.140 --> 00:36:59.980 align:middle line:84% So in the example we just saw, these two are the same. 00:36:59.980 --> 00:37:02.760 align:middle line:90% So they just cancel. 00:37:02.760 --> 00:37:06.990 align:middle line:84% But more generally, you want to multiply by the quotient. 00:37:06.990 --> 00:37:10.928 align:middle line:84% And you can think of this as the product rule in disguise. 00:37:10.928 --> 00:37:12.720 align:middle line:84% Like if you just move this probability of A 00:37:12.720 --> 00:37:16.957 align:middle line:84% over to this side, now you're just 00:37:16.957 --> 00:37:19.290 align:middle line:84% computing the probability of the intersection of A and B 00:37:19.290 --> 00:37:20.530 align:middle line:90% on both sides. 00:37:20.530 --> 00:37:23.280 align:middle line:90% Does that make sense? 00:37:23.280 --> 00:37:26.720 align:middle line:90% But it can often be useful-- 00:37:26.720 --> 00:37:28.720 align:middle line:84% even though it's not really saying anything new, 00:37:28.720 --> 00:37:31.140 align:middle line:84% it can often be useful to think about it in this way. 00:37:31.140 --> 00:37:35.610 align:middle line:84% If you want to reverse the direction of an inference, 00:37:35.610 --> 00:37:38.190 align:middle line:84% you want to multiply by the quotient of the two 00:37:38.190 --> 00:37:40.190 align:middle line:84% probabilities of the events that you care about. 00:37:40.190 --> 00:37:45.672 align:middle line:90% 00:37:45.672 --> 00:37:51.180 align:middle line:84% And one particularly useful consequence of this 00:37:51.180 --> 00:37:52.530 align:middle line:90% is the following. 00:37:52.530 --> 00:37:56.970 align:middle line:84% So suppose we have two events, B and C. 00:37:56.970 --> 00:38:00.470 align:middle line:84% And we're trying to condition them both on the same event. 00:38:00.470 --> 00:38:04.010 align:middle line:90% 00:38:04.010 --> 00:38:10.890 align:middle line:84% So what is the probability of C given A-- 00:38:10.890 --> 00:38:16.775 align:middle line:84% or B given A divided by the probability of C given A? 00:38:16.775 --> 00:38:21.780 align:middle line:90% 00:38:21.780 --> 00:38:23.550 align:middle line:84% And in particular, often we would 00:38:23.550 --> 00:38:26.410 align:middle line:84% take C to be the complement of B or something like that. 00:38:26.410 --> 00:38:29.910 align:middle line:84% And that will then help us to compute this probability of B 00:38:29.910 --> 00:38:35.100 align:middle line:84% given A. So as it turns out, this 00:38:35.100 --> 00:38:36.840 align:middle line:84% is going to be the probability of A given 00:38:36.840 --> 00:38:43.590 align:middle line:84% B times the probability of B divided 00:38:43.590 --> 00:38:49.470 align:middle line:90% by probability of A given C-- 00:38:49.470 --> 00:39:04.630 align:middle line:90% oops-- probability of C. 00:39:04.630 --> 00:39:06.470 align:middle line:84% So does that make sense to everybody? 00:39:06.470 --> 00:39:14.200 align:middle line:90% 00:39:14.200 --> 00:39:18.610 align:middle line:84% And the reason that this in particular might be useful-- 00:39:18.610 --> 00:39:20.770 align:middle line:84% in order to compute the probability of B, 00:39:20.770 --> 00:39:27.070 align:middle line:84% if we use B complement to C, this right-hand side 00:39:27.070 --> 00:39:29.320 align:middle line:84% here does not depend on the probability 00:39:29.320 --> 00:39:32.300 align:middle line:84% of A. We don't have to compute the probability of A. 00:39:32.300 --> 00:39:34.972 align:middle line:84% So maybe if A is some really nasty event 00:39:34.972 --> 00:39:37.180 align:middle line:84% and we don't actually want to compute its probability 00:39:37.180 --> 00:39:39.200 align:middle line:84% but we can still condition on it, 00:39:39.200 --> 00:39:41.950 align:middle line:84% we might still be able to use this formula more easily than we 00:39:41.950 --> 00:39:44.020 align:middle line:90% could use this one. 00:39:44.020 --> 00:39:46.100 align:middle line:84% Does that make sense to everybody? 00:39:46.100 --> 00:39:51.312 align:middle line:90% 00:39:51.312 --> 00:39:57.830 align:middle line:84% So let's take a look at a couple of applications of Bayes' rule. 00:39:57.830 --> 00:40:02.760 align:middle line:90% 00:40:02.760 --> 00:40:07.710 align:middle line:84% Or actually, first-- mm, first, maybe I 00:40:07.710 --> 00:40:11.820 align:middle line:84% should give you a brief lesson in statistics. 00:40:11.820 --> 00:40:24.480 align:middle line:84% So terminology-- the probability of A given B, 00:40:24.480 --> 00:40:41.540 align:middle line:84% this would often be called the likelihood of A given B. 00:40:41.540 --> 00:40:45.030 align:middle line:84% So if any of you have ever seen maximum likelihood estimations 00:40:45.030 --> 00:40:47.580 align:middle line:84% or anything like that, that's where this comes from. 00:40:47.580 --> 00:40:49.940 align:middle line:84% Or this is where that comes from, sorry. 00:40:49.940 --> 00:40:52.540 align:middle line:90% 00:40:52.540 --> 00:41:13.540 align:middle line:84% Probability of B we call the prior probability of B. 00:41:13.540 --> 00:41:20.230 align:middle line:84% And the left-hand side, the probability of B given A, 00:41:20.230 --> 00:41:38.050 align:middle line:84% is the posterior of B. Actually, we don't usually 00:41:38.050 --> 00:41:39.485 align:middle line:90% specify it with that. 00:41:39.485 --> 00:41:39.985 align:middle line:90% Oops. 00:41:39.985 --> 00:41:44.200 align:middle line:90% 00:41:44.200 --> 00:41:45.790 align:middle line:84% So this is how you would normally 00:41:45.790 --> 00:41:49.260 align:middle line:84% talk about it in statistics or something like that-- 00:41:49.260 --> 00:41:52.630 align:middle line:84% the likelihood of A given B, the prior probability, 00:41:52.630 --> 00:41:54.870 align:middle line:84% and the posterior probability-- because basically 00:41:54.870 --> 00:41:58.510 align:middle line:84% what you're doing is you're starting with some prior. 00:41:58.510 --> 00:42:00.360 align:middle line:84% If you don't know anything, this is 00:42:00.360 --> 00:42:03.870 align:middle line:84% what you think the probability of B should be. 00:42:03.870 --> 00:42:08.890 align:middle line:84% And now you make some observation A. 00:42:08.890 --> 00:42:10.650 align:middle line:84% And so now you update, and you get 00:42:10.650 --> 00:42:15.940 align:middle line:84% a posterior probability of B conditioned on that observation. 00:42:15.940 --> 00:42:16.850 align:middle line:90% Does that make sense? 00:42:16.850 --> 00:42:22.920 align:middle line:90% 00:42:22.920 --> 00:42:29.075 align:middle line:84% OK, so why don't we take a look at a couple of examples then? 00:42:29.075 --> 00:42:37.560 align:middle line:90% 00:42:37.560 --> 00:42:39.313 align:middle line:84% Hopefully everybody is OK with those. 00:42:39.313 --> 00:42:43.500 align:middle line:90% 00:42:43.500 --> 00:42:46.665 align:middle line:84% So first example, suppose I've got two coins. 00:42:46.665 --> 00:42:49.280 align:middle line:90% 00:42:49.280 --> 00:42:50.820 align:middle line:90% I've got one fair coin. 00:42:50.820 --> 00:42:52.850 align:middle line:84% I toss this coin, and it'll come up heads 00:42:52.850 --> 00:42:56.920 align:middle line:84% with 50% probability, tails with 50% probability. 00:42:56.920 --> 00:42:58.620 align:middle line:90% And I've also got a trick coin. 00:42:58.620 --> 00:43:01.070 align:middle line:90% It's double heads. 00:43:01.070 --> 00:43:02.350 align:middle line:90% I flip it, I always get heads. 00:43:02.350 --> 00:43:06.236 align:middle line:90% 00:43:06.236 --> 00:43:09.650 align:middle line:84% Suppose I pick one of these coins uniformly at random, 00:43:09.650 --> 00:43:13.495 align:middle line:84% I flip that coin, and it comes up heads. 