WEBVTT 00:00:00.000 --> 00:00:00.040 align:middle line:90% 00:00:00.040 --> 00:00:02.460 align:middle line:84% The following content is provided under a Creative 00:00:02.460 --> 00:00:03.870 align:middle line:90% Commons license. 00:00:03.870 --> 00:00:06.320 align:middle line:84% Your support will help MIT OpenCourseWare 00:00:06.320 --> 00:00:10.560 align:middle line:84% continue to offer high quality educational resources for free. 00:00:10.560 --> 00:00:13.300 align:middle line:84% To make a donation or view additional materials 00:00:13.300 --> 00:00:17.210 align:middle line:84% from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:17.210 --> 00:00:17.865 align:middle line:90% at ocw.mit.edu. 00:00:17.865 --> 00:00:22.440 align:middle line:90% 00:00:22.440 --> 00:00:28.500 align:middle line:84% PROFESSOR: OK, so welcome to 6.041/6.431, 00:00:28.500 --> 00:00:31.750 align:middle line:84% the class on probability models and the like. 00:00:31.750 --> 00:00:32.740 align:middle line:90% I'm John Tsitsiklis. 00:00:32.740 --> 00:00:35.480 align:middle line:84% I will be teaching this class, and I'm 00:00:35.480 --> 00:00:39.760 align:middle line:84% looking forward to this being an enjoyable and also useful 00:00:39.760 --> 00:00:41.060 align:middle line:90% experience. 00:00:41.060 --> 00:00:43.370 align:middle line:84% We have a fair amount of staff involved 00:00:43.370 --> 00:00:46.280 align:middle line:84% in this course, your recitation instructors 00:00:46.280 --> 00:00:50.140 align:middle line:84% and also a bunch of TAs, but I want to single out our head 00:00:50.140 --> 00:00:54.450 align:middle line:84% TA, Uzoma, who is the key person in this class. 00:00:54.450 --> 00:00:56.550 align:middle line:84% Everything has to go through him. 00:00:56.550 --> 00:00:59.080 align:middle line:84% If he doesn't know in which recitation section 00:00:59.080 --> 00:01:03.600 align:middle line:84% you are, then simply you do not exist, so keep that in mind. 00:01:03.600 --> 00:01:04.099 align:middle line:90% All right. 00:01:04.099 --> 00:01:07.770 align:middle line:84% So we want to jump right into the subject, 00:01:07.770 --> 00:01:09.630 align:middle line:84% but I'm going to take just a few minutes 00:01:09.630 --> 00:01:12.600 align:middle line:84% to talk about a few administrative details 00:01:12.600 --> 00:01:14.580 align:middle line:90% and how the course is run. 00:01:14.580 --> 00:01:17.650 align:middle line:84% So we're going to have lectures twice a week 00:01:17.650 --> 00:01:20.300 align:middle line:84% and I'm going to use old fashioned transparencies. 00:01:20.300 --> 00:01:23.270 align:middle line:84% Now, you get copies of these slides with plenty of space 00:01:23.270 --> 00:01:25.760 align:middle line:90% for you to keep notes on them. 00:01:25.760 --> 00:01:29.670 align:middle line:84% A useful way of making good use of the slides 00:01:29.670 --> 00:01:33.920 align:middle line:84% is to use them as a sort of mnemonic summary of what 00:01:33.920 --> 00:01:35.720 align:middle line:90% happens in lecture. 00:01:35.720 --> 00:01:37.510 align:middle line:84% Not everything that I'm going to say 00:01:37.510 --> 00:01:40.490 align:middle line:84% is, of course, on the slides, but by looking them 00:01:40.490 --> 00:01:42.760 align:middle line:84% you get the sense of what's happening right now. 00:01:42.760 --> 00:01:45.070 align:middle line:84% And it may be a good idea to review them 00:01:45.070 --> 00:01:47.061 align:middle line:90% before you go to recitation. 00:01:47.061 --> 00:01:48.310 align:middle line:90% So what happens in recitation? 00:01:48.310 --> 00:01:51.090 align:middle line:84% In recitation, your recitation instructor 00:01:51.090 --> 00:01:53.990 align:middle line:84% is going to maybe review some of the theory 00:01:53.990 --> 00:01:57.150 align:middle line:84% and then solve some problems for you. 00:01:57.150 --> 00:01:59.260 align:middle line:84% And then you have tutorials where 00:01:59.260 --> 00:02:02.750 align:middle line:84% you meet in very small groups together with your TA. 00:02:02.750 --> 00:02:05.330 align:middle line:84% And what happens in tutorials is that you actually 00:02:05.330 --> 00:02:07.960 align:middle line:84% do the problem solving with the help of your TA 00:02:07.960 --> 00:02:12.290 align:middle line:84% and the help of your classmates in your tutorial section. 00:02:12.290 --> 00:02:14.340 align:middle line:84% Now probability is a tricky subject. 00:02:14.340 --> 00:02:16.750 align:middle line:84% You may be reading the text, listening to lectures, 00:02:16.750 --> 00:02:19.630 align:middle line:84% everything makes perfect sense, and so on, 00:02:19.630 --> 00:02:23.100 align:middle line:84% but until you actually sit down and try to solve problems, 00:02:23.100 --> 00:02:26.160 align:middle line:84% you don't quite appreciate the subtleties and the difficulties 00:02:26.160 --> 00:02:27.310 align:middle line:90% that are involved. 00:02:27.310 --> 00:02:30.550 align:middle line:84% So problem solving is a key part of this class. 00:02:30.550 --> 00:02:33.916 align:middle line:84% And tutorials are extremely useful just for this reason 00:02:33.916 --> 00:02:35.290 align:middle line:84% because that's where you actually 00:02:35.290 --> 00:02:38.120 align:middle line:84% get the practice of solving problems on your own, 00:02:38.120 --> 00:02:42.090 align:middle line:84% as opposed to seeing someone else who's solving them 00:02:42.090 --> 00:02:43.510 align:middle line:90% for you. 00:02:43.510 --> 00:02:47.270 align:middle line:84% OK but, mechanics, a key part of what's going to happen today 00:02:47.270 --> 00:02:51.500 align:middle line:84% is that you will turn in your schedule forms that 00:02:51.500 --> 00:02:55.350 align:middle line:84% are at the end of the handout that you have in your hands. 00:02:55.350 --> 00:02:59.820 align:middle line:84% Then, the TAs will be working frantically through the night, 00:02:59.820 --> 00:03:03.660 align:middle line:84% and they're going to be producing a list of who 00:03:03.660 --> 00:03:05.700 align:middle line:90% goes into what section. 00:03:05.700 --> 00:03:09.330 align:middle line:84% And when that happens, any person in this class, 00:03:09.330 --> 00:03:11.410 align:middle line:84% with probability 90%, is going to be 00:03:11.410 --> 00:03:15.520 align:middle line:84% happy with their assignment and, with probability 10%, 00:03:15.520 --> 00:03:17.670 align:middle line:90% they're going to be unhappy. 00:03:17.670 --> 00:03:20.860 align:middle line:84% Now, unhappy people have an option, though. 00:03:20.860 --> 00:03:23.230 align:middle line:84% You can resubmit your form together 00:03:23.230 --> 00:03:25.280 align:middle line:84% with your full schedule and constraints, 00:03:25.280 --> 00:03:28.150 align:middle line:84% give it back to the head TA, who will then 00:03:28.150 --> 00:03:32.160 align:middle line:84% do some further juggling and reassign people, 00:03:32.160 --> 00:03:36.040 align:middle line:84% and after that happens, 90% of those unhappy people 00:03:36.040 --> 00:03:37.570 align:middle line:90% will become happy. 00:03:37.570 --> 00:03:42.270 align:middle line:84% And 10% of them will be less unhappy. 00:03:42.270 --> 00:03:42.840 align:middle line:90% OK. 00:03:42.840 --> 00:03:46.350 align:middle line:84% So what's the probability that a random person 00:03:46.350 --> 00:03:49.800 align:middle line:84% is going to be unhappy at the end of this process? 00:03:49.800 --> 00:03:50.780 align:middle line:90% It's 1%. 00:03:50.780 --> 00:03:51.330 align:middle line:90% Excellent. 00:03:51.330 --> 00:03:51.829 align:middle line:90% Good. 00:03:51.829 --> 00:03:53.200 align:middle line:90% Maybe you don't need this class. 00:03:53.200 --> 00:03:54.340 align:middle line:90% OK, so 1%. 00:03:54.340 --> 00:03:56.810 align:middle line:84% We have about 100 people in this class, 00:03:56.810 --> 00:03:59.590 align:middle line:84% so there's going to be about one unhappy person. 00:03:59.590 --> 00:04:02.630 align:middle line:84% I mean, anywhere you look in life, in any group 00:04:02.630 --> 00:04:05.370 align:middle line:84% you look at, there's always one unhappy person, right? 00:04:05.370 --> 00:04:09.060 align:middle line:90% So, what can we do about it? 00:04:09.060 --> 00:04:09.660 align:middle line:90% All right. 00:04:09.660 --> 00:04:11.690 align:middle line:84% Another important part about mechanics 00:04:11.690 --> 00:04:13.290 align:middle line:84% is to read carefully the statement 00:04:13.290 --> 00:04:15.050 align:middle line:84% that we have about collaboration, 00:04:15.050 --> 00:04:17.019 align:middle line:90% academic honesty, and all that. 00:04:17.019 --> 00:04:19.070 align:middle line:84% You're encouraged, it's a very good idea 00:04:19.070 --> 00:04:21.140 align:middle line:90% to work with other students. 00:04:21.140 --> 00:04:24.550 align:middle line:84% You can consult sources that are out there, but when 00:04:24.550 --> 00:04:26.870 align:middle line:84% you sit down and write your solutions 00:04:26.870 --> 00:04:29.510 align:middle line:84% you have to do that by setting things aside 00:04:29.510 --> 00:04:32.050 align:middle line:90% and just write them on your own. 00:04:32.050 --> 00:04:37.040 align:middle line:84% You cannot copy something that somebody else has given to you. 00:04:37.040 --> 00:04:39.790 align:middle line:84% One reason is that we're not going 00:04:39.790 --> 00:04:43.750 align:middle line:84% to like it when it happens, and then another reason 00:04:43.750 --> 00:04:46.212 align:middle line:84% is that you're not going to do yourself any favor. 00:04:46.212 --> 00:04:48.045 align:middle line:84% Really the only way to do well in this class 00:04:48.045 --> 00:04:51.620 align:middle line:84% is to get a lot of practice by solving problems yourselves. 00:04:51.620 --> 00:04:53.470 align:middle line:84% So if you don't do that on your own, 00:04:53.470 --> 00:04:56.880 align:middle line:84% then when quiz and exam time comes, 00:04:56.880 --> 00:04:59.070 align:middle line:84% things are going to be difficult. 00:04:59.070 --> 00:05:01.100 align:middle line:84% So, as I mentioned here, we're going 00:05:01.100 --> 00:05:04.520 align:middle line:84% to have recitation sections, that some of them 00:05:04.520 --> 00:05:08.220 align:middle line:84% are for 6.041 students, some are for 6.431 students, 00:05:08.220 --> 00:05:10.270 align:middle line:84% the graduate section of the class. 00:05:10.270 --> 00:05:14.170 align:middle line:84% Now undergraduates can sit in the graduate recitation 00:05:14.170 --> 00:05:14.690 align:middle line:90% sections. 00:05:14.690 --> 00:05:16.850 align:middle line:84% What's going to happen there is that things 00:05:16.850 --> 00:05:18.780 align:middle line:84% may be just a little faster and you 00:05:18.780 --> 00:05:21.930 align:middle line:84% may be covering a problem that's a little more advanced 00:05:21.930 --> 00:05:24.670 align:middle line:84% and is not covered in the undergrad sections. 00:05:24.670 --> 00:05:27.470 align:middle line:84% But if you sit in the graduate section, 00:05:27.470 --> 00:05:29.040 align:middle line:84% and you're an undergraduate, you're 00:05:29.040 --> 00:05:33.130 align:middle line:84% still just responsible for the undergraduate material. 00:05:33.130 --> 00:05:35.500 align:middle line:84% That is, you can just do the undergraduate work 00:05:35.500 --> 00:05:37.429 align:middle line:84% in the class, but maybe be exposed 00:05:37.429 --> 00:05:38.470 align:middle line:90% at the different section. 00:05:38.470 --> 00:05:41.070 align:middle line:90% 00:05:41.070 --> 00:05:43.036 align:middle line:90% OK. 00:05:43.036 --> 00:05:46.220 align:middle line:84% A few words about the style of this class. 00:05:46.220 --> 00:05:50.760 align:middle line:84% We want to focus on basic ideas and concepts. 00:05:50.760 --> 00:05:52.620 align:middle line:84% There's going to be lots of formulas, 00:05:52.620 --> 00:05:54.710 align:middle line:84% but what we try to do in this class 00:05:54.710 --> 00:05:56.530 align:middle line:84% is to actually have you understand 00:05:56.530 --> 00:05:58.190 align:middle line:90% what those formulas mean. 00:05:58.190 --> 00:06:01.260 align:middle line:84% And, in a year from now when almost all of the formulas 00:06:01.260 --> 00:06:03.510 align:middle line:84% have been wiped out from your memory, 00:06:03.510 --> 00:06:05.610 align:middle line:84% you still have the basic concepts. 00:06:05.610 --> 00:06:08.690 align:middle line:84% You can understand them, so when you look things up again, 00:06:08.690 --> 00:06:12.820 align:middle line:90% they will still make sense. 00:06:12.820 --> 00:06:15.650 align:middle line:84% It's not the plug and chug kind of class 00:06:15.650 --> 00:06:19.210 align:middle line:84% where you're given a list of formulas, you're given numbers, 00:06:19.210 --> 00:06:21.470 align:middle line:84% and you plug in and you get answers. 00:06:21.470 --> 00:06:24.950 align:middle line:84% The really hard part is usually to choose which 00:06:24.950 --> 00:06:26.280 align:middle line:90% formulas you're going to use. 00:06:26.280 --> 00:06:28.900 align:middle line:84% You need judgment, you need intuition. 00:06:28.900 --> 00:06:32.400 align:middle line:84% Lots of probability problems, at least the interesting ones, 00:06:32.400 --> 00:06:34.450 align:middle line:84% often have lots of different solutions. 00:06:34.450 --> 00:06:37.440 align:middle line:84% Some are extremely long, some are extremely short. 00:06:37.440 --> 00:06:40.430 align:middle line:84% The extremely short ones usually involve some kind 00:06:40.430 --> 00:06:43.860 align:middle line:84% of deeper understanding of what's going on so that you 00:06:43.860 --> 00:06:46.350 align:middle line:90% can pick a shortcut and use it. 00:06:46.350 --> 00:06:49.110 align:middle line:84% And hopefully you are going to develop this skill 00:06:49.110 --> 00:06:51.630 align:middle line:90% during this class. 00:06:51.630 --> 00:06:56.360 align:middle line:84% Now, I could spend a lot of time in this lecture talking 00:06:56.360 --> 00:06:58.570 align:middle line:84% about why the subject is important. 00:06:58.570 --> 00:07:02.270 align:middle line:84% I'll keep it short because I think it's almost obvious. 00:07:02.270 --> 00:07:05.650 align:middle line:84% Anything that happens in life is uncertain. 00:07:05.650 --> 00:07:08.910 align:middle line:84% There's uncertainty anywhere, so whatever you try to do, 00:07:08.910 --> 00:07:11.830 align:middle line:84% you need to have some way of dealing or thinking 00:07:11.830 --> 00:07:13.930 align:middle line:90% about this uncertainty. 00:07:13.930 --> 00:07:16.280 align:middle line:84% And the way to do that in a systematic way 00:07:16.280 --> 00:07:17.955 align:middle line:84% is by using the models that are given 00:07:17.955 --> 00:07:20.110 align:middle line:90% to us by probability theory. 00:07:20.110 --> 00:07:21.800 align:middle line:84% So if you're an engineer and you're 00:07:21.800 --> 00:07:24.630 align:middle line:84% dealing with a communication system or signal processing, 00:07:24.630 --> 00:07:28.440 align:middle line:84% basically you're facing a fight against noise. 00:07:28.440 --> 00:07:30.380 align:middle line:90% Noise is random, is uncertain. 00:07:30.380 --> 00:07:31.450 align:middle line:90% How do you model it? 00:07:31.450 --> 00:07:33.120 align:middle line:90% How do you deal with it? 00:07:33.120 --> 00:07:35.210 align:middle line:84% If you're a manager, I guess you're 00:07:35.210 --> 00:07:38.410 align:middle line:84% dealing with customer demand, which is, of course, random. 00:07:38.410 --> 00:07:40.950 align:middle line:84% Or you're dealing with the stock market, 00:07:40.950 --> 00:07:42.820 align:middle line:90% which is definitely random. 00:07:42.820 --> 00:07:48.190 align:middle line:84% Or you play the casino, which is, again, random, and so on. 00:07:48.190 --> 00:07:50.800 align:middle line:84% And the same goes for pretty much any other field 00:07:50.800 --> 00:07:52.880 align:middle line:90% that you can think of. 00:07:52.880 --> 00:07:57.190 align:middle line:84% But, independent of which field you're coming from, 00:07:57.190 --> 00:08:00.630 align:middle line:84% the basic concepts and tools are really all the same. 00:08:00.630 --> 00:08:03.100 align:middle line:84% So you may see in bookstores that there 00:08:03.100 --> 00:08:05.630 align:middle line:84% are books, probability for scientists, 00:08:05.630 --> 00:08:07.670 align:middle line:84% probability for engineers, probability 00:08:07.670 --> 00:08:11.440 align:middle line:84% for social scientists, probability for astrologists. 00:08:11.440 --> 00:08:13.960 align:middle line:84% Well, what all those books have inside 00:08:13.960 --> 00:08:16.790 align:middle line:84% them is exactly the same models, the same equations, 00:08:16.790 --> 00:08:18.040 align:middle line:90% the same problems. 00:08:18.040 --> 00:08:21.510 align:middle line:84% They just make them somewhat different word problems. 00:08:21.510 --> 00:08:24.780 align:middle line:84% The basic concepts are just one and the same, 00:08:24.780 --> 00:08:28.420 align:middle line:84% and we'll take this as an excuse for not 00:08:28.420 --> 00:08:31.960 align:middle line:84% going too much into specific domain applications. 00:08:31.960 --> 00:08:33.830 align:middle line:84% We will have problems and examples 00:08:33.830 --> 00:08:36.150 align:middle line:84% that are motivated, in some loose sense, 00:08:36.150 --> 00:08:38.140 align:middle line:90% from real world situations. 00:08:38.140 --> 00:08:40.510 align:middle line:84% But we're not really trying in this class 00:08:40.510 --> 00:08:46.220 align:middle line:84% to develop the skills for domain-specific problems. 00:08:46.220 --> 00:08:49.660 align:middle line:84% Rather, we're going to try to stick to general understanding 00:08:49.660 --> 00:08:52.260 align:middle line:90% of the subject. 00:08:52.260 --> 00:08:52.760 align:middle line:90% OK. 00:08:52.760 --> 00:08:57.280 align:middle line:84% So the next slide, of which you do have in your handout, 00:08:57.280 --> 00:09:01.080 align:middle line:84% gives you a few more details about the class. 00:09:01.080 --> 00:09:03.590 align:middle line:84% Maybe one thing to comment here is that you 00:09:03.590 --> 00:09:06.370 align:middle line:90% do need to read the text. 00:09:06.370 --> 00:09:08.560 align:middle line:84% And with calculus books, perhaps you 00:09:08.560 --> 00:09:11.100 align:middle line:84% can live with a just a two page summary of all 00:09:11.100 --> 00:09:13.190 align:middle line:84% of the interesting formulas in calculus, 00:09:13.190 --> 00:09:18.050 align:middle line:84% and you can get by just with those formulas. 00:09:18.050 --> 00:09:21.080 align:middle line:84% But here, because we want to develop concepts and intuition, 00:09:21.080 --> 00:09:24.700 align:middle line:84% actually reading words, as opposed to just browsing 00:09:24.700 --> 00:09:27.430 align:middle line:84% through equations, does make a difference. 00:09:27.430 --> 00:09:30.250 align:middle line:84% In the beginning, the class is kind of easy. 00:09:30.250 --> 00:09:32.420 align:middle line:84% When we deal with discrete probability, 00:09:32.420 --> 00:09:36.420 align:middle line:84% that's the material until our first quiz, and some of you 00:09:36.420 --> 00:09:39.790 align:middle line:84% may get by without being too systematic about following 00:09:39.790 --> 00:09:40.710 align:middle line:90% the material. 00:09:40.710 --> 00:09:43.970 align:middle line:84% But it does get substantially harder afterwards. 00:09:43.970 --> 00:09:47.020 align:middle line:84% And I would keep restating that you 00:09:47.020 --> 00:09:52.460 align:middle line:84% do have to read the text to really understand the material. 00:09:52.460 --> 00:09:52.980 align:middle line:90% OK. 00:09:52.980 --> 00:09:57.850 align:middle line:84% So now we can start with the real part of the lecture. 00:09:57.850 --> 00:10:01.670 align:middle line:90% Let us set the goals for today. 00:10:01.670 --> 00:10:04.320 align:middle line:84% So probability, or probability theory, 00:10:04.320 --> 00:10:08.130 align:middle line:84% is a framework for dealing with uncertainty, 00:10:08.130 --> 00:10:10.250 align:middle line:84% for dealing with situations in which we 00:10:10.250 --> 00:10:12.200 align:middle line:90% have some kind of randomness. 00:10:12.200 --> 00:10:15.860 align:middle line:84% So what we want to do is, by the end of today's lecture, 00:10:15.860 --> 00:10:18.780 align:middle line:84% to give you anything that you need 00:10:18.780 --> 00:10:21.980 align:middle line:84% to know how to set up what does it take to set up 00:10:21.980 --> 00:10:23.970 align:middle line:90% a probabilistic model. 