00:43:13.495 --> 00:43:16.895 align:middle line:90% 00:43:16.895 --> 00:43:19.280 align:middle line:84% What is the probability that I picked the fair coin 00:43:19.280 --> 00:43:21.475 align:middle line:84% to begin with, as opposed to the biased one? 00:43:21.475 --> 00:43:27.080 align:middle line:90% 00:43:27.080 --> 00:43:28.095 align:middle line:90% Any ideas? 00:43:28.095 --> 00:43:30.595 align:middle line:84% Does anybody have any guesses as to what the probability is? 00:43:30.595 --> 00:43:36.470 align:middle line:90% 00:43:36.470 --> 00:43:40.100 align:middle line:84% Who wants to just draw the tree diagram? 00:43:40.100 --> 00:43:40.800 align:middle line:90% A few people. 00:43:40.800 --> 00:43:42.650 align:middle line:84% People aren't as enthusiastic about it 00:43:42.650 --> 00:43:43.890 align:middle line:90% as they were on Tuesday. 00:43:43.890 --> 00:43:46.170 align:middle line:90% That's slightly unfortunate. 00:43:46.170 --> 00:43:48.100 align:middle line:90% Let's do it anyway. 00:43:48.100 --> 00:43:54.180 align:middle line:84% Now, let's make precise the events that we care about. 00:43:54.180 --> 00:44:01.545 align:middle line:84% So let's let H be the event that we flipped heads. 00:44:01.545 --> 00:44:06.280 align:middle line:90% 00:44:06.280 --> 00:44:13.185 align:middle line:84% And F is the event picked the fair coin. 00:44:13.185 --> 00:44:23.506 align:middle line:90% 00:44:23.506 --> 00:44:25.390 align:middle line:84% So what is the probability that we're 00:44:25.390 --> 00:44:32.170 align:middle line:84% trying to compute in terms of these two quantities? 00:44:32.170 --> 00:44:33.280 align:middle line:90% Yeah? 00:44:33.280 --> 00:44:35.590 align:middle line:84% AUDIENCE: Probability of F given H. 00:44:35.590 --> 00:44:37.950 align:middle line:84% BRYNMOR CHAPMAN: Probability of F given H, yes. 00:44:37.950 --> 00:44:44.370 align:middle line:90% 00:44:44.370 --> 00:44:49.740 align:middle line:84% OK, so how might we compute this? 00:44:49.740 --> 00:44:51.450 align:middle line:84% People didn't seem too enthusiastic 00:44:51.450 --> 00:44:52.510 align:middle line:90% about the tree diagram. 00:44:52.510 --> 00:44:53.885 align:middle line:84% So what else could we do instead? 00:44:53.885 --> 00:45:02.760 align:middle line:90% 00:45:02.760 --> 00:45:04.535 align:middle line:84% Does anybody see a way to use Bayes' rule? 00:45:04.535 --> 00:45:08.760 align:middle line:90% 00:45:08.760 --> 00:45:09.780 align:middle line:90% Yeah? 00:45:09.780 --> 00:45:11.080 align:middle line:90% Or just a stretch? 00:45:11.080 --> 00:45:11.580 align:middle line:90% Sorry. 00:45:11.580 --> 00:45:13.044 align:middle line:90% [CHUCKLES] 00:45:13.044 --> 00:45:16.935 align:middle line:90% 00:45:16.935 --> 00:45:17.435 align:middle line:90% Yeah? 00:45:17.435 --> 00:45:21.824 align:middle line:84% AUDIENCE: The probability of H [INAUDIBLE]. 00:45:21.824 --> 00:45:24.510 align:middle line:84% BRYNMOR CHAPMAN: Yeah, the probability of H given F 00:45:24.510 --> 00:45:25.450 align:middle line:90% is easy. 00:45:25.450 --> 00:45:27.270 align:middle line:90% That's 50%. 00:45:27.270 --> 00:45:30.850 align:middle line:84% So that's an easy known quantity. 00:45:30.850 --> 00:45:35.000 align:middle line:84% So if we use Bayes' rule, that might make it easier to compute. 00:45:35.000 --> 00:45:37.550 align:middle line:90% 00:45:37.550 --> 00:45:43.730 align:middle line:84% So actually, why don't we use the slightly more complicated 00:45:43.730 --> 00:45:45.180 align:middle line:90% version of Bayes' rule? 00:45:45.180 --> 00:45:47.810 align:middle line:84% So let's compute the ratio of the probability 00:45:47.810 --> 00:45:50.270 align:middle line:84% that we picked the fair coin given that we observed 00:45:50.270 --> 00:45:56.060 align:middle line:84% a heads versus the probability of picking the biased coin given 00:45:56.060 --> 00:45:59.270 align:middle line:90% that we observed a heads. 00:45:59.270 --> 00:46:09.230 align:middle line:84% So the probability of F given H over probability of not F 00:46:09.230 --> 00:46:15.235 align:middle line:90% given H. So what was that? 00:46:15.235 --> 00:46:32.810 align:middle line:90% 00:46:32.810 --> 00:46:37.380 align:middle line:84% So Bayes' rule is about flipping the order. 00:46:37.380 --> 00:46:39.380 align:middle line:84% So instead of F conditioned on H, 00:46:39.380 --> 00:46:44.760 align:middle line:84% we want H conditioned on F, multiplied by what? 00:46:44.760 --> 00:46:51.250 align:middle line:90% 00:46:51.250 --> 00:46:51.750 align:middle line:90% Yeah? 00:46:51.750 --> 00:46:52.780 align:middle line:90% AUDIENCE: Probability of F. 00:46:52.780 --> 00:46:54.447 align:middle line:84% BRYNMOR CHAPMAN: Yeah, probability of f. 00:46:54.447 --> 00:46:58.600 align:middle line:90% 00:46:58.600 --> 00:47:01.090 align:middle line:84% And on the bottom, we can do the same thing. 00:47:01.090 --> 00:47:15.796 align:middle line:84% Probability of H given not F times the probability of not F. 00:47:15.796 --> 00:47:19.750 align:middle line:84% So this expression is much easier to work with. 00:47:19.750 --> 00:47:21.480 align:middle line:84% What's the probability of H given F? 00:47:21.480 --> 00:47:26.710 align:middle line:90% 00:47:26.710 --> 00:47:27.490 align:middle line:90% Yep? 00:47:27.490 --> 00:47:28.292 align:middle line:90% AUDIENCE: One half. 00:47:28.292 --> 00:47:29.625 align:middle line:90% BRYNMOR CHAPMAN: One half, yeah. 00:47:29.625 --> 00:47:33.970 align:middle line:90% 00:47:33.970 --> 00:47:35.970 align:middle line:84% What about the probability of H given not F? 00:47:35.970 --> 00:47:38.530 align:middle line:90% 00:47:38.530 --> 00:47:39.100 align:middle line:90% Yeah? 00:47:39.100 --> 00:47:39.870 align:middle line:90% AUDIENCE: One. 00:47:39.870 --> 00:47:40.745 align:middle line:90% BRYNMOR CHAPMAN: One. 00:47:40.745 --> 00:47:43.170 align:middle line:90% 00:47:43.170 --> 00:47:45.545 align:middle line:90% And probability of F and not F? 00:47:45.545 --> 00:47:50.460 align:middle line:90% 00:47:50.460 --> 00:47:52.650 align:middle line:84% Those are each one half by assumption. 00:47:52.650 --> 00:47:58.060 align:middle line:90% 00:47:58.060 --> 00:47:59.760 align:middle line:90% So those just cancel. 00:47:59.760 --> 00:48:02.835 align:middle line:90% And we are left with one half. 00:48:02.835 --> 00:48:07.405 align:middle line:90% 00:48:07.405 --> 00:48:11.290 align:middle line:84% So now, what is this probability? 00:48:11.290 --> 00:48:22.590 align:middle line:90% 00:48:22.590 --> 00:48:26.226 align:middle line:84% We computed the ratio of this probability to its complement. 00:48:26.226 --> 00:48:29.040 align:middle line:90% 00:48:29.040 --> 00:48:31.595 align:middle line:84% So what does that tell us the probability is? 00:48:31.595 --> 00:48:36.240 align:middle line:90% 00:48:36.240 --> 00:48:36.777 align:middle line:90% Yeah? 00:48:36.777 --> 00:48:37.610 align:middle line:90% AUDIENCE: One third. 00:48:37.610 --> 00:48:38.985 align:middle line:84% BRYNMOR CHAPMAN: One third, yeah. 00:48:38.985 --> 00:48:55.520 align:middle line:90% 00:48:55.520 --> 00:49:02.792 align:middle line:84% So observe here that if our prior changes, 00:49:02.792 --> 00:49:06.920 align:middle line:84% if the chance that we are picking the fair coin to begin 00:49:06.920 --> 00:49:12.310 align:middle line:84% with changes, this posterior probability also changes. 00:49:12.310 --> 00:49:15.626 align:middle line:90% 00:49:15.626 --> 00:49:23.738 align:middle line:84% So if I originally pick the fair coin with very high probability 00:49:23.738 --> 00:49:28.760 align:middle line:84% and then I observe a heads, do you think the probability 00:49:28.760 --> 00:49:32.090 align:middle line:84% that the posterior probability should 00:49:32.090 --> 00:49:33.595 align:middle line:90% be the same, higher, or lower? 00:49:33.595 --> 00:49:41.445 align:middle line:90% 00:49:41.445 --> 00:49:42.700 align:middle line:90% AUDIENCE: Just a question. 00:49:42.700 --> 00:49:43.400 align:middle line:90% BRYNMOR CHAPMAN: Oh, yeah? 00:49:43.400 --> 00:49:44.065 align:middle line:84% AUDIENCE: How did you get a third? 00:49:44.065 --> 00:49:45.620 align:middle line:90% BRYNMOR CHAPMAN: Oh, question. 00:49:45.620 --> 00:49:47.860 align:middle line:90% How did we get a third? 00:49:47.860 --> 00:49:52.490 align:middle line:84% So the ratio of this to its complement is one half. 00:49:52.490 --> 00:49:53.842 align:middle line:90% They sum to 1. 00:49:53.842 --> 00:49:55.050 align:middle line:90% So they're one third and 2/3. 00:49:55.050 --> 00:50:02.110 align:middle line:90% 00:50:02.110 --> 00:50:03.330 align:middle line:90% Yeah? 00:50:03.330 --> 00:50:06.580 align:middle line:84% AUDIENCE: Can you write the numerical values of H and F? 00:50:06.580 --> 00:50:11.440 align:middle line:84% BRYNMOR CHAPMAN: The numerical values of H and F. 00:50:11.440 --> 00:50:15.970 align:middle line:90% So H and F are events. 00:50:15.970 --> 00:50:18.950 align:middle line:90% They're not numbers exactly. 00:50:18.950 --> 00:50:24.190 align:middle line:84% So H is the event that whichever coin I flipped, 00:50:24.190 --> 00:50:27.676 align:middle line:90% it got a heads on it. 00:50:27.676 --> 00:50:33.100 align:middle line:84% And F is the event that the coin I picked was the fair one. 00:50:33.100 --> 00:50:35.290 align:middle line:90% So they're not really numbers. 00:50:35.290 --> 00:50:40.230 align:middle line:84% They're just sets of outcomes that could have happened. 00:50:40.230 --> 00:50:40.730 align:middle line:90% Yeah? 00:50:40.730 --> 00:50:41.850 align:middle line:84% AUDIENCE: [INAUDIBLE] the assumptions. 00:50:41.850 --> 00:50:43.183 align:middle line:90% Can you go over the assumptions? 00:50:43.183 --> 00:50:45.474 align:middle line:84% BRYNMOR CHAPMAN: Oh, the assumptions. 00:50:45.474 --> 00:51:00.455 align:middle line:84% So pick fair or biased uniformly, and then flip it. 00:51:00.455 --> 00:51:03.450 align:middle line:90% 00:51:03.450 --> 00:51:07.450 align:middle line:84% So "fair" means it comes up with 50/50 heads or tails. 00:51:07.450 --> 00:51:09.200 align:middle line:90% Biased, it's a two-headed coin. 00:51:09.200 --> 00:51:15.600 align:middle line:90% 00:51:15.600 --> 00:51:16.450 align:middle line:90% Yeah? 00:51:16.450 --> 00:51:17.520 align:middle line:84% AUDIENCE: Could you re-explain where 00:51:17.520 --> 00:51:18.665 align:middle line:90% the one third came from again? 00:51:18.665 --> 00:51:21.207 align:middle line:84% BRYNMOR CHAPMAN: Explain where the one third came from again. 00:51:21.207 --> 00:51:28.080 align:middle line:84% So the probability of F given H over not F given H, these 00:51:28.080 --> 00:51:29.710 align:middle line:90% are complementary probabilities. 00:51:29.710 --> 00:51:31.500 align:middle line:90% They should sum to 1. 00:51:31.500 --> 00:51:37.845 align:middle line:84% So we've figured that the ratio right is one half. 00:51:37.845 --> 00:51:40.470 align:middle line:90% 00:51:40.470 --> 00:51:45.050 align:middle line:84% So if we want them to sum to 1, then they should be one third 00:51:45.050 --> 00:51:47.420 align:middle line:90% and 2/3. 00:51:47.420 --> 00:51:49.580 align:middle line:90% Does that make sense? 00:51:49.580 --> 00:51:51.380 align:middle line:90% Yeah. 00:51:51.380 --> 00:51:53.810 align:middle line:90% OK. 00:51:53.810 --> 00:51:54.590 align:middle line:90% Yeah? 00:51:54.590 --> 00:51:57.720 align:middle line:84% AUDIENCE: [INAUDIBLE] do we care which direction it's biased, 00:51:57.720 --> 00:51:58.220 align:middle line:90% or-- 00:51:58.220 --> 00:51:59.178 align:middle line:90% BRYNMOR CHAPMAN: Sorry? 00:51:59.178 --> 00:52:01.160 align:middle line:84% AUDIENCE: If it's biased [INAUDIBLE] 00:52:01.160 --> 00:52:02.370 align:middle line:90% do we care [INAUDIBLE]? 00:52:02.370 --> 00:52:03.840 align:middle line:90% BRYNMOR CHAPMAN: Oh, question. 00:52:03.840 --> 00:52:06.710 align:middle line:84% If it's biased, do we care in which direction it's biased? 00:52:06.710 --> 00:52:07.485 align:middle line:90% Yes, absolutely. 00:52:07.485 --> 00:52:11.450 align:middle line:84% [CHUCKLES] So if it's a double-tailed coin instead 00:52:11.450 --> 00:52:15.060 align:middle line:84% of a double-heads coin, if I observe a heads, 00:52:15.060 --> 00:52:18.890 align:middle line:84% it must be the fair coin because the double-tails coin cannot get 00:52:18.890 --> 00:52:20.270 align:middle line:90% a heads. 00:52:20.270 --> 00:52:22.000 align:middle line:84% So then my probability is just 1. 00:52:22.000 --> 00:52:33.630 align:middle line:90% 00:52:33.630 --> 00:52:36.670 align:middle line:84% OK, are people reasonably happy with this example? 00:52:36.670 --> 00:52:38.045 align:middle line:84% Shall we move on to a second one? 00:52:38.045 --> 00:52:41.880 align:middle line:90% 00:52:41.880 --> 00:52:44.950 align:middle line:90% OK, second example-- 00:52:44.950 --> 00:52:46.180 align:middle line:90% COVID testing. 00:52:46.180 --> 00:52:49.920 align:middle line:84% Woo-hoo, everybody loves COVID testing, right? 00:52:49.920 --> 00:52:51.870 align:middle line:84% Oh, I suppose you-- most of you probably 00:52:51.870 --> 00:52:55.680 align:middle line:84% weren't here when there were those daily or near-daily nasal 00:52:55.680 --> 00:52:57.420 align:middle line:90% swabs, right? 00:52:57.420 --> 00:52:58.725 align:middle line:90% Well, those were fun times. 00:52:58.725 --> 00:53:01.800 align:middle line:90% 00:53:01.800 --> 00:53:04.120 align:middle line:84% So in the next couple of examples, 00:53:04.120 --> 00:53:09.210 align:middle line:84% we're going to see some rather counterintuitive behavior that 00:53:09.210 --> 00:53:12.163 align:middle line:90% arises from Bayes' rule. 00:53:12.163 --> 00:53:16.540 align:middle line:84% So just as we were discussing last time, 00:53:16.540 --> 00:53:20.310 align:middle line:84% you want to throw away intuitive reasoning 00:53:20.310 --> 00:53:24.220 align:middle line:90% and just go with the numbers. 00:53:24.220 --> 00:53:35.380 align:middle line:90% So suppose 10% of MIT has COVID. 00:53:35.380 --> 00:53:42.446 align:middle line:90% 00:53:42.446 --> 00:53:46.296 align:middle line:84% And everybody has to take a COVID test. 00:53:46.296 --> 00:54:01.100 align:middle line:84% They have a false-positive rate of 30% 00:54:01.100 --> 00:54:14.020 align:middle line:90% and false-negative rate of 10%. 00:54:14.020 --> 00:54:18.830 align:middle line:84% So false-positive rate means if I am healthy 00:54:18.830 --> 00:54:22.130 align:middle line:84% and I take a COVID test, there's a 30% chance 00:54:22.130 --> 00:54:23.630 align:middle line:84% that it's going to be wrong and it's 00:54:23.630 --> 00:54:26.330 align:middle line:90% going to say that I have COVID. 00:54:26.330 --> 00:54:30.720 align:middle line:84% So it's falsely reporting a positive result. 00:54:30.720 --> 00:54:32.390 align:middle line:84% And false negative is the opposite. 00:54:32.390 --> 00:54:34.790 align:middle line:84% So if I'm sick and I take a COVID test, 00:54:34.790 --> 00:54:37.700 align:middle line:84% there is a 10% chance that it does not detect it, 00:54:37.700 --> 00:54:39.790 align:middle line:90% and it reports that I'm healthy. 00:54:39.790 --> 00:54:42.650 align:middle line:90% Does this setup make sense? 00:54:42.650 --> 00:54:43.150 align:middle line:90% Yeah? 00:54:43.150 --> 00:54:45.490 align:middle line:84% AUDIENCE: Are the false positives included in the 10%? 00:54:45.490 --> 00:54:47.300 align:middle line:90% Or is the 10% the actual-- 00:54:47.300 --> 00:54:49.700 align:middle line:84% BRYNMOR CHAPMAN: Oh, question was, 00:54:49.700 --> 00:54:51.710 align:middle line:84% are the false positives included in the 10%? 00:54:51.710 --> 00:54:53.920 align:middle line:84% Or is the 10% when you actually have it? 00:54:53.920 --> 00:54:57.280 align:middle line:84% So this 10% is conditioned on being sick. 00:54:57.280 --> 00:55:01.990 align:middle line:84% The conditional probability that it gets it wrong is 10%. 