00:10:23.970 --> 00:10:28.150 align:middle line:84% And what are the basic rules of the game for dealing 00:10:28.150 --> 00:10:30.520 align:middle line:90% with probabilistic models? 00:10:30.520 --> 00:10:32.230 align:middle line:84% So, by the end of this lecture, you 00:10:32.230 --> 00:10:35.290 align:middle line:84% will have essentially recovered half of this semester's 00:10:35.290 --> 00:10:36.860 align:middle line:90% tuition, right? 00:10:36.860 --> 00:10:40.100 align:middle line:84% So we're going to talk about probabilistic models in more 00:10:40.100 --> 00:10:40.820 align:middle line:90% detail-- 00:10:40.820 --> 00:10:42.740 align:middle line:84% the sample space, which is basically 00:10:42.740 --> 00:10:44.500 align:middle line:84% a description of all the things that 00:10:44.500 --> 00:10:47.410 align:middle line:84% may happen during a random experiment, 00:10:47.410 --> 00:10:50.590 align:middle line:84% and the probability law, which describes our beliefs 00:10:50.590 --> 00:10:54.130 align:middle line:84% about which outcomes are more likely to occur compared 00:10:54.130 --> 00:10:56.080 align:middle line:90% to other outcomes. 00:10:56.080 --> 00:10:59.130 align:middle line:84% Probability laws have to obey certain properties that we 00:10:59.130 --> 00:11:00.640 align:middle line:90% call the axioms of probability. 00:11:00.640 --> 00:11:03.570 align:middle line:84% So the main part of today's lecture 00:11:03.570 --> 00:11:05.380 align:middle line:84% is to describe those axioms, which 00:11:05.380 --> 00:11:08.540 align:middle line:84% are the rules of the game, and consider 00:11:08.540 --> 00:11:12.770 align:middle line:90% a few really trivial examples. 00:11:12.770 --> 00:11:15.370 align:middle line:84% OK, so let's start with our agenda. 00:11:15.370 --> 00:11:17.600 align:middle line:84% The first piece in a probabilistic model 00:11:17.600 --> 00:11:21.850 align:middle line:84% is a description of the sample space of an experiment. 00:11:21.850 --> 00:11:25.960 align:middle line:84% So we do an experiment, and by experiment we 00:11:25.960 --> 00:11:30.270 align:middle line:84% just mean that just something happens out there. 00:11:30.270 --> 00:11:34.200 align:middle line:84% And that something that happens, it could be flipping a coin, 00:11:34.200 --> 00:11:38.910 align:middle line:84% or it could be rolling a dice, or it could be 00:11:38.910 --> 00:11:41.550 align:middle line:90% doing something in a card game. 00:11:41.550 --> 00:11:44.190 align:middle line:84% So we fix a particular experiment. 00:11:44.190 --> 00:11:48.780 align:middle line:84% And we come up with a list of all the possible things that 00:11:48.780 --> 00:11:51.090 align:middle line:84% may happen during this experiment. 00:11:51.090 --> 00:11:54.880 align:middle line:84% So we write down a list of all the possible outcomes. 00:11:54.880 --> 00:11:57.610 align:middle line:84% So here's a list of all the possible outcomes 00:11:57.610 --> 00:11:59.050 align:middle line:90% of the experiment. 00:11:59.050 --> 00:12:01.720 align:middle line:84% I use the word "list," but, if you 00:12:01.720 --> 00:12:04.160 align:middle line:84% want to be a little more formal, it's better 00:12:04.160 --> 00:12:06.730 align:middle line:90% to think of that list as a set. 00:12:06.730 --> 00:12:08.630 align:middle line:90% So we have a set. 00:12:08.630 --> 00:12:11.000 align:middle line:90% That set is our sample space. 00:12:11.000 --> 00:12:14.530 align:middle line:84% And it's a set whose elements are the possible outcomes 00:12:14.530 --> 00:12:15.920 align:middle line:90% of the experiment. 00:12:15.920 --> 00:12:18.530 align:middle line:84% So, for example, if you're dealing with flipping a coin, 00:12:18.530 --> 00:12:22.380 align:middle line:84% your sample space would be heads, this is one outcome, 00:12:22.380 --> 00:12:24.450 align:middle line:90% tails is one outcome. 00:12:24.450 --> 00:12:26.460 align:middle line:84% And this set, which has two elements, 00:12:26.460 --> 00:12:29.170 align:middle line:84% is the sample space of the experiment. 00:12:29.170 --> 00:12:29.670 align:middle line:90% OK. 00:12:29.670 --> 00:12:32.570 align:middle line:84% What do we need to think about when we're 00:12:32.570 --> 00:12:34.430 align:middle line:90% setting up the sample space? 00:12:34.430 --> 00:12:36.690 align:middle line:84% First, the list should be mutually exclusive, 00:12:36.690 --> 00:12:37.830 align:middle line:90% collectively exhaustive. 00:12:37.830 --> 00:12:39.150 align:middle line:90% What does that mean? 00:12:39.150 --> 00:12:41.070 align:middle line:84% Collectively exhaustive means that, 00:12:41.070 --> 00:12:43.740 align:middle line:84% no matter what happens in the experiment, 00:12:43.740 --> 00:12:47.700 align:middle line:84% you're going to get one of the outcomes inside here. 00:12:47.700 --> 00:12:51.010 align:middle line:84% So you have not forgotten any of the possibilities of what 00:12:51.010 --> 00:12:53.020 align:middle line:90% may happen in the experiment. 00:12:53.020 --> 00:12:56.730 align:middle line:84% Mutually exclusive means that if this happens, 00:12:56.730 --> 00:12:58.870 align:middle line:90% then that cannot happen. 00:12:58.870 --> 00:13:00.660 align:middle line:84% So at the end of the experiment, you 00:13:00.660 --> 00:13:05.610 align:middle line:84% should be able to point out to me just one, exactly one, 00:13:05.610 --> 00:13:10.540 align:middle line:84% of these outcomes and say, this is the outcome that happened. 00:13:10.540 --> 00:13:11.040 align:middle line:90% OK. 00:13:11.040 --> 00:13:13.690 align:middle line:84% So these are sort of basic requirements. 00:13:13.690 --> 00:13:16.540 align:middle line:84% There's another requirement which is a little more loose. 00:13:16.540 --> 00:13:18.130 align:middle line:84% When you set up your sample space, 00:13:18.130 --> 00:13:19.930 align:middle line:84% sometimes you do have some freedom 00:13:19.930 --> 00:13:24.900 align:middle line:84% about the details of how you're going to describe it. 00:13:24.900 --> 00:13:27.020 align:middle line:84% And the question is, how much detail 00:13:27.020 --> 00:13:28.730 align:middle line:90% are you going to include? 00:13:28.730 --> 00:13:31.410 align:middle line:84% So let's take this coin flipping experiment 00:13:31.410 --> 00:13:34.070 align:middle line:84% and think of the following sample space. 00:13:34.070 --> 00:13:37.550 align:middle line:84% One possible outcome is heads, a second possible outcome 00:13:37.550 --> 00:13:43.750 align:middle line:84% is tails and it's raining, and the third possible outcome 00:13:43.750 --> 00:13:45.500 align:middle line:90% is tails and it's not raining. 00:13:45.500 --> 00:13:49.180 align:middle line:90% 00:13:49.180 --> 00:13:53.530 align:middle line:84% So this is another possible sample space for the experiment 00:13:53.530 --> 00:13:56.910 align:middle line:90% where I flip a coin just once. 00:13:56.910 --> 00:13:58.330 align:middle line:90% It's a legitimate one. 00:13:58.330 --> 00:14:01.430 align:middle line:84% These three possibilities are mutually exclusive 00:14:01.430 --> 00:14:03.470 align:middle line:90% and collectively exhaustive. 00:14:03.470 --> 00:14:05.410 align:middle line:84% Which one is the right sample space? 00:14:05.410 --> 00:14:08.440 align:middle line:90% Is it this one or that one? 00:14:08.440 --> 00:14:12.020 align:middle line:84% Well, if you think that my coin flipping inside this room 00:14:12.020 --> 00:14:15.230 align:middle line:84% is completely unrelated to the weather outside, 00:14:15.230 --> 00:14:18.470 align:middle line:84% then you're going to stick with this sample space. 00:14:18.470 --> 00:14:22.080 align:middle line:84% If, on the other hand, you have some superstitious belief 00:14:22.080 --> 00:14:25.510 align:middle line:84% that maybe rain has an effect on my coins, 00:14:25.510 --> 00:14:29.520 align:middle line:84% you might work with the sample space of this kind. 00:14:29.520 --> 00:14:32.040 align:middle line:84% So you probably wouldn't do that, 00:14:32.040 --> 00:14:35.370 align:middle line:84% but it's a legitimate option, strictly speaking. 00:14:35.370 --> 00:14:38.650 align:middle line:84% Now this example is a little bit on the frivolous side, 00:14:38.650 --> 00:14:40.600 align:middle line:84% but the issue that comes up here is 00:14:40.600 --> 00:14:43.270 align:middle line:84% a basic one that shows up anywhere 00:14:43.270 --> 00:14:44.700 align:middle line:90% in science and engineering. 00:14:44.700 --> 00:14:48.150 align:middle line:84% Whenever you're dealing with a model or with a situation, 00:14:48.150 --> 00:14:50.645 align:middle line:84% there are zillions of details in that situation. 00:14:50.645 --> 00:14:52.550 align:middle line:84% And when you come up with a model, 00:14:52.550 --> 00:14:57.170 align:middle line:84% you choose some of those details that you keep in your model, 00:14:57.170 --> 00:15:00.060 align:middle line:84% and some that you say, well, these are irrelevant. 00:15:00.060 --> 00:15:03.780 align:middle line:84% Or maybe there are small effects, I can neglect them, 00:15:03.780 --> 00:15:05.970 align:middle line:84% and you keep them outside your model. 00:15:05.970 --> 00:15:08.660 align:middle line:84% So when you go to the real world, 00:15:08.660 --> 00:15:11.860 align:middle line:84% there's definitely an element of art and some judgment 00:15:11.860 --> 00:15:15.430 align:middle line:84% that you need to do in order to set up an appropriate sample 00:15:15.430 --> 00:15:15.930 align:middle line:90% space. 00:15:15.930 --> 00:15:20.270 align:middle line:90% 00:15:20.270 --> 00:15:23.310 align:middle line:90% So, an easy example now. 00:15:23.310 --> 00:15:25.770 align:middle line:84% So of course, the elementary examples 00:15:25.770 --> 00:15:29.420 align:middle line:90% are coins, cards, and dice. 00:15:29.420 --> 00:15:30.840 align:middle line:90% So let's deal with dice. 00:15:30.840 --> 00:15:34.550 align:middle line:84% But to keep the diagram small, instead of a six-sided die, 00:15:34.550 --> 00:15:38.270 align:middle line:84% we're going to think about the die that only has four faces. 