00:55:01.990 --> 00:55:04.690 align:middle line:84% And the false-positive rate is conditioned on being-- 00:55:04.690 --> 00:55:07.430 align:middle line:84% oh, sorry, false negative is conditioned on being healthy. 00:55:07.430 --> 00:55:09.940 align:middle line:84% False positive is conditioned on being sick. 00:55:09.940 --> 00:55:11.830 align:middle line:90% No. 00:55:11.830 --> 00:55:12.823 align:middle line:90% No, positive is sick. 00:55:12.823 --> 00:55:14.240 align:middle line:84% Yeah, it was right the first time. 00:55:14.240 --> 00:55:14.770 align:middle line:90% Sorry. 00:55:14.770 --> 00:55:17.470 align:middle line:90% I'm confused. 00:55:17.470 --> 00:55:21.170 align:middle line:84% Yeah, so these are both talking about conditional probabilities. 00:55:21.170 --> 00:55:22.730 align:middle line:84% Conditioned on being healthy or sick, 00:55:22.730 --> 00:55:25.147 align:middle line:84% what's the probability that the test is accurate or wrong? 00:55:25.147 --> 00:55:29.390 align:middle line:90% 00:55:29.390 --> 00:55:32.070 align:middle line:90% So let's define some events. 00:55:32.070 --> 00:55:35.193 align:middle line:90% Let's say that H is healthy. 00:55:35.193 --> 00:55:40.080 align:middle line:90% 00:55:40.080 --> 00:55:42.016 align:middle line:90% And S can be its complement-- 00:55:42.016 --> 00:55:44.890 align:middle line:90% 00:55:44.890 --> 00:55:45.390 align:middle line:90% sick. 00:55:45.390 --> 00:55:48.180 align:middle line:90% 00:55:48.180 --> 00:55:54.570 align:middle line:84% Plus is going to be positive result. 00:55:54.570 --> 00:56:05.134 align:middle line:84% And minus is going to be a negative result. 00:56:05.134 --> 00:56:11.940 align:middle line:84% So suppose I take this COVID test and it comes out positive. 00:56:11.940 --> 00:56:13.715 align:middle line:84% What's the probability that I have COVID? 00:56:13.715 --> 00:56:22.020 align:middle line:90% 00:56:22.020 --> 00:56:23.522 align:middle line:84% So first, what is the probability 00:56:23.522 --> 00:56:24.730 align:middle line:90% that we're trying to compute? 00:56:24.730 --> 00:56:34.350 align:middle line:90% 00:56:34.350 --> 00:56:34.850 align:middle line:90% Yeah? 00:56:34.850 --> 00:56:35.975 align:middle line:90% AUDIENCE: Probability of S. 00:56:35.975 --> 00:56:41.510 align:middle line:84% BRYNMOR CHAPMAN: Probability of S. Unconditionally? 00:56:41.510 --> 00:56:44.533 align:middle line:90% 00:56:44.533 --> 00:56:45.950 align:middle line:84% AUDIENCE: If [INAUDIBLE] positive. 00:56:45.950 --> 00:56:49.740 align:middle line:84% BRYNMOR CHAPMAN: Yeah, so S given positive. 00:56:49.740 --> 00:56:53.340 align:middle line:84% The unconditional probability of S is just 10% by assumption. 00:56:53.340 --> 00:56:57.230 align:middle line:84% What we're actually looking for is the probability 00:56:57.230 --> 00:57:01.010 align:middle line:84% that I'm sick given that I tested positive because that's 00:57:01.010 --> 00:57:02.530 align:middle line:90% the observation that we made. 00:57:02.530 --> 00:57:05.780 align:middle line:90% 00:57:05.780 --> 00:57:07.375 align:middle line:90% So how would we compute this? 00:57:07.375 --> 00:57:10.248 align:middle line:90% 00:57:10.248 --> 00:57:12.415 align:middle line:84% Well, let's do the same thing we just did over here. 00:57:12.415 --> 00:57:19.070 align:middle line:90% 00:57:19.070 --> 00:57:26.236 align:middle line:84% Let's look at the ratio of this probability to its complement 00:57:26.236 --> 00:57:29.170 align:middle line:90% and apply Bayes' rule. 00:57:29.170 --> 00:57:33.550 align:middle line:84% So we're going to flip this conditional here-- 00:57:33.550 --> 00:57:38.260 align:middle line:84% positive given sick times probability of sick 00:57:38.260 --> 00:57:45.850 align:middle line:84% divided by probability of positive given healthy times 00:57:45.850 --> 00:57:47.985 align:middle line:90% probability of healthy. 00:57:47.985 --> 00:57:53.770 align:middle line:90% 00:57:53.770 --> 00:57:58.160 align:middle line:84% Now, those four probabilities are probabilities that we know. 00:57:58.160 --> 00:57:58.770 align:middle line:90% What are they? 00:57:58.770 --> 00:58:06.820 align:middle line:90% 00:58:06.820 --> 00:58:09.795 align:middle line:84% What's the probability of plus given that I'm sick? 00:58:09.795 --> 00:58:13.480 align:middle line:90% 00:58:13.480 --> 00:58:15.114 align:middle line:90% Yeah? 00:58:15.114 --> 00:58:16.090 align:middle line:90% AUDIENCE: 90. 00:58:16.090 --> 00:58:18.340 align:middle line:90% BRYNMOR CHAPMAN: Yeah, 90%. 00:58:18.340 --> 00:58:22.380 align:middle line:90% So let's write that as 0.9. 00:58:22.380 --> 00:58:28.110 align:middle line:84% What's the prior probability that I'm sick? 00:58:28.110 --> 00:58:29.050 align:middle line:90% Yeah? 00:58:29.050 --> 00:58:29.657 align:middle line:90% AUDIENCE: 10. 00:58:29.657 --> 00:58:30.490 align:middle line:90% BRYNMOR CHAPMAN: 10. 00:58:30.490 --> 00:58:30.990 align:middle line:90% 10%. 00:58:30.990 --> 00:58:33.840 align:middle line:90% 00:58:33.840 --> 00:58:37.890 align:middle line:84% And now what's the probability of a positive result conditioned 00:58:37.890 --> 00:58:38.600 align:middle line:90% on being healthy? 00:58:38.600 --> 00:58:45.630 align:middle line:90% 00:58:45.630 --> 00:58:46.660 align:middle line:90% Yeah? 00:58:46.660 --> 00:58:47.580 align:middle line:90% AUDIENCE: 0.3. 00:58:47.580 --> 00:58:51.720 align:middle line:84% BRYNMOR CHAPMAN: Yeah, that's the false positive rate-- 00:58:51.720 --> 00:58:55.470 align:middle line:84% 0.3 times-- and what's the probability that I'm healthy, 00:58:55.470 --> 00:58:57.095 align:middle line:84% the prior probability that I'm healthy? 00:58:57.095 --> 00:59:04.855 align:middle line:90% 00:59:04.855 --> 00:59:07.230 align:middle line:84% Well, that's just going to be the complement of the event 00:59:07.230 --> 00:59:08.050 align:middle line:90% that I'm sick. 00:59:08.050 --> 00:59:14.436 align:middle line:90% So that's going to be 0.9. 00:59:14.436 --> 00:59:17.070 align:middle line:90% So these 0.9's cancel. 00:59:17.070 --> 00:59:18.910 align:middle line:90% This is going to be one third. 00:59:18.910 --> 00:59:22.340 align:middle line:90% 00:59:22.340 --> 00:59:26.670 align:middle line:84% So I am one third as likely to be sick as healthy. 00:59:26.670 --> 00:59:28.790 align:middle line:84% So the probability that I'm sick conditioned 00:59:28.790 --> 00:59:32.160 align:middle line:84% on getting the positive result is actually only 1 in 4. 00:59:32.160 --> 00:59:35.210 align:middle line:90% 00:59:35.210 --> 00:59:43.690 align:middle line:84% So probability S given plus equals one fourth. 00:59:43.690 --> 00:59:48.950 align:middle line:90% 00:59:48.950 --> 00:59:52.340 align:middle line:84% Does that make sense to everybody? 00:59:52.340 --> 00:59:56.480 align:middle line:84% So it might seem a little bit weird, right? 00:59:56.480 --> 00:59:58.640 align:middle line:90% This test is a pretty good one. 00:59:58.640 --> 01:00:01.280 align:middle line:84% It's usually giving the right answer. 01:00:01.280 --> 01:00:03.240 align:middle line:90% But I took the test. 01:00:03.240 --> 01:00:06.900 align:middle line:84% I got a positive result. But I'm still probably healthy. 01:00:06.900 --> 01:00:07.720 align:middle line:90% Why is that? 01:00:07.720 --> 01:00:11.300 align:middle line:90% 01:00:11.300 --> 01:00:13.490 align:middle line:84% Can anybody explain what's going on here? 01:00:13.490 --> 01:00:14.210 align:middle line:90% Yeah? 01:00:14.210 --> 01:00:18.130 align:middle line:84% AUDIENCE: There's a slim chance that you have COVID. 01:00:18.130 --> 01:00:19.970 align:middle line:90% BRYNMOR CHAPMAN: Yeah, exactly. 01:00:19.970 --> 01:00:22.940 align:middle line:84% So there's a very slim chance that I had COVID to begin with. 01:00:22.940 --> 01:00:25.910 align:middle line:84% So it's actually much higher now that I've got a positive result. 01:00:25.910 --> 01:00:31.840 align:middle line:84% But the prior was overwhelmingly in favor of being healthy. 01:00:31.840 --> 01:00:34.780 align:middle line:84% And so it's still very likely that I'm 01:00:34.780 --> 01:00:39.625 align:middle line:90% healthy, just slightly less so. 01:00:39.625 --> 01:00:44.200 align:middle line:84% So often, people underestimate the impact 01:00:44.200 --> 01:00:47.776 align:middle line:90% of the prior in this way. 01:00:47.776 --> 01:00:51.620 align:middle line:84% Of course, in the real world, you generally don't do this. 01:00:51.620 --> 01:00:55.180 align:middle line:84% So if you get a positive result, you can be more confident in it 01:00:55.180 --> 01:00:58.932 align:middle line:84% because normally you only test when you have symptoms. 01:00:58.932 --> 01:01:02.600 align:middle line:84% So the probability that you're sick, 01:01:02.600 --> 01:01:05.022 align:middle line:84% given that you tested positive and you have symptoms, 01:01:05.022 --> 01:01:07.355 align:middle line:84% that's probably going to be much higher than one fourth. 01:01:07.355 --> 01:01:15.340 align:middle line:90% 01:01:15.340 --> 01:01:18.480 align:middle line:90% Are people happy with that? 01:01:18.480 --> 01:01:19.470 align:middle line:90% Yeah? 01:01:19.470 --> 01:01:24.743 align:middle line:84% OK, why don't we look at another interesting application called 01:01:24.743 --> 01:01:25.493 align:middle line:90% Simpson's paradox? 01:01:25.493 --> 01:01:35.742 align:middle line:90% 01:01:35.742 --> 01:01:39.660 align:middle line:84% So there's actually kind of a funny story behind this one. 01:01:39.660 --> 01:01:45.390 align:middle line:84% So about 50 years ago, Berkeley, UC Berkeley, 01:01:45.390 --> 01:01:51.390 align:middle line:84% got sued because it was claimed that they were being very sexist 01:01:51.390 --> 01:01:54.630 align:middle line:90% in their admissions policies. 01:01:54.630 --> 01:02:01.500 align:middle line:84% And the claim was that the admissions rate was much higher 01:02:01.500 --> 01:02:04.620 align:middle line:90% for men than it was for women. 01:02:04.620 --> 01:02:07.440 align:middle line:90% So that's clearly bad, right? 01:02:07.440 --> 01:02:09.900 align:middle line:84% But then Berkeley counterclaimed by saying 01:02:09.900 --> 01:02:11.860 align:middle line:84% that if you look at each individual department, 01:02:11.860 --> 01:02:13.440 align:middle line:90% it's actually the opposite. 01:02:13.440 --> 01:02:19.112 align:middle line:84% Women are being admitted with much higher rates than men. 01:02:19.112 --> 01:02:23.300 align:middle line:90% So what's going on? 01:02:23.300 --> 01:02:24.510 align:middle line:90% Was somebody lying? 01:02:24.510 --> 01:02:25.645 align:middle line:90% Can these both be true? 01:02:25.645 --> 01:02:29.030 align:middle line:90% 01:02:29.030 --> 01:02:31.910 align:middle line:90% Any ideas? 01:02:31.910 --> 01:02:33.180 align:middle line:90% Yeah? 01:02:33.180 --> 01:02:36.130 align:middle line:84% AUDIENCE: Maybe the sizes of the departments. 01:02:36.130 --> 01:02:38.400 align:middle line:90% BRYNMOR CHAPMAN: Yeah, exactly. 01:02:38.400 --> 01:02:43.370 align:middle line:84% So the observation was that the sizes of the departments 01:02:43.370 --> 01:02:45.396 align:middle line:90% could be very different. 01:02:45.396 --> 01:02:49.310 align:middle line:90% And what else? 01:02:49.310 --> 01:02:50.677 align:middle line:90% Yeah? 01:02:50.677 --> 01:02:53.660 align:middle line:90% AUDIENCE: [INAUDIBLE]. 01:02:53.660 --> 01:02:56.150 align:middle line:84% BRYNMOR CHAPMAN: Yeah, so the admissions rates-- 01:02:56.150 --> 01:02:58.490 align:middle line:84% the overall admissions rates for the departments 01:02:58.490 --> 01:03:01.410 align:middle line:90% can also be very different. 01:03:01.410 --> 01:03:06.830 align:middle line:84% And so maybe there's some departments-- 01:03:06.830 --> 01:03:08.880 align:middle line:84% let's take a look at a very simple example. 01:03:08.880 --> 01:03:13.503 align:middle line:84% So suppose we have two departments, EE and CS. 01:03:13.503 --> 01:03:18.146 align:middle line:90% 01:03:18.146 --> 01:03:22.965 align:middle line:84% And maybe both of them mildly favor women. 01:03:22.965 --> 01:03:25.554 align:middle line:90% 01:03:25.554 --> 01:03:29.150 align:middle line:84% But one of them is much more popular with men. 01:03:29.150 --> 01:03:31.130 align:middle line:84% One of them is much more popular with women. 01:03:31.130 --> 01:03:35.510 align:middle line:84% And they have very different admissions rates generally. 01:03:35.510 --> 01:03:38.080 align:middle line:90% 01:03:38.080 --> 01:03:43.930 align:middle line:84% So maybe we have a hundred men and a hundred women 01:03:43.930 --> 01:03:48.430 align:middle line:84% applying to this two-department UC Berkeley. 01:03:48.430 --> 01:04:00.850 align:middle line:84% And maybe we have 99 women and one man apply to CS. 01:04:00.850 --> 01:04:03.280 align:middle line:84% And in double E, it's the opposite-- 01:04:03.280 --> 01:04:06.270 align:middle line:90% one woman and 99 men. 01:04:06.270 --> 01:04:11.479 align:middle line:90% 01:04:11.479 --> 01:04:18.120 align:middle line:84% And maybe in CS, we're a bunch of snobs. 01:04:18.120 --> 01:04:19.110 align:middle line:90% We hate everybody. 01:04:19.110 --> 01:04:21.920 align:middle line:90% We take one applicant. 01:04:21.920 --> 01:04:24.020 align:middle line:90% So maybe we take-- 01:04:24.020 --> 01:04:28.765 align:middle line:90% one woman is accepted. 01:04:28.765 --> 01:04:29.890 align:middle line:90% Everybody else is rejected. 01:04:29.890 --> 01:04:33.040 align:middle line:90% 01:04:33.040 --> 01:04:35.670 align:middle line:84% And double E, on the other hand, has no standards at all. 01:04:35.670 --> 01:04:38.120 align:middle line:84% They'll just take anybody with a pulse. 01:04:38.120 --> 01:04:40.020 align:middle line:84% But unfortunately, one of these men is dead. 01:04:40.020 --> 01:04:44.005 align:middle line:84% So they take one woman and 98 men. 01:04:44.005 --> 01:04:51.217 align:middle line:90% 01:04:51.217 --> 01:04:53.300 align:middle line:84% So now, if you look at the individual department's 01:04:53.300 --> 01:04:59.760 align:middle line:84% admissions rates, CS has accepted a woman. 01:04:59.760 --> 01:05:03.480 align:middle line:84% So their admissions rate for women is non-zero. 01:05:03.480 --> 01:05:04.760 align:middle line:90% They haven't accepted any men. 01:05:04.760 --> 01:05:07.940 align:middle line:84% So they've got a zero admissions rate for men. 01:05:07.940 --> 01:05:11.500 align:middle line:90% So it's higher for women. 01:05:11.500 --> 01:05:15.230 align:middle line:84% And in double E, they've accepted all of the women 01:05:15.230 --> 01:05:16.460 align:middle line:90% but not all of the men. 01:05:16.460 --> 01:05:18.520 align:middle line:90% So again, it's higher for women. 01:05:18.520 --> 01:05:21.672 align:middle line:84% But if you look at what's happened overall, 01:05:21.672 --> 01:05:25.800 align:middle line:84% we've got two women being accepted and 98 men. 01:05:25.800 --> 01:05:29.272 align:middle line:90% 01:05:29.272 --> 01:05:32.410 align:middle line:84% So you can argue that, rather than being 01:05:32.410 --> 01:05:37.460 align:middle line:84% a problem with the University or the departments, 01:05:37.460 --> 01:05:42.012 align:middle line:90% it's more of a systemic problem. 01:05:42.012 --> 01:05:44.560 align:middle line:84% Maybe they're encouraging men and women 01:05:44.560 --> 01:05:47.210 align:middle line:84% to apply to different departments at different rates. 01:05:47.210 --> 01:05:49.820 align:middle line:90% And it's all a bit messy. 