00:15:38.270 --> 00:15:40.220 align:middle line:84% So you can do that with a tetrahedron, 00:15:40.220 --> 00:15:41.150 align:middle line:90% doesn't really matter. 00:15:41.150 --> 00:15:43.520 align:middle line:84% Basically, it's a die that when you roll it, 00:15:43.520 --> 00:15:47.360 align:middle line:84% you get a result which is one, two, three or four. 00:15:47.360 --> 00:15:50.590 align:middle line:84% However, the experiment that I'm going to think about 00:15:50.590 --> 00:15:55.770 align:middle line:84% will consist of two rolls of a dice. 00:15:55.770 --> 00:15:57.600 align:middle line:90% A crucial point here-- 00:15:57.600 --> 00:15:59.890 align:middle line:84% I'm rolling the die twice, but I'm 00:15:59.890 --> 00:16:03.680 align:middle line:84% thinking of this as just one experiment, not 00:16:03.680 --> 00:16:08.540 align:middle line:84% two different experiments, not a repetition twice 00:16:08.540 --> 00:16:10.110 align:middle line:90% of the same experiment. 00:16:10.110 --> 00:16:12.040 align:middle line:90% So it's one big experiment. 00:16:12.040 --> 00:16:14.210 align:middle line:84% During that big experiment various things 00:16:14.210 --> 00:16:17.150 align:middle line:84% could happen, such as I'm rolling the die once, 00:16:17.150 --> 00:16:20.384 align:middle line:84% and then I'm rolling the die twice. 00:16:20.384 --> 00:16:22.450 align:middle line:90% OK. 00:16:22.450 --> 00:16:25.280 align:middle line:84% So what's the sample space for that experiment? 00:16:25.280 --> 00:16:28.700 align:middle line:84% Well, the sample space consists of the possible outcomes. 00:16:28.700 --> 00:16:32.170 align:middle line:84% One possible outcome is that your first roll 00:16:32.170 --> 00:16:36.670 align:middle line:84% resulted in two and the second roll resulted in three. 00:16:36.670 --> 00:16:40.690 align:middle line:84% In which case, the outcome that you get is this one, 00:16:40.690 --> 00:16:42.840 align:middle line:90% a two followed by three. 00:16:42.840 --> 00:16:45.840 align:middle line:90% This is one possible outcome. 00:16:45.840 --> 00:16:49.280 align:middle line:84% The way I'm describing things, this outcome 00:16:49.280 --> 00:16:51.750 align:middle line:84% is to be distinguished from this outcome 00:16:51.750 --> 00:16:56.656 align:middle line:84% here, where a three is followed by two. 00:16:56.656 --> 00:16:59.680 align:middle line:84% If you're playing backgammon, it doesn't matter 00:16:59.680 --> 00:17:02.250 align:middle line:90% which one of the two happened. 00:17:02.250 --> 00:17:05.540 align:middle line:84% But if you're dealing with a probabilistic model 00:17:05.540 --> 00:17:08.310 align:middle line:84% that you want to keep track of everything that happens 00:17:08.310 --> 00:17:11.359 align:middle line:84% in this composite experiment, there 00:17:11.359 --> 00:17:14.000 align:middle line:84% are good reasons for distinguishing 00:17:14.000 --> 00:17:15.859 align:middle line:90% between these two outcomes. 00:17:15.859 --> 00:17:18.609 align:middle line:84% I mean, when this happens, it's definitely something 00:17:18.609 --> 00:17:20.220 align:middle line:90% different from that happening. 00:17:20.220 --> 00:17:23.319 align:middle line:84% A two followed by a three is different from a three followed 00:17:23.319 --> 00:17:24.349 align:middle line:90% by a two. 00:17:24.349 --> 00:17:27.700 align:middle line:84% So this is the correct sample space for this experiment 00:17:27.700 --> 00:17:29.890 align:middle line:90% where we roll the die twice. 00:17:29.890 --> 00:17:33.240 align:middle line:84% It has a total of 16 elements and it's, of course, 00:17:33.240 --> 00:17:35.840 align:middle line:90% a finite set. 00:17:35.840 --> 00:17:39.850 align:middle line:84% Sometimes, instead of describing sample spaces 00:17:39.850 --> 00:17:43.950 align:middle line:84% in terms of lists, or sets, or diagrams of this kind, 00:17:43.950 --> 00:17:46.550 align:middle line:84% it's useful to describe the experiment 00:17:46.550 --> 00:17:48.660 align:middle line:90% in some sequential way. 00:17:48.660 --> 00:17:50.370 align:middle line:84% Whenever you have an experiment that 00:17:50.370 --> 00:17:53.170 align:middle line:84% consists of multiple stages, it might 00:17:53.170 --> 00:17:57.860 align:middle line:84% be useful, at least visually, to give a diagram that shows you 00:17:57.860 --> 00:17:59.940 align:middle line:90% how those stages evolve. 00:17:59.940 --> 00:18:03.720 align:middle line:84% And that's what we do by using a sequential description 00:18:03.720 --> 00:18:06.960 align:middle line:84% or a tree-based description by drawing 00:18:06.960 --> 00:18:09.690 align:middle line:84% a tree of the possible evolutions 00:18:09.690 --> 00:18:11.250 align:middle line:90% during our experiment. 00:18:11.250 --> 00:18:14.890 align:middle line:84% So in this tree, I'm thinking of a first stage in which I 00:18:14.890 --> 00:18:18.655 align:middle line:84% roll the first die, and there are four possible results, one, 00:18:18.655 --> 00:18:20.520 align:middle line:90% two, three and four.and 4. 00:18:20.520 --> 00:18:24.310 align:middle line:84% And, given what happened, let's say in the first roll, 00:18:24.310 --> 00:18:26.050 align:middle line:90% suppose I got a one. 00:18:26.050 --> 00:18:28.310 align:middle line:84% Then I'm rolling the second dice, 00:18:28.310 --> 00:18:30.320 align:middle line:84% and there are four possibilities for what 00:18:30.320 --> 00:18:32.060 align:middle line:90% may happen to the second die. 00:18:32.060 --> 00:18:36.010 align:middle line:84% And the possible results are one, tow, three and four again. 00:18:36.010 --> 00:18:38.860 align:middle line:84% So what's the relation between the two diagrams? 00:18:38.860 --> 00:18:41.150 align:middle line:84% Well, for example, the outcome two 00:18:41.150 --> 00:18:46.940 align:middle line:84% followed by three corresponds to this path on the tree. 00:18:46.940 --> 00:18:50.550 align:middle line:84% So this path corresponds to two followed by a three. 00:18:50.550 --> 00:18:53.910 align:middle line:84% Any path is associated to a particular outcome, 00:18:53.910 --> 00:18:57.360 align:middle line:84% any outcome is associated to a particular path. 00:18:57.360 --> 00:18:59.320 align:middle line:84% And, instead of paths, you may want 00:18:59.320 --> 00:19:01.990 align:middle line:84% to think in terms of the leaves of this diagram. 00:19:01.990 --> 00:19:04.870 align:middle line:84% Same thing, think of each one of the leaves 00:19:04.870 --> 00:19:07.980 align:middle line:90% as being one possible outcome. 00:19:07.980 --> 00:19:10.340 align:middle line:84% And of course we have 16 outcomes here, 00:19:10.340 --> 00:19:12.790 align:middle line:90% we have 16 outcomes here. 00:19:12.790 --> 00:19:15.920 align:middle line:84% Maybe you noticed the subtlety that I used in my language. 00:19:15.920 --> 00:19:18.810 align:middle line:84% I said I rolled the first dice and the result 00:19:18.810 --> 00:19:20.580 align:middle line:90% that I get is a two. 00:19:20.580 --> 00:19:22.660 align:middle line:90% I didn't use the word "outcome." 00:19:22.660 --> 00:19:24.940 align:middle line:84% I want to reserve the word "outcome" 00:19:24.940 --> 00:19:28.810 align:middle line:84% to mean the overall outcome at the end 00:19:28.810 --> 00:19:30.570 align:middle line:90% of the overall experiment. 00:19:30.570 --> 00:19:36.300 align:middle line:84% So "2, 3" is the outcome of the experiment. 00:19:36.300 --> 00:19:38.910 align:middle line:84% The experiment consisted of stages. 00:19:38.910 --> 00:19:40.970 align:middle line:84% Two was the result in the first stage, 00:19:40.970 --> 00:19:43.370 align:middle line:84% three was the result in the second stage. 00:19:43.370 --> 00:19:45.390 align:middle line:84% You put all those results together, 00:19:45.390 --> 00:19:47.520 align:middle line:90% and you get your outcome. 00:19:47.520 --> 00:19:50.220 align:middle line:84% OK, perhaps we are splitting hairs here, 00:19:50.220 --> 00:19:56.470 align:middle line:84% but it's useful to keep the concepts right. 00:19:56.470 --> 00:19:58.540 align:middle line:84% What's special about this example 00:19:58.540 --> 00:20:00.840 align:middle line:84% is that, besides being trivial, it has 00:20:00.840 --> 00:20:03.230 align:middle line:90% a sample space which is finite. 00:20:03.230 --> 00:20:06.000 align:middle line:84% There's 16 possible total outcomes. 00:20:06.000 --> 00:20:09.210 align:middle line:84% Not every experiment has a finite sample space. 00:20:09.210 --> 00:20:12.840 align:middle line:84% Here's an experiment in which the sample space is infinite. 00:20:12.840 --> 00:20:17.690 align:middle line:84% So you are playing darts and the target is this square. 00:20:17.690 --> 00:20:20.910 align:middle line:84% And you're perfect at that game, so you're 00:20:20.910 --> 00:20:26.010 align:middle line:84% sure that your darts will always fall inside the square. 00:20:26.010 --> 00:20:29.550 align:middle line:84% So, but where exactly your dart would fall inside that square, 00:20:29.550 --> 00:20:31.180 align:middle line:90% that itself is random. 00:20:31.180 --> 00:20:32.880 align:middle line:84% We don't know what it's going to be. 00:20:32.880 --> 00:20:34.300 align:middle line:90% It's uncertain. 00:20:34.300 --> 00:20:37.220 align:middle line:84% So all the possible points inside the square 00:20:37.220 --> 00:20:39.710 align:middle line:84% are possible outcomes of the experiment. 00:20:39.710 --> 00:20:43.060 align:middle line:84% So a typical outcome of the experiment is going to a pair 00:20:43.060 --> 00:20:47.180 align:middle line:84% of numbers, x,y, where x and y are real numbers between zero 00:20:47.180 --> 00:20:48.280 align:middle line:90% and one. 00:20:48.280 --> 00:20:51.170 align:middle line:84% Now there's infinitely many real numbers, 00:20:51.170 --> 00:20:54.100 align:middle line:84% there's infinitely many points in the square, 00:20:54.100 --> 00:20:56.700 align:middle line:84% so this is an example in which our sample 00:20:56.700 --> 00:20:58.740 align:middle line:90% space is an infinite set. 00:20:58.740 --> 00:21:01.670 align:middle line:90% 00:21:01.670 --> 00:21:06.910 align:middle line:84% OK, so we're going to revisit this example a little later. 