01:05:49.820 --> 01:05:53.630 align:middle line:90% And also, this is 50 years ago. 01:05:53.630 --> 01:05:56.140 align:middle line:84% So they only really considered men and women. 01:05:56.140 --> 01:06:01.760 align:middle line:84% Didn't really consider anybody who identified as anything else. 01:06:01.760 --> 01:06:04.220 align:middle line:84% But we're not saying anything about that. 01:06:04.220 --> 01:06:06.770 align:middle line:84% This is just to illustrate the maths. 01:06:06.770 --> 01:06:10.800 align:middle line:84% So hopefully it's not terribly offensive. 01:06:10.800 --> 01:06:16.200 align:middle line:84% But yeah, even though, for each department, 01:06:16.200 --> 01:06:18.490 align:middle line:84% the women are doing better than men, 01:06:18.490 --> 01:06:21.030 align:middle line:84% the men are doing much better overall. 01:06:21.030 --> 01:06:24.718 align:middle line:84% So that's what we call Simpson's paradox. 01:06:24.718 --> 01:06:26.510 align:middle line:84% Does anybody have any questions about that? 01:06:26.510 --> 01:06:33.930 align:middle line:90% 01:06:33.930 --> 01:06:38.353 align:middle line:90% OK, so last example. 01:06:38.353 --> 01:06:42.450 align:middle line:90% 01:06:42.450 --> 01:06:46.530 align:middle line:84% Has anybody seen The Dragon Prince? 01:06:46.530 --> 01:06:47.860 align:middle line:90% A couple of hands. 01:06:47.860 --> 01:06:52.230 align:middle line:90% OK, well, so last example. 01:06:52.230 --> 01:06:55.200 align:middle line:84% Suppose a cake has gone missing from the bakery. 01:06:55.200 --> 01:07:00.460 align:middle line:84% And Barius the baker has a prime suspect. 01:07:00.460 --> 01:07:02.375 align:middle line:84% He thinks that Prince Ezran stole the cake. 01:07:02.375 --> 01:07:05.640 align:middle line:90% 01:07:05.640 --> 01:07:10.610 align:middle line:84% And he wants to present evidence to the king. 01:07:10.610 --> 01:07:16.340 align:middle line:84% He would like to tell the king that this deviant is always 01:07:16.340 --> 01:07:18.360 align:middle line:84% stealing jelly tarts from my bakery. 01:07:18.360 --> 01:07:21.590 align:middle line:84% That makes it much more likely that he stole the cake, right? 01:07:21.590 --> 01:07:23.930 align:middle line:84% Jelly tart thieves are 10 times as likely 01:07:23.930 --> 01:07:25.585 align:middle line:90% as randos to steal cakes. 01:07:25.585 --> 01:07:28.250 align:middle line:90% 01:07:28.250 --> 01:07:30.170 align:middle line:90% That seems fairly reasonable. 01:07:30.170 --> 01:07:32.210 align:middle line:84% Does anybody think that this is a good reason 01:07:32.210 --> 01:07:37.490 align:middle line:84% to admit the jelly tart thievery into evidence? 01:07:37.490 --> 01:07:41.190 align:middle line:84% Who thinks that it's not a compelling reason? 01:07:41.190 --> 01:07:41.690 align:middle line:90% No? 01:07:41.690 --> 01:07:43.790 align:middle line:90% Who has no idea? 01:07:43.790 --> 01:07:44.820 align:middle line:90% More hands. 01:07:44.820 --> 01:07:45.830 align:middle line:90% OK. 01:07:45.830 --> 01:07:51.030 align:middle line:84% So maybe Prince Ezran's counterclaim is, well, 01:07:51.030 --> 01:07:55.020 align:middle line:84% if you look at all jelly tart thieves, only one in, say, 01:07:55.020 --> 01:07:58.340 align:middle line:84% 2,500 of them actually go on to steal cakes. 01:07:58.340 --> 01:08:06.070 align:middle line:84% So even though the probability of being a cake thief has 01:08:06.070 --> 01:08:08.770 align:middle line:84% increased dramatically from the baseline in terms 01:08:08.770 --> 01:08:12.610 align:middle line:84% of a multiplicative factor, it can't actually have increased 01:08:12.610 --> 01:08:18.130 align:middle line:84% very much because it went from 1 in 25,000 to 1 in 2,500. 01:08:18.130 --> 01:08:23.029 align:middle line:84% So the probative value here is essentially 0. 01:08:23.029 --> 01:08:25.899 align:middle line:84% You're trying to prove guilt beyond a reasonable doubt. 01:08:25.899 --> 01:08:31.160 align:middle line:84% And so a 1-in-2,500-chance increase is kind of negligible. 01:08:31.160 --> 01:08:34.580 align:middle line:84% But on the other hand, if you admit this into evidence, 01:08:34.580 --> 01:08:36.923 align:middle line:84% the king is going to be biased against Ezran. 01:08:36.923 --> 01:08:38.840 align:middle line:84% He's going to be like, oh, this kid's a thief. 01:08:38.840 --> 01:08:39.680 align:middle line:90% This is terrible. 01:08:39.680 --> 01:08:43.289 align:middle line:84% We got to punish him, regardless of whether he stole the cake. 01:08:43.289 --> 01:08:45.819 align:middle line:90% 01:08:45.819 --> 01:08:48.200 align:middle line:90% So who's right? 01:08:48.200 --> 01:08:49.609 align:middle line:90% Is the baker right? 01:08:49.609 --> 01:08:51.510 align:middle line:90% Or is the suspect right? 01:08:51.510 --> 01:08:55.090 align:middle line:90% 01:08:55.090 --> 01:08:56.220 align:middle line:90% Who thinks baker is right? 01:08:56.220 --> 01:09:00.100 align:middle line:90% 01:09:00.100 --> 01:09:03.420 align:middle line:90% Who thinks suspect is right? 01:09:03.420 --> 01:09:04.220 align:middle line:90% Who thinks neither? 01:09:04.220 --> 01:09:05.544 align:middle line:90% [CHUCKLES] 01:09:05.544 --> 01:09:08.850 align:middle line:90% 01:09:08.850 --> 01:09:10.415 align:middle line:90% Yeah, I've got a few hands. 01:09:10.415 --> 01:09:10.915 align:middle line:90% OK. 01:09:10.915 --> 01:09:14.490 align:middle line:90% 01:09:14.490 --> 01:09:18.330 align:middle line:84% Well, both parties here are trying 01:09:18.330 --> 01:09:24.705 align:middle line:84% to reason about a conditional probability. 01:09:24.705 --> 01:09:27.069 align:middle line:84% What is that conditional probability? 01:09:27.069 --> 01:09:34.649 align:middle line:90% 01:09:34.649 --> 01:09:35.460 align:middle line:90% Anybody? 01:09:35.460 --> 01:09:40.229 align:middle line:90% 01:09:40.229 --> 01:09:43.200 align:middle line:84% Well, maybe we should start by writing out some events. 01:09:43.200 --> 01:09:53.690 align:middle line:84% So let's let J be the event suspect steals jelly tarts. 01:09:53.690 --> 01:09:59.070 align:middle line:90% 01:09:59.070 --> 01:10:01.545 align:middle line:84% C can be the event suspect steals cake. 01:10:01.545 --> 01:10:10.220 align:middle line:90% 01:10:10.220 --> 01:10:13.645 align:middle line:84% And X can be the event cake was stolen. 01:10:13.645 --> 01:10:22.580 align:middle line:90% 01:10:22.580 --> 01:10:25.460 align:middle line:84% So both the baker and the suspect 01:10:25.460 --> 01:10:29.000 align:middle line:84% were making claims about conditional probabilities. 01:10:29.000 --> 01:10:30.710 align:middle line:84% What specific conditional probability 01:10:30.710 --> 01:10:33.670 align:middle line:84% in terms of these events were they making claims about? 01:10:33.670 --> 01:10:48.255 align:middle line:90% 01:10:48.255 --> 01:10:48.755 align:middle line:90% Yeah? 01:10:48.755 --> 01:10:50.547 align:middle line:84% AUDIENCE: Can you repeat the assumptions? 01:10:50.547 --> 01:10:51.505 align:middle line:90% BRYNMOR CHAPMAN: Sorry? 01:10:51.505 --> 01:10:53.380 align:middle line:84% AUDIENCE: Can you repeat the assumptions? 01:10:53.380 --> 01:10:54.480 align:middle line:84% BRYNMOR CHAPMAN: Repeat the assertions? 01:10:54.480 --> 01:10:55.397 align:middle line:90% AUDIENCE: Assumptions. 01:10:55.397 --> 01:10:56.960 align:middle line:84% BRYNMOR CHAPMAN: Oh, the assumptions. 01:10:56.960 --> 01:10:59.120 align:middle line:84% So the assumptions were that a cake has 01:10:59.120 --> 01:11:00.710 align:middle line:90% gone missing from the bakery. 01:11:00.710 --> 01:11:04.390 align:middle line:90% The baker has a prime suspect. 01:11:04.390 --> 01:11:08.830 align:middle line:84% And the baker wants to admit into evidence that this suspect 01:11:08.830 --> 01:11:11.960 align:middle line:90% is always stealing jelly tarts. 01:11:11.960 --> 01:11:17.470 align:middle line:84% And the reasoning is that jelly tart thieves 01:11:17.470 --> 01:11:21.195 align:middle line:84% are much more likely to steal cakes than random other people. 01:11:21.195 --> 01:11:24.145 align:middle line:90% 01:11:24.145 --> 01:11:28.400 align:middle line:84% And the suspect counters by saying, well, sure, 01:11:28.400 --> 01:11:31.750 align:middle line:84% but only 1 in 2,500 jelly tart thieves is actually a cake 01:11:31.750 --> 01:11:34.465 align:middle line:90% thief. 01:11:34.465 --> 01:11:41.572 align:middle line:84% So the actual increase is less than 1 in 2,500. 01:11:41.572 --> 01:11:44.030 align:middle line:84% So what are the probabilities that they're reasoning about? 01:11:44.030 --> 01:11:46.010 align:middle line:84% What are the conditional probabilities? 01:11:46.010 --> 01:11:49.295 align:middle line:90% 01:11:49.295 --> 01:11:49.795 align:middle line:90% Yeah? 01:11:49.795 --> 01:11:51.046 align:middle line:90% AUDIENCE: [INAUDIBLE]. 01:11:51.046 --> 01:11:54.790 align:middle line:90% 01:11:54.790 --> 01:11:56.800 align:middle line:90% BRYNMOR CHAPMAN: Yeah. 01:11:56.800 --> 01:11:58.160 align:middle line:90% Well, you're both right. 01:11:58.160 --> 01:12:01.470 align:middle line:84% Sorry, it's-- [CHUCKLES] geometry. 01:12:01.470 --> 01:12:09.610 align:middle line:84% They're both trying to reason about probability of C given J. 01:12:09.610 --> 01:12:12.180 align:middle line:84% So we're conditioning on the suspect having stolen jelly 01:12:12.180 --> 01:12:13.180 align:middle line:90% tarts. 01:12:13.180 --> 01:12:17.085 align:middle line:84% And they're both trying to make claims about, 01:12:17.085 --> 01:12:19.445 align:middle line:84% what is the probability of C conditioned on that? 01:12:19.445 --> 01:12:25.794 align:middle line:90% 01:12:25.794 --> 01:12:30.810 align:middle line:84% So, well, the baker is arguing that probability of C 01:12:30.810 --> 01:12:39.383 align:middle line:84% given J divided by probability of C given not J is high. 01:12:39.383 --> 01:12:44.050 align:middle line:90% 01:12:44.050 --> 01:12:49.033 align:middle line:84% So because this is about 10, J is very relevant to the case. 01:12:49.033 --> 01:12:50.075 align:middle line:90% So it should be admitted. 01:12:50.075 --> 01:12:53.241 align:middle line:90% 01:12:53.241 --> 01:13:00.380 align:middle line:84% The suspect is saying that probability of C given J is low. 01:13:00.380 --> 01:13:03.390 align:middle line:90% 01:13:03.390 --> 01:13:08.460 align:middle line:84% So J cannot actually dramatically increase 01:13:08.460 --> 01:13:10.990 align:middle line:90% the probability. 01:13:10.990 --> 01:13:13.750 align:middle line:84% The increase is going to be less than this probability. 01:13:13.750 --> 01:13:18.240 align:middle line:90% 01:13:18.240 --> 01:13:19.050 align:middle line:90% Yeah? 01:13:19.050 --> 01:13:25.440 align:middle line:84% AUDIENCE: [INAUDIBLE] single person, a single piece. 01:13:25.440 --> 01:13:31.260 align:middle line:84% If this were an actual event, wouldn't these probabilities 01:13:31.260 --> 01:13:34.630 align:middle line:84% only really work in aggregate [INAUDIBLE]? 01:13:34.630 --> 01:13:39.070 align:middle line:90% 01:13:39.070 --> 01:13:42.450 align:middle line:84% BRYNMOR CHAPMAN: Yeah, so that's a good point. 01:13:42.450 --> 01:13:45.240 align:middle line:84% So the point was that we're talking 01:13:45.240 --> 01:13:48.420 align:middle line:84% about one individual suspect, one individual cake, one 01:13:48.420 --> 01:13:49.950 align:middle line:90% individual case. 01:13:49.950 --> 01:13:55.000 align:middle line:84% So as we were saying before when we first started looking 01:13:55.000 --> 01:13:59.000 align:middle line:84% at Bayes' rule, either it happened or it didn't. 01:13:59.000 --> 01:14:01.570 align:middle line:84% You could say that it doesn't make a whole lot of sense 01:14:01.570 --> 01:14:04.100 align:middle line:90% to talk about probabilities. 01:14:04.100 --> 01:14:09.820 align:middle line:84% But you're trying to convince the king or a jury or somebody. 01:14:09.820 --> 01:14:13.180 align:middle line:84% So this is kind of similar to when 01:14:13.180 --> 01:14:16.882 align:middle line:84% you present DNA evidence in court or something like that. 01:14:16.882 --> 01:14:18.340 align:middle line:84% The fact that there's a match means 01:14:18.340 --> 01:14:19.930 align:middle line:84% that there's overwhelming probability 01:14:19.930 --> 01:14:21.850 align:middle line:84% that the suspect was at the crime scene 01:14:21.850 --> 01:14:23.080 align:middle line:90% or something like that. 01:14:23.080 --> 01:14:28.180 align:middle line:84% So you do still talk about probabilities. 01:14:28.180 --> 01:14:30.760 align:middle line:84% But you could think of it as like, how confident are you 01:14:30.760 --> 01:14:34.600 align:middle line:84% in the inference that the suspect did something? 01:14:34.600 --> 01:14:36.712 align:middle line:90% Does that make sense? 01:14:36.712 --> 01:14:38.430 align:middle line:90% AUDIENCE: Yeah. 01:14:38.430 --> 01:14:39.390 align:middle line:90% BRYNMOR CHAPMAN: OK. 01:14:39.390 --> 01:14:52.522 align:middle line:90% 01:14:52.522 --> 01:14:57.600 align:middle line:84% So if you were the judge in this case determining 01:14:57.600 --> 01:15:00.360 align:middle line:84% what should be presented to the king, 01:15:00.360 --> 01:15:01.820 align:middle line:90% what would your verdict be? 01:15:01.820 --> 01:15:05.340 align:middle line:90% 01:15:05.340 --> 01:15:09.040 align:middle line:84% Do we allow jelly tart thievery into evidence or no? 01:15:09.040 --> 01:15:10.355 align:middle line:90% AUDIENCE: Draw a tree diagram. 01:15:10.355 --> 01:15:11.355 align:middle line:90% BRYNMOR CHAPMAN: Pardon? 01:15:11.355 --> 01:15:12.951 align:middle line:90% AUDIENCE: Draw a tree diagram. 01:15:12.951 --> 01:15:14.951 align:middle line:84% BRYNMOR CHAPMAN: [CHUCKLES] Draw a tree diagram. 01:15:14.951 --> 01:15:16.784 align:middle line:84% Well, that's not actually necessary anymore. 01:15:16.784 --> 01:15:19.270 align:middle line:84% We've already figured out what the probabilities are. 01:15:19.270 --> 01:15:24.720 align:middle line:84% The question is, is the probative value good enough 01:15:24.720 --> 01:15:28.270 align:middle line:84% to justify putting it into evidence? 01:15:28.270 --> 01:15:31.110 align:middle line:84% Or is the probative value small enough 01:15:31.110 --> 01:15:34.780 align:middle line:84% that the bias is going to be more of a factor? 01:15:34.780 --> 01:15:35.550 align:middle line:90% Yeah? 01:15:35.550 --> 01:15:41.385 align:middle line:84% AUDIENCE: [INAUDIBLE] the evidence about the jelly part 01:15:41.385 --> 01:15:45.460 align:middle line:84% because, also, like, an MIT student, for instance, 01:15:45.460 --> 01:15:47.760 align:middle line:84% given an amazing student, there's 01:15:47.760 --> 01:15:51.880 align:middle line:84% a very low chance of them being a cake thief [INAUDIBLE]. 01:15:51.880 --> 01:15:55.600 align:middle line:90% 01:15:55.600 --> 01:15:57.070 align:middle line:84% I guess this is what I'm wondering 01:15:57.070 --> 01:16:01.390 align:middle line:84% about an individual instance where it doesn't really 01:16:01.390 --> 01:16:02.210 align:middle line:90% count for them. 01:16:02.210 --> 01:16:05.530 align:middle line:84% BRYNMOR CHAPMAN: Yeah, so it will not generally 01:16:05.530 --> 01:16:06.590 align:middle line:90% capture the full story. 01:16:06.590 --> 01:16:08.740 align:middle line:84% And this is why lawyers in a courtroom 01:16:08.740 --> 01:16:13.610 align:middle line:84% are trying to find all kinds of evidence to bolster their cases. 01:16:13.610 --> 01:16:15.730 align:middle line:84% So, yeah, something like that might be something 01:16:15.730 --> 01:16:20.440 align:middle line:84% that the defense would want to present. 