00:21:06.910 --> 00:21:11.120 align:middle line:84% So these are two examples of what the sample space might 00:21:11.120 --> 00:21:13.730 align:middle line:90% be in simple experiments. 00:21:13.730 --> 00:21:17.350 align:middle line:84% Now, the more important order of business 00:21:17.350 --> 00:21:20.030 align:middle line:84% is now to look at those possible outcomes 00:21:20.030 --> 00:21:21.800 align:middle line:84% and to make some statements about 00:21:21.800 --> 00:21:23.910 align:middle line:90% their relative likelihoods. 00:21:23.910 --> 00:21:29.060 align:middle line:84% Which outcome is more likely to occur compared to the others? 00:21:29.060 --> 00:21:33.580 align:middle line:84% And the way we do this is by assigning probabilities 00:21:33.580 --> 00:21:36.210 align:middle line:90% to the outcomes. 00:21:36.210 --> 00:21:38.590 align:middle line:90% Well, not exactly. 00:21:38.590 --> 00:21:42.440 align:middle line:84% Suppose that all you were to do was to assign probabilities 00:21:42.440 --> 00:21:44.320 align:middle line:90% to individual outcomes. 00:21:44.320 --> 00:21:49.760 align:middle line:84% If you go back to this example, and you consider one particular 00:21:49.760 --> 00:21:52.250 align:middle line:90% outcome-- let's say this point-- 00:21:52.250 --> 00:21:55.900 align:middle line:84% what would be the probability that you hit exactly this point 00:21:55.900 --> 00:21:58.640 align:middle line:90% to infinite precision? 00:21:58.640 --> 00:22:01.070 align:middle line:84% Intuitively, that probability would be zero. 00:22:01.070 --> 00:22:06.090 align:middle line:84% So any individual point in this diagram in any reasonable model 00:22:06.090 --> 00:22:08.520 align:middle line:90% should have zero probability. 00:22:08.520 --> 00:22:11.870 align:middle line:84% So if you just tell me that any individual outcome has 00:22:11.870 --> 00:22:13.920 align:middle line:84% zero probability, you're not really 00:22:13.920 --> 00:22:17.030 align:middle line:90% telling me much to work with. 00:22:17.030 --> 00:22:19.890 align:middle line:84% For that reason, what instead we're going to do 00:22:19.890 --> 00:22:23.760 align:middle line:84% is to assign probabilities to subsets of the sample 00:22:23.760 --> 00:22:26.065 align:middle line:84% space, as opposed to assigning probabilities 00:22:26.065 --> 00:22:29.170 align:middle line:90% to individual outcomes. 00:22:29.170 --> 00:22:32.410 align:middle line:90% So here's the picture. 00:22:32.410 --> 00:22:36.230 align:middle line:84% We have our sample space, which is omega, 00:22:36.230 --> 00:22:39.690 align:middle line:84% and we consider some subset of the sample space. 00:22:39.690 --> 00:22:45.140 align:middle line:84% Call it A. And I want to assign a number, 00:22:45.140 --> 00:22:50.720 align:middle line:84% a numerical probability, to this particular subset which 00:22:50.720 --> 00:22:56.841 align:middle line:84% represents my belief about how likely this set is to occur. 00:22:56.841 --> 00:22:57.340 align:middle line:90% OK. 00:22:57.340 --> 00:22:59.710 align:middle line:90% What do we mean "to occur?" 00:22:59.710 --> 00:23:01.720 align:middle line:84% And I'm introducing here a language 00:23:01.720 --> 00:23:03.770 align:middle line:84% that's being used in probability theory. 00:23:03.770 --> 00:23:06.580 align:middle line:84% When we talk about subsets of the sample space, 00:23:06.580 --> 00:23:10.470 align:middle line:84% we usually call them events, as opposed to subsets. 00:23:10.470 --> 00:23:13.250 align:middle line:84% And the reason is because it works nicely 00:23:13.250 --> 00:23:16.710 align:middle line:84% with the language that describes what's going on. 00:23:16.710 --> 00:23:19.010 align:middle line:90% So the outcome is a point. 00:23:19.010 --> 00:23:20.540 align:middle line:90% The outcome is random. 00:23:20.540 --> 00:23:25.720 align:middle line:84% The outcome may be inside this set, in which case 00:23:25.720 --> 00:23:31.270 align:middle line:84% we say that event A occurred, if we get an outcome inside here. 00:23:31.270 --> 00:23:35.120 align:middle line:84% Or the outcome may fall outside the set, in which case 00:23:35.120 --> 00:23:38.530 align:middle line:84% we say that event A did not occur. 00:23:38.530 --> 00:23:42.310 align:middle line:84% So we're going to assign probabilities to events. 00:23:42.310 --> 00:23:45.630 align:middle line:84% And now, how should we do this assignment? 00:23:45.630 --> 00:23:48.840 align:middle line:84% Well, probabilities are meant to describe your beliefs 00:23:48.840 --> 00:23:52.880 align:middle line:84% about which sets are more likely to occur versus other sets. 00:23:52.880 --> 00:23:56.080 align:middle line:84% So there's many ways that you can assign those probabilities. 00:23:56.080 --> 00:23:59.290 align:middle line:84% But there are some ground rules for this game. 00:23:59.290 --> 00:24:03.310 align:middle line:84% First, we want probabilities to be numbers between zero and one 00:24:03.310 --> 00:24:06.740 align:middle line:84% because that's the usual convention. 00:24:06.740 --> 00:24:08.680 align:middle line:84% So a probability of zero means we're 00:24:08.680 --> 00:24:10.820 align:middle line:84% certain that something is not going to happen. 00:24:10.820 --> 00:24:12.460 align:middle line:84% Probability of one means that we're 00:24:12.460 --> 00:24:14.870 align:middle line:84% essentially certain that something's going to happen. 00:24:14.870 --> 00:24:17.450 align:middle line:84% So we want numbers between zero and one. 00:24:17.450 --> 00:24:19.740 align:middle line:90% We also want a few other things. 00:24:19.740 --> 00:24:21.740 align:middle line:84% And those few other things are going 00:24:21.740 --> 00:24:25.060 align:middle line:84% to be encapsulated in a set of axioms. 00:24:25.060 --> 00:24:27.610 align:middle line:84% What "axioms" means in this context, 00:24:27.610 --> 00:24:31.740 align:middle line:84% it's the ground rules that any legitimate probabilistic model 00:24:31.740 --> 00:24:33.410 align:middle line:90% should obey. 00:24:33.410 --> 00:24:37.080 align:middle line:84% You have a choice of what kind of probabilities you use. 00:24:37.080 --> 00:24:39.420 align:middle line:84% But, no matter what you use, they 00:24:39.420 --> 00:24:43.660 align:middle line:84% should obey certain consistency properties because if they 00:24:43.660 --> 00:24:45.220 align:middle line:84% obey those properties, then you can 00:24:45.220 --> 00:24:47.370 align:middle line:84% go ahead and do useful calculations 00:24:47.370 --> 00:24:49.360 align:middle line:90% and do some useful reasoning. 00:24:49.360 --> 00:24:51.010 align:middle line:90% So what are these properties? 00:24:51.010 --> 00:24:55.060 align:middle line:84% First, probabilities should be non-negative. 00:24:55.060 --> 00:24:56.590 align:middle line:90% OK? 00:24:56.590 --> 00:24:57.530 align:middle line:90% That's our convention. 00:24:57.530 --> 00:25:00.350 align:middle line:84% We want probabilities to be numbers between zero and one. 00:25:00.350 --> 00:25:02.130 align:middle line:84% So they should certainly be non-negative. 00:25:02.130 --> 00:25:04.170 align:middle line:84% The probability that event A occurs 00:25:04.170 --> 00:25:06.135 align:middle line:90% should be a non-negative number. 00:25:06.135 --> 00:25:08.110 align:middle line:90% What's the second axiom? 00:25:08.110 --> 00:25:13.760 align:middle line:84% The probability of the entire sample space is equal to one. 00:25:13.760 --> 00:25:15.590 align:middle line:90% Why does this make sense? 00:25:15.590 --> 00:25:20.120 align:middle line:84% Well, the outcome is certain to be an element of the sample 00:25:20.120 --> 00:25:23.720 align:middle line:84% space because we set up a sample space, which is collectively 00:25:23.720 --> 00:25:24.660 align:middle line:90% exhaustive. 00:25:24.660 --> 00:25:27.559 align:middle line:84% No matter what the outcome is, it's 00:25:27.559 --> 00:25:29.350 align:middle line:84% going to be an element of the sample space. 00:25:29.350 --> 00:25:33.710 align:middle line:84% We're certain that event omega is going to occur. 00:25:33.710 --> 00:25:36.700 align:middle line:84% Therefore, we represent this certainty 00:25:36.700 --> 00:25:41.520 align:middle line:84% by saying that the probability of omega is equal to one. 00:25:41.520 --> 00:25:47.180 align:middle line:90% Pretty straightforward so far. 00:25:47.180 --> 00:25:52.240 align:middle line:84% The more interesting axiom is the third rule. 00:25:52.240 --> 00:25:55.580 align:middle line:84% Before getting into it, just a quick reminder. 00:25:55.580 --> 00:26:01.950 align:middle line:84% If you have two sets, A and B, the intersection of A and B 00:26:01.950 --> 00:26:07.220 align:middle line:84% consists of those elements that belong both to A and B. 00:26:07.220 --> 00:26:09.580 align:middle line:90% And we denote it this way. 00:26:09.580 --> 00:26:11.060 align:middle line:84% When you think probabilistically, 00:26:11.060 --> 00:26:15.090 align:middle line:84% the way to think of intersection is by using the word "and." 00:26:15.090 --> 00:26:18.890 align:middle line:84% This event, this intersection, is the event 00:26:18.890 --> 00:26:22.450 align:middle line:90% that A occurred and B occurred. 00:26:22.450 --> 00:26:25.200 align:middle line:84% If I get an outcome inside here, A has occurred 00:26:25.200 --> 00:26:27.950 align:middle line:84% and B has occurred at the same time. 00:26:27.950 --> 00:26:31.150 align:middle line:84% So you may find the word "and" to be a little more convenient 00:26:31.150 --> 00:26:33.680 align:middle line:90% than the word "intersection." 00:26:33.680 --> 00:26:35.950 align:middle line:84% And similarly, we have some notation 00:26:35.950 --> 00:26:42.280 align:middle line:84% for the union of two events, which we write this way. 00:26:42.280 --> 00:26:44.930 align:middle line:84% The union of two sets, or two events, 00:26:44.930 --> 00:26:47.080 align:middle line:84% is the collection of all the elements 00:26:47.080 --> 00:26:49.990 align:middle line:84% that belong either to the first set, or to the second, 00:26:49.990 --> 00:26:51.400 align:middle line:90% or to both. 00:26:51.400 --> 00:26:54.970 align:middle line:84% When you talk about events, you can use the word "or." 00:26:54.970 --> 00:26:59.990 align:middle line:84% So this is the event that A occurred or B occurred. 