01:16:20.440 --> 01:16:22.940 align:middle line:84% Or the prosecution might be like, oh, 01:16:22.940 --> 01:16:24.530 align:middle line:84% but even though it's an MIT student, 01:16:24.530 --> 01:16:29.070 align:middle line:84% it's an MIT EECS student who do steal cakes all the time, so-- 01:16:29.070 --> 01:16:31.658 align:middle line:84% [CHUCKLES] I'm getting some nods. 01:16:31.658 --> 01:16:32.700 align:middle line:90% That's mildly concerning. 01:16:32.700 --> 01:16:34.496 align:middle line:90% [LAUGHING] 01:16:34.496 --> 01:16:35.440 align:middle line:90% 01:16:35.440 --> 01:16:39.610 align:middle line:84% So, yeah, it's going to be very difficult to capture 01:16:39.610 --> 01:16:41.680 align:middle line:90% the entire picture. 01:16:41.680 --> 01:16:44.560 align:middle line:84% For right now, we're just trying to limit the scope to, 01:16:44.560 --> 01:16:46.730 align:middle line:84% these are the things that are known. 01:16:46.730 --> 01:16:48.560 align:middle line:90% We know that a cake is stolen. 01:16:48.560 --> 01:16:51.330 align:middle line:84% We know that this suspect steals jelly tarts. 01:16:51.330 --> 01:16:52.180 align:middle line:90% But that's it. 01:16:52.180 --> 01:16:54.930 align:middle line:84% We want to try and figure out what 01:16:54.930 --> 01:17:00.450 align:middle line:84% the probative value of this evidence is just based on those. 01:17:00.450 --> 01:17:03.390 align:middle line:90% Does that make sense? 01:17:03.390 --> 01:17:08.400 align:middle line:84% So who thinks the jelly tarts should be admitted? 01:17:08.400 --> 01:17:09.155 align:middle line:90% Who thinks no? 01:17:09.155 --> 01:17:11.700 align:middle line:90% 01:17:11.700 --> 01:17:15.760 align:middle line:84% OK, I'm getting more hands this time, less abstentions. 01:17:15.760 --> 01:17:18.835 align:middle line:90% Does anybody want to say why? 01:17:18.835 --> 01:17:19.335 align:middle line:90% Yeah? 01:17:19.335 --> 01:17:23.770 align:middle line:84% AUDIENCE: Probability of C given [INAUDIBLE] low [INAUDIBLE] 01:17:23.770 --> 01:17:26.340 align:middle line:84% low probability of C given [INAUDIBLE]. 01:17:26.340 --> 01:17:29.392 align:middle line:84% So that could account for an inflated probability as well. 01:17:29.392 --> 01:17:32.130 align:middle line:84% BRYNMOR CHAPMAN: Yeah, so the observation 01:17:32.130 --> 01:17:35.800 align:middle line:84% was the probability of C given not J is also very low. 01:17:35.800 --> 01:17:37.770 align:middle line:84% And in fact, it is much lower than C 01:17:37.770 --> 01:17:43.500 align:middle line:84% given J. So that's where the ratio there, probability 01:17:43.500 --> 01:17:48.390 align:middle line:84% of C given J over C given not J, is coming from. 01:17:48.390 --> 01:17:51.330 align:middle line:84% Probability of C given J is 1 in 2,500. 01:17:51.330 --> 01:17:54.650 align:middle line:84% Probability of C given not J is 1 in 25,000. 01:17:54.650 --> 01:17:57.350 align:middle line:84% So that's where you're getting this factor of 10. 01:17:57.350 --> 01:18:00.920 align:middle line:84% So you agree with the defense, that it should not 01:18:00.920 --> 01:18:04.220 align:middle line:90% be admitted into evidence. 01:18:04.220 --> 01:18:06.270 align:middle line:84% Does anybody have any counterclaims? 01:18:06.270 --> 01:18:10.460 align:middle line:90% 01:18:10.460 --> 01:18:12.020 align:middle line:90% Yeah? 01:18:12.020 --> 01:18:14.660 align:middle line:84% AUDIENCE: So wouldn't you consider the part 01:18:14.660 --> 01:18:19.443 align:middle line:84% where the thief also steals where the cake is stolen before? 01:18:19.443 --> 01:18:20.360 align:middle line:90% BRYNMOR CHAPMAN: Yeah. 01:18:20.360 --> 01:18:21.234 align:middle line:90% [CHUCKLES] 01:18:21.234 --> 01:18:23.853 align:middle line:90% AUDIENCE: [INAUDIBLE]. 01:18:23.853 --> 01:18:25.770 align:middle line:84% BRYNMOR CHAPMAN: Yeah, so that's a good point. 01:18:25.770 --> 01:18:27.728 align:middle line:84% We've kind of forgotten about something, right? 01:18:27.728 --> 01:18:29.038 align:middle line:90% [CHUCKLES] 01:18:29.038 --> 01:18:32.990 align:middle line:90% 01:18:32.990 --> 01:18:34.115 align:middle line:90% The cake was stolen, right? 01:18:34.115 --> 01:18:35.532 align:middle line:84% We don't care that the probability 01:18:35.532 --> 01:18:36.810 align:middle line:90% of cakes being stolen is low. 01:18:36.810 --> 01:18:37.540 align:middle line:90% It happened. 01:18:37.540 --> 01:18:40.640 align:middle line:90% [LAUGHS] Right? 01:18:40.640 --> 01:18:46.380 align:middle line:84% So we don't really care about these probabilities at all. 01:18:46.380 --> 01:18:48.255 align:middle line:84% What is the actual probability we care about? 01:18:48.255 --> 01:18:52.960 align:middle line:90% 01:18:52.960 --> 01:18:57.690 align:middle line:84% What we really want is probability of C given, what? 01:18:57.690 --> 01:19:02.530 align:middle line:90% 01:19:02.530 --> 01:19:04.630 align:middle line:90% Yeah? 01:19:04.630 --> 01:19:07.450 align:middle line:84% AUDIENCE: Given that they stole the jelly tarts 01:19:07.450 --> 01:19:08.367 align:middle line:90% and they stole a cake? 01:19:08.367 --> 01:19:10.450 align:middle line:84% BRYNMOR CHAPMAN: Well, not that they stole a cake, 01:19:10.450 --> 01:19:11.710 align:middle line:90% but that a cake was stolen. 01:19:11.710 --> 01:19:12.460 align:middle line:90% Yeah. 01:19:12.460 --> 01:19:21.460 align:middle line:84% So C given J and X. So that's what we really care about. 01:19:21.460 --> 01:19:26.345 align:middle line:84% The fact that the cake is stolen is really important. 01:19:26.345 --> 01:19:28.220 align:middle line:84% If you don't know whether the cake is stolen, 01:19:28.220 --> 01:19:29.553 align:middle line:90% why are you even having a trial? 01:19:29.553 --> 01:19:32.770 align:middle line:90% 01:19:32.770 --> 01:19:35.770 align:middle line:84% And as it turns out, this probability 01:19:35.770 --> 01:19:37.230 align:middle line:90% is something like 80%. 01:19:37.230 --> 01:19:40.960 align:middle line:90% 01:19:40.960 --> 01:19:43.795 align:middle line:84% Jelly tart thieves found in bakeries with missing cakes, 01:19:43.795 --> 01:19:45.170 align:middle line:84% usually they're the guilty party. 01:19:45.170 --> 01:19:48.600 align:middle line:90% 01:19:48.600 --> 01:19:52.900 align:middle line:84% So, yeah, I've kind of made this a little bit more palatable. 01:19:52.900 --> 01:19:58.630 align:middle line:84% But this is based on an actual courtroom case. 01:19:58.630 --> 01:20:01.470 align:middle line:84% This is the kind of thing that people-- 01:20:01.470 --> 01:20:04.770 align:middle line:84% people try and use and end up misusing probabilities 01:20:04.770 --> 01:20:06.790 align:middle line:84% and conditional probabilities all the time. 01:20:06.790 --> 01:20:09.990 align:middle line:84% This was actually from a very high profile case, the OJ 01:20:09.990 --> 01:20:15.180 align:middle line:84% Simpson case, where both the defense and the prosecution 01:20:15.180 --> 01:20:18.270 align:middle line:84% attorneys, they're supposed to be experts. 01:20:18.270 --> 01:20:20.250 align:middle line:84% But they were both making bogus claims 01:20:20.250 --> 01:20:23.610 align:middle line:90% about conditional probabilities. 01:20:23.610 --> 01:20:26.590 align:middle line:84% It's very easy to make these kinds of mistakes. 01:20:26.590 --> 01:20:29.133 align:middle line:90% So fall back on fundamentals. 01:20:29.133 --> 01:20:31.300 align:middle line:84% When you're in doubt, fall back on the fundamentals. 01:20:31.300 --> 01:20:33.570 align:middle line:84% Even if you're not, it's probably a good idea anyway. 01:20:33.570 --> 01:20:36.190 align:middle line:84% OK, feel free to come up if you have any questions. 01:20:36.190 --> 01:20:39.380 align:middle line:84% And otherwise, we will see you next week. 01:20:39.380 --> 01:20:44.000 align:middle line:90%