00:26:59.990 --> 00:27:03.700 align:middle line:84% And this "or" means that it could also be that both of them 00:27:03.700 --> 00:27:04.200 align:middle line:90% occurred. 00:27:04.200 --> 00:27:08.650 align:middle line:90% 00:27:08.650 --> 00:27:09.150 align:middle line:90% OK. 00:27:09.150 --> 00:27:10.870 align:middle line:84% So now that we have this notation, 00:27:10.870 --> 00:27:13.835 align:middle line:90% what does the third axiom say? 00:27:13.835 --> 00:27:19.830 align:middle line:84% The third axiom says that if we have two events, A and B, 00:27:19.830 --> 00:27:23.140 align:middle line:90% that have no common elements-- 00:27:23.140 --> 00:27:29.180 align:middle line:84% so here's A, here's B, and perhaps this 00:27:29.180 --> 00:27:31.140 align:middle line:90% is our big sample space. 00:27:31.140 --> 00:27:33.470 align:middle line:84% The two events have no common elements. 00:27:33.470 --> 00:27:36.510 align:middle line:84% So the intersection of the two events is the empty set. 00:27:36.510 --> 00:27:38.930 align:middle line:84% There's nothing in their intersection. 00:27:38.930 --> 00:27:42.780 align:middle line:84% Then, the total probability of A together with B 00:27:42.780 --> 00:27:46.600 align:middle line:84% has to be equal to the sum of the individual probabilities. 00:27:46.600 --> 00:27:50.020 align:middle line:84% So the probability that A occurs or B occurs 00:27:50.020 --> 00:27:52.930 align:middle line:84% is equal to the probability that A occurs plus the probability 00:27:52.930 --> 00:27:55.040 align:middle line:90% that B occurs. 00:27:55.040 --> 00:27:58.860 align:middle line:84% So think of probability as being cream cheese. 00:27:58.860 --> 00:28:03.020 align:middle line:84% You have one pound of cream cheese, the total probability 00:28:03.020 --> 00:28:05.340 align:middle line:84% assigned to the entire sample space. 00:28:05.340 --> 00:28:12.780 align:middle line:84% And that cream cheese is spread out over this set. 00:28:12.780 --> 00:28:15.760 align:middle line:84% The probability of A is how much cream cheese 00:28:15.760 --> 00:28:20.120 align:middle line:84% sits on top of A. Probability of B is how much sits on top of B. 00:28:20.120 --> 00:28:25.370 align:middle line:84% The probability of A union B is the total amount of cream 00:28:25.370 --> 00:28:28.660 align:middle line:84% cheese sitting on top of this and that, 00:28:28.660 --> 00:28:31.510 align:middle line:84% which is obviously the sum of how much is sitting here 00:28:31.510 --> 00:28:33.220 align:middle line:90% and how much is sitting there. 00:28:33.220 --> 00:28:35.670 align:middle line:84% So probabilities behave like cream cheese, 00:28:35.670 --> 00:28:38.450 align:middle line:90% or they behave like mass. 00:28:38.450 --> 00:28:47.690 align:middle line:84% For example, if you think of some material object, 00:28:47.690 --> 00:28:50.510 align:middle line:84% the mass of this set consisting of two pieces 00:28:50.510 --> 00:28:53.120 align:middle line:84% is obviously the sum of the two masses. 00:28:53.120 --> 00:28:55.680 align:middle line:84% So this property is a very intuitive one. 00:28:55.680 --> 00:28:58.282 align:middle line:84% It's a pretty natural one to have. 00:28:58.282 --> 00:29:00.640 align:middle line:90% OK. 00:29:00.640 --> 00:29:03.880 align:middle line:84% Are these axioms enough for what we want to do? 00:29:03.880 --> 00:29:07.670 align:middle line:84% I mentioned a while ago that we want probabilities to be 00:29:07.670 --> 00:29:10.069 align:middle line:90% numbers between zero and one. 00:29:10.069 --> 00:29:12.110 align:middle line:84% Here's an axiom that tells you that probabilities 00:29:12.110 --> 00:29:13.710 align:middle line:90% are non-negative. 00:29:13.710 --> 00:29:15.590 align:middle line:84% Should we have another axiom that 00:29:15.590 --> 00:29:21.670 align:middle line:84% tells us that probabilities are less than or equal to one? 00:29:21.670 --> 00:29:23.150 align:middle line:90% It's a desirable property. 00:29:23.150 --> 00:29:26.090 align:middle line:84% We would like to have it in our hands. 00:29:26.090 --> 00:29:29.030 align:middle line:90% OK, why is it not in that list? 00:29:29.030 --> 00:29:32.680 align:middle line:84% Well, the people who are in the axiom making business 00:29:32.680 --> 00:29:34.610 align:middle line:84% are mathematicians and mathematicians 00:29:34.610 --> 00:29:36.390 align:middle line:90% tend to be pretty laconic. 00:29:36.390 --> 00:29:40.020 align:middle line:84% You don't say something if you don't have to say it. 00:29:40.020 --> 00:29:42.580 align:middle line:90% And this is the case here. 00:29:42.580 --> 00:29:46.410 align:middle line:84% We don't need that extra axiom because we can derive it 00:29:46.410 --> 00:29:48.440 align:middle line:90% from the existing axioms. 00:29:48.440 --> 00:29:50.590 align:middle line:90% Here's how it goes. 00:29:50.590 --> 00:29:55.180 align:middle line:84% One is the probability over the entire sample space. 00:29:55.180 --> 00:29:57.450 align:middle line:84% Here we're using the second axiom. 00:29:57.450 --> 00:30:00.310 align:middle line:90% 00:30:00.310 --> 00:30:05.690 align:middle line:84% Now the sample space consists of A together 00:30:05.690 --> 00:30:07.680 align:middle line:90% with the complement of A. OK? 00:30:07.680 --> 00:30:11.200 align:middle line:90% 00:30:11.200 --> 00:30:13.000 align:middle line:84% When I write the complement of A, 00:30:13.000 --> 00:30:16.800 align:middle line:84% I mean the complement of A inside of the set omega. 00:30:16.800 --> 00:30:21.700 align:middle line:84% So we have omega, here's A, here's the complement of A, 00:30:21.700 --> 00:30:24.660 align:middle line:90% and the overall set is omega. 00:30:24.660 --> 00:30:25.350 align:middle line:90% OK. 00:30:25.350 --> 00:30:27.520 align:middle line:90% Now, what's the next step? 00:30:27.520 --> 00:30:28.650 align:middle line:90% What should I do next? 00:30:28.650 --> 00:30:31.320 align:middle line:90% Which axiom should I use? 00:30:31.320 --> 00:30:35.950 align:middle line:84% We use axiom three because a set and the complement of that set 00:30:35.950 --> 00:30:36.730 align:middle line:90% are disjoint. 00:30:36.730 --> 00:30:38.770 align:middle line:84% They don't have any common elements. 00:30:38.770 --> 00:30:42.810 align:middle line:84% So axiom three applies and tells me 00:30:42.810 --> 00:30:45.530 align:middle line:84% that this is the probability of A 00:30:45.530 --> 00:30:48.150 align:middle line:84% plus the probability of A complement. 00:30:48.150 --> 00:30:52.090 align:middle line:84% In particular, the probability of A 00:30:52.090 --> 00:30:56.850 align:middle line:84% is equal to one minus the probability of A complement, 00:30:56.850 --> 00:31:00.321 align:middle line:84% and this is less than or equal to one. 00:31:00.321 --> 00:31:00.820 align:middle line:90% Why? 00:31:00.820 --> 00:31:03.430 align:middle line:90% 00:31:03.430 --> 00:31:06.670 align:middle line:84% Because probabilities are non-negative, 00:31:06.670 --> 00:31:09.810 align:middle line:90% by the first axiom. 00:31:09.810 --> 00:31:10.310 align:middle line:90% OK. 00:31:10.310 --> 00:31:12.440 align:middle line:84% So we got the conclusion that we wanted. 00:31:12.440 --> 00:31:15.520 align:middle line:84% Probabilities are always less than or equal to one, 00:31:15.520 --> 00:31:18.250 align:middle line:84% and this is a simple consequence of the three axioms 00:31:18.250 --> 00:31:20.230 align:middle line:90% that we have. 00:31:20.230 --> 00:31:23.130 align:middle line:84% This is a really nice argument because it actually 00:31:23.130 --> 00:31:26.560 align:middle line:90% uses each one of those axioms. 00:31:26.560 --> 00:31:28.160 align:middle line:84% The argument is simple, but you have 00:31:28.160 --> 00:31:30.490 align:middle line:84% to use all of these three properties 00:31:30.490 --> 00:31:33.050 align:middle line:84% to get the conclusion that you want. 00:31:33.050 --> 00:31:33.720 align:middle line:90% OK. 00:31:33.720 --> 00:31:37.140 align:middle line:84% So we can get interesting things out of our axioms. 00:31:37.140 --> 00:31:40.050 align:middle line:84% Can we get some more interesting ones? 00:31:40.050 --> 00:31:44.540 align:middle line:84% How about the union of three sets? 00:31:44.540 --> 00:31:47.000 align:middle line:84% What kind of probability should it have? 00:31:47.000 --> 00:31:52.870 align:middle line:84% So here's an event consisting of three pieces. 00:31:52.870 --> 00:31:55.500 align:middle line:84% And I want to say something about the probability 00:31:55.500 --> 00:32:01.080 align:middle line:84% of A union B union C. What I would like to say 00:32:01.080 --> 00:32:04.940 align:middle line:84% is that this probability is equal to the sum of the three 00:32:04.940 --> 00:32:07.140 align:middle line:90% individual probabilities. 00:32:07.140 --> 00:32:08.860 align:middle line:90% How can I do it? 00:32:08.860 --> 00:32:10.870 align:middle line:84% I have an axiom that tells me that I 00:32:10.870 --> 00:32:12.760 align:middle line:90% can do it for two events. 00:32:12.760 --> 00:32:15.370 align:middle line:84% I don't have an axiom for three events. 00:32:15.370 --> 00:32:18.730 align:middle line:84% Well, maybe I can massage things and still 00:32:18.730 --> 00:32:20.620 align:middle line:90% be able to use that axiom. 00:32:20.620 --> 00:32:22.700 align:middle line:90% And here's the trick. 00:32:22.700 --> 00:32:26.150 align:middle line:84% The union of three sets, you can think of it 00:32:26.150 --> 00:32:30.500 align:middle line:84% as forming the union of the first two sets 00:32:30.500 --> 00:32:35.670 align:middle line:84% and then taking the union with the third set. 00:32:35.670 --> 00:32:36.530 align:middle line:90% OK? 00:32:36.530 --> 00:32:39.450 align:middle line:84% So taking unions, you can take the unions in any order 00:32:39.450 --> 00:32:40.440 align:middle line:90% that you want. 00:32:40.440 --> 00:32:44.580 align:middle line:84% So here we have the union of two sets. 00:32:44.580 --> 00:32:49.150 align:middle line:84% Now, ABC are disjoint, by assumption 00:32:49.150 --> 00:32:51.780 align:middle line:90% or that's how I drew it. 00:32:51.780 --> 00:32:55.760 align:middle line:84% So if A, B, and C are disjoint, then A union B 00:32:55.760 --> 00:32:59.030 align:middle line:84% is disjoint from C. So here we have 00:32:59.030 --> 00:33:01.400 align:middle line:90% the union of two disjoint sets. 00:33:01.400 --> 00:33:05.810 align:middle line:84% So by the additivity axiom, the probability of that the union 00:33:05.810 --> 00:33:08.960 align:middle line:84% is going to be the probability of the first set 00:33:08.960 --> 00:33:12.000 align:middle line:84% plus the probability of the second set. 00:33:12.000 --> 00:33:15.580 align:middle line:84% And now I can use the additivity axiom once more 00:33:15.580 --> 00:33:17.520 align:middle line:84% to write that this is probability 00:33:17.520 --> 00:33:21.240 align:middle line:84% of A plus probability of B plus probability 00:33:21.240 --> 00:33:26.290 align:middle line:84% of C. So by using this axiom which was stated for two sets, 00:33:26.290 --> 00:33:29.580 align:middle line:84% we can actually derive a similar property 00:33:29.580 --> 00:33:32.450 align:middle line:84% for the union of three disjoint sets. 00:33:32.450 --> 00:33:34.960 align:middle line:84% And then you can repeat this argument as many times 00:33:34.960 --> 00:33:35.940 align:middle line:90% as you want. 00:33:35.940 --> 00:33:38.750 align:middle line:84% It's valid for the union of ten disjoint sets, 00:33:38.750 --> 00:33:41.240 align:middle line:84% for the union of a hundred disjoint sets, 00:33:41.240 --> 00:33:44.910 align:middle line:84% for the union of any finite number of sets. 00:33:44.910 --> 00:33:53.210 align:middle line:84% So if A1 up to An are disjoint, then the probability 00:33:53.210 --> 00:33:59.185 align:middle line:84% of A1 union An is equal to the sum of the probabilities 00:33:59.185 --> 00:34:01.500 align:middle line:90% of the individual sets. 00:34:01.500 --> 00:34:04.180 align:middle line:90% 00:34:04.180 --> 00:34:05.740 align:middle line:90% OK. 00:34:05.740 --> 00:34:10.790 align:middle line:84% Special case of this is when we're dealing with finite sets. 00:34:10.790 --> 00:34:14.300 align:middle line:84% Suppose I have just a finite set of outcomes. 00:34:14.300 --> 00:34:17.150 align:middle line:84% I put them together in a set and I'm 00:34:17.150 --> 00:34:19.630 align:middle line:84% interested in the probability of that set. 00:34:19.630 --> 00:34:22.050 align:middle line:90% So here's our sample space. 00:34:22.050 --> 00:34:26.620 align:middle line:84% There's lots of outcomes, but I'm taking a few of these 00:34:26.620 --> 00:34:30.120 align:middle line:90% and I form a set out of them. 00:34:30.120 --> 00:34:34.760 align:middle line:84% This is a set consisting of, in this picture, three elements. 00:34:34.760 --> 00:34:38.260 align:middle line:84% In general, it consists of k elements. 00:34:38.260 --> 00:34:41.340 align:middle line:84% Now, a finite set, I can write it 00:34:41.340 --> 00:34:44.889 align:middle line:84% as a union of single element sets. 00:34:44.889 --> 00:34:49.080 align:middle line:84% So this set here is the union of this one element set, 00:34:49.080 --> 00:34:51.469 align:middle line:84% together with this one element set together 00:34:51.469 --> 00:34:53.980 align:middle line:90% with that one element set. 00:34:53.980 --> 00:34:55.909 align:middle line:84% So the total probability of this set 00:34:55.909 --> 00:35:00.180 align:middle line:84% is going to be the sum of the probabilities of the one 00:35:00.180 --> 00:35:02.510 align:middle line:90% element sets. 00:35:02.510 --> 00:35:05.840 align:middle line:84% Now, probability of a one element 00:35:05.840 --> 00:35:09.320 align:middle line:84% set, you need to use the brackets here 00:35:09.320 --> 00:35:12.260 align:middle line:84% because probabilities are assigned to sets. 00:35:12.260 --> 00:35:15.560 align:middle line:84% But this gets kind of tedious, so here one abuses 00:35:15.560 --> 00:35:18.510 align:middle line:84% notation a little bit and we get rid of those 00:35:18.510 --> 00:35:20.740 align:middle line:84% brackets and just write probability 00:35:20.740 --> 00:35:24.030 align:middle line:84% of this single, individual outcome. 00:35:24.030 --> 00:35:26.770 align:middle line:84% In any case, conclusion from this exercise 00:35:26.770 --> 00:35:32.010 align:middle line:84% is that the total probability of a finite collection 00:35:32.010 --> 00:35:35.790 align:middle line:84% of possible outcomes, the total probability 00:35:35.790 --> 00:35:38.360 align:middle line:84% is equal to the sum of the probabilities 00:35:38.360 --> 00:35:42.190 align:middle line:90% of individual elements. 00:35:42.190 --> 00:35:46.460 align:middle line:84% So these are basically the axioms of probability theory. 00:35:46.460 --> 00:35:49.970 align:middle line:84% Or, well, they're almost the axioms. 00:35:49.970 --> 00:35:53.060 align:middle line:84% There are some subtleties that are involved here. 00:35:53.060 --> 00:35:58.480 align:middle line:84% One subtlety is that this axiom here doesn't quite 00:35:58.480 --> 00:36:01.340 align:middle line:84% do the job for everything we would like to do. 00:36:01.340 --> 00:36:05.080 align:middle line:84% And we're going to come back to this at the end of the lecture. 00:36:05.080 --> 00:36:10.380 align:middle line:84% A second subtlety has to do with weird sets. 00:36:10.380 --> 00:36:13.270 align:middle line:84% We said that an event is a subset of the sample space 00:36:13.270 --> 00:36:16.712 align:middle line:84% and we assign probabilities to events. 00:36:16.712 --> 00:36:19.760 align:middle line:84% Does this mean that we are going to assign probability 00:36:19.760 --> 00:36:23.500 align:middle line:84% to every possible subset of the sample space? 00:36:23.500 --> 00:36:26.660 align:middle line:84% Ideally, we would wish to do that. 00:36:26.660 --> 00:36:29.580 align:middle line:84% Unfortunately, this is not always possible. 00:36:29.580 --> 00:36:34.510 align:middle line:84% If you take a sample space, such as the square, 00:36:34.510 --> 00:36:36.950 align:middle line:84% the square has nice subsets, those 00:36:36.950 --> 00:36:40.220 align:middle line:84% that you can describe by cutting it with lines and so on. 00:36:40.220 --> 00:36:45.270 align:middle line:84% But it does have some very ugly subsets, as well, 00:36:45.270 --> 00:36:47.210 align:middle line:84% that are impossible to visualize, 00:36:47.210 --> 00:36:50.030 align:middle line:84% impossible to imagine, but they do exist. 00:36:50.030 --> 00:36:53.130 align:middle line:84% And those very weird sets are such that there's 00:36:53.130 --> 00:36:56.220 align:middle line:84% no way to assign probabilities to them in a way that's 00:36:56.220 --> 00:36:58.500 align:middle line:84% consistent with the axioms of probability. 00:36:58.500 --> 00:36:59.000 align:middle line:90% OK. 00:36:59.000 --> 00:37:01.460 align:middle line:84% So this is a very, very fine point 00:37:01.460 --> 00:37:05.940 align:middle line:84% that you can immediately forget for the rest of this class. 00:37:05.940 --> 00:37:08.040 align:middle line:84% You will only encounter these sets 00:37:08.040 --> 00:37:12.340 align:middle line:84% if you end up doing doctoral work on the theoretical aspects 00:37:12.340 --> 00:37:15.910 align:middle line:90% of probability theory. 00:37:15.910 --> 00:37:18.350 align:middle line:84% So it's just a mathematical subtlety 00:37:18.350 --> 00:37:20.740 align:middle line:84% that some very weird sets do not have 00:37:20.740 --> 00:37:22.560 align:middle line:90% probabilities assigned to them. 00:37:22.560 --> 00:37:24.780 align:middle line:84% But we're not going to encounter these sets 00:37:24.780 --> 00:37:26.885 align:middle line:84% and they do not show up in any applications. 00:37:26.885 --> 00:37:29.340 align:middle line:90% 00:37:29.340 --> 00:37:29.840 align:middle line:90% OK. 00:37:29.840 --> 00:37:32.410 align:middle line:84% So now let's revisit our examples. 00:37:32.410 --> 00:37:34.800 align:middle line:84% Let's go back to the die example. 00:37:34.800 --> 00:37:36.950 align:middle line:90% We have our sample space. 00:37:36.950 --> 00:37:40.830 align:middle line:84% Now we need to assign a probability law. 00:37:40.830 --> 00:37:43.260 align:middle line:84% There's lots of possible probability laws 00:37:43.260 --> 00:37:44.690 align:middle line:90% that you can assign. 00:37:44.690 --> 00:37:48.200 align:middle line:84% I'm picking one here, arbitrarily, 00:37:48.200 --> 00:37:50.960 align:middle line:84% in which I say that every possible outcome has 00:37:50.960 --> 00:37:55.440 align:middle line:90% the same probability of 1/16. 00:37:55.440 --> 00:37:56.040 align:middle line:90% OK. 00:37:56.040 --> 00:37:58.010 align:middle line:90% Why do I make this model? 00:37:58.010 --> 00:38:01.990 align:middle line:84% Well, empirically, if you have well-manufactured dice, 00:38:01.990 --> 00:38:04.540 align:middle line:90% they tend to behave that way. 00:38:04.540 --> 00:38:06.870 align:middle line:84% We will be coming back to this kind of story 00:38:06.870 --> 00:38:08.500 align:middle line:90% later in this class. 00:38:08.500 --> 00:38:12.790 align:middle line:84% But I'm not saying that this is the only probability 00:38:12.790 --> 00:38:13.720 align:middle line:90% law that there can be. 00:38:13.720 --> 00:38:17.460 align:middle line:84% You might have weird dice in which certain outcomes are 00:38:17.460 --> 00:38:19.280 align:middle line:90% more likely than others. 00:38:19.280 --> 00:38:21.730 align:middle line:84% But to keep things simple, let's take every outcome 00:38:21.730 --> 00:38:24.870 align:middle line:84% to have the same probability of 1/16. 00:38:24.870 --> 00:38:26.790 align:middle line:90% OK. 00:38:26.790 --> 00:38:29.110 align:middle line:84% Now that we have in our hands a sample space 00:38:29.110 --> 00:38:31.350 align:middle line:84% and the probability law, we can actually 00:38:31.350 --> 00:38:33.250 align:middle line:90% solve any problem there is. 00:38:33.250 --> 00:38:36.070 align:middle line:84% We can answer any question that could be posed to us. 00:38:36.070 --> 00:38:38.230 align:middle line:84% For example, what's the probability 00:38:38.230 --> 00:38:43.590 align:middle line:84% that the outcome, which is this pair, is either 1,1 or 1,2. 00:38:43.590 --> 00:38:50.160 align:middle line:84% We're talking here about this particular event, 1,1 or 1,2. 00:38:50.160 --> 00:38:53.300 align:middle line:84% So it's an event consisting of these two items. 00:38:53.300 --> 00:38:55.880 align:middle line:84% According to what we were just discussing, 00:38:55.880 --> 00:38:58.720 align:middle line:84% the probability of a finite collection of outcomes 00:38:58.720 --> 00:39:01.170 align:middle line:84% is the sum of their individual probabilities. 00:39:01.170 --> 00:39:03.710 align:middle line:84% Each one of them has probability of 1/16, 00:39:03.710 --> 00:39:07.720 align:middle line:84% so the probability of this is 2/16. 00:39:07.720 --> 00:39:12.240 align:middle line:84% How about the probability of the event that x is equal to one. 00:39:12.240 --> 00:39:14.500 align:middle line:84% x is the first roll, so that's the probability 00:39:14.500 --> 00:39:18.120 align:middle line:84% that the first roll is equal to one. 00:39:18.120 --> 00:39:22.340 align:middle line:84% Notice the syntax that's being used here. 00:39:22.340 --> 00:39:26.200 align:middle line:84% Probabilities are assigned to subsets, to sets, 00:39:26.200 --> 00:39:30.800 align:middle line:84% so we think of this as meaning the set of all outcomes 00:39:30.800 --> 00:39:33.660 align:middle line:90% such that x is equal to one. 00:39:33.660 --> 00:39:35.210 align:middle line:90% How do you answer this question? 00:39:35.210 --> 00:39:36.680 align:middle line:84% You go back to the picture and you 00:39:36.680 --> 00:39:40.810 align:middle line:84% try to visualize or identify this event of interest. 00:39:40.810 --> 00:39:45.570 align:middle line:84% x is equal to one corresponds to this event here. 00:39:45.570 --> 00:39:48.950 align:middle line:84% These are all the outcomes at which x is equal to one. 00:39:48.950 --> 00:39:50.100 align:middle line:90% There's four outcomes. 00:39:50.100 --> 00:39:54.180 align:middle line:84% Each one has probability 1/16, so the answer is 4/16. 00:39:54.180 --> 00:39:56.760 align:middle line:90% 00:39:56.760 --> 00:39:57.820 align:middle line:90% OK. 00:39:57.820 --> 00:40:06.482 align:middle line:84% How about the probability that x plus y is odd? 00:40:06.482 --> 00:40:07.100 align:middle line:90% OK. 00:40:07.100 --> 00:40:09.840 align:middle line:84% That will take a little bit more work. 00:40:09.840 --> 00:40:11.590 align:middle line:84% But you go to the sample space and you 00:40:11.590 --> 00:40:16.010 align:middle line:84% identify all the outcomes at which the sum is an odd number. 00:40:16.010 --> 00:40:24.480 align:middle line:84% So that's a place where the sum is odd, these are other places, 00:40:24.480 --> 00:40:28.610 align:middle line:84% and I guess that exhausts all the possible outcomes 00:40:28.610 --> 00:40:31.780 align:middle line:90% at which we have an odd sum. 00:40:31.780 --> 00:40:32.890 align:middle line:90% We count them. 00:40:32.890 --> 00:40:34.030 align:middle line:90% How many are there? 00:40:34.030 --> 00:40:35.540 align:middle line:84% There's a total of eight of them. 00:40:35.540 --> 00:40:40.490 align:middle line:84% Each one has probability 1/16, total probability is 8/16. 00:40:40.490 --> 00:40:41.620 align:middle line:90% And harder question. 00:40:41.620 --> 00:40:44.310 align:middle line:84% What is the probability that the minimum of the two rolls 00:40:44.310 --> 00:40:45.782 align:middle line:90% is equal to 2? 00:40:45.782 --> 00:40:47.240 align:middle line:84% This is something that you probably 00:40:47.240 --> 00:40:51.640 align:middle line:84% couldn't do in your head without the help of a diagram. 00:40:51.640 --> 00:40:54.780 align:middle line:84% But once you have a diagram, things are simple. 00:40:54.780 --> 00:40:55.760 align:middle line:90% You ask the question. 00:40:55.760 --> 00:40:59.600 align:middle line:84% OK, this is an event, that the minimum of the two rolls 00:40:59.600 --> 00:41:01.140 align:middle line:90% is equal to two. 00:41:01.140 --> 00:41:03.150 align:middle line:90% This can happen in several ways. 00:41:03.150 --> 00:41:05.250 align:middle line:84% What are the several ways that it can happen? 00:41:05.250 --> 00:41:07.980 align:middle line:84% Go to the diagram and try to identify them. 00:41:07.980 --> 00:41:11.620 align:middle line:84% So the minimum is equal to two if both of them are two's. 00:41:11.620 --> 00:41:14.230 align:middle line:90% 00:41:14.230 --> 00:41:18.780 align:middle line:84% Or it could be that x is two and y is bigger, or y is two 00:41:18.780 --> 00:41:21.900 align:middle line:90% and x is bigger. 00:41:21.900 --> 00:41:23.150 align:middle line:90% OK. 00:41:23.150 --> 00:41:28.670 align:middle line:84% I guess we rediscover that yellow and blue make green, 00:41:28.670 --> 00:41:31.670 align:middle line:84% so we see here that there's a total 00:41:31.670 --> 00:41:34.630 align:middle line:90% of five possible outcomes. 00:41:34.630 --> 00:41:37.645 align:middle line:84% The probability of this event is 5/16. 00:41:37.645 --> 00:41:41.250 align:middle line:90% 00:41:41.250 --> 00:41:46.590 align:middle line:84% Simple example, but the procedure 00:41:46.590 --> 00:41:48.680 align:middle line:84% that we followed in this example actually 00:41:48.680 --> 00:41:54.240 align:middle line:84% applies to any probability model you might ever encounter. 00:41:54.240 --> 00:41:56.630 align:middle line:84% You set up your sample space, you 00:41:56.630 --> 00:41:58.970 align:middle line:84% make a statement that describes the probability 00:41:58.970 --> 00:42:01.510 align:middle line:84% law over that sample space, then somebody asks you 00:42:01.510 --> 00:42:03.640 align:middle line:90% questions about various events. 00:42:03.640 --> 00:42:07.010 align:middle line:84% You go to your pictures, identify those events, 00:42:07.010 --> 00:42:10.075 align:middle line:84% pin them down, and then start kind 00:42:10.075 --> 00:42:12.990 align:middle line:84% of counting and calculating the total probability 00:42:12.990 --> 00:42:16.560 align:middle line:84% for those outcomes that you're considering. 00:42:16.560 --> 00:42:19.590 align:middle line:84% This example is a special case of what is called 00:42:19.590 --> 00:42:22.780 align:middle line:90% the discrete uniform law. 00:42:22.780 --> 00:42:24.970 align:middle line:84% The model obeys the discrete uniform law 00:42:24.970 --> 00:42:28.340 align:middle line:84% if all outcomes are equally likely. 00:42:28.340 --> 00:42:30.040 align:middle line:90% It doesn't have to be that way. 00:42:30.040 --> 00:42:33.290 align:middle line:84% That's just one example of a probability law. 00:42:33.290 --> 00:42:37.830 align:middle line:84% But when things are that way, if all outcomes are equally likely 00:42:37.830 --> 00:42:46.770 align:middle line:84% and we have N of them, and you have a set A that has little n 00:42:46.770 --> 00:42:50.870 align:middle line:84% elements, then each one of those elements 00:42:50.870 --> 00:42:53.840 align:middle line:84% has probability one over capital N 00:42:53.840 --> 00:42:56.450 align:middle line:84% since all outcomes are equally likely. 00:42:56.450 --> 00:42:58.360 align:middle line:84% And for our probabilities to add up to one, 00:42:58.360 --> 00:43:01.000 align:middle line:84% each one must have this much probability, 00:43:01.000 --> 00:43:02.620 align:middle line:90% and there's little n elements. 00:43:02.620 --> 00:43:06.120 align:middle line:84% That gives you the probability of the event of interest. 00:43:06.120 --> 00:43:08.710 align:middle line:84% So problems like the one in the previous slide 00:43:08.710 --> 00:43:10.840 align:middle line:84% and more generally of the type described here 00:43:10.840 --> 00:43:13.460 align:middle line:84% under discrete uniform law, these problems 00:43:13.460 --> 00:43:15.270 align:middle line:90% reduce to just counting. 00:43:15.270 --> 00:43:17.500 align:middle line:84% How many elements are there in my sample space? 00:43:17.500 --> 00:43:21.160 align:middle line:84% How many elements are there inside the event of interest? 00:43:21.160 --> 00:43:24.110 align:middle line:84% Counting is generally simple, but for some problems 00:43:24.110 --> 00:43:25.950 align:middle line:90% it gets pretty complicated. 00:43:25.950 --> 00:43:28.170 align:middle line:84% And in a couple of weeks, we're going 00:43:28.170 --> 00:43:30.390 align:middle line:84% to have to spend the whole lecture just 00:43:30.390 --> 00:43:33.280 align:middle line:84% on the subject of how to count systematically. 00:43:33.280 --> 00:43:36.730 align:middle line:84% Now the procedure we followed in the previous example 00:43:36.730 --> 00:43:38.440 align:middle line:84% is the same as the procedure you would 00:43:38.440 --> 00:43:41.330 align:middle line:84% follow in continuous probability problems. 00:43:41.330 --> 00:43:43.230 align:middle line:84% So, going back to our dart problem, 00:43:43.230 --> 00:43:46.550 align:middle line:84% we get the random point inside the square. 00:43:46.550 --> 00:43:48.030 align:middle line:90% That's our sample space. 00:43:48.030 --> 00:43:50.360 align:middle line:84% We need to assign a probability law. 00:43:50.360 --> 00:43:53.480 align:middle line:84% For lack of imagination, I'm taking the probability law 00:43:53.480 --> 00:43:56.280 align:middle line:90% to be the area of a subset. 00:43:56.280 --> 00:44:00.260 align:middle line:84% So if we have two subsets of the sample space 00:44:00.260 --> 00:44:04.890 align:middle line:84% that have equal areas, then I'm postulating that they 00:44:04.890 --> 00:44:06.386 align:middle line:90% are equally likely to occur. 00:44:06.386 --> 00:44:09.010 align:middle line:84% The probably that they fall here is the same as the probability 00:44:09.010 --> 00:44:11.430 align:middle line:90% that they fall there. 00:44:11.430 --> 00:44:13.670 align:middle line:84% The model doesn't have to be that way. 00:44:13.670 --> 00:44:15.980 align:middle line:84% But if I have sort of complete ignorance 00:44:15.980 --> 00:44:18.230 align:middle line:84% of which points are more likely than others, 00:44:18.230 --> 00:44:21.430 align:middle line:84% that might be the reasonable model to use. 00:44:21.430 --> 00:44:24.680 align:middle line:84% So equal areas mean equal probabilities. 00:44:24.680 --> 00:44:26.910 align:middle line:84% If the area is twice as large, the probability 00:44:26.910 --> 00:44:28.830 align:middle line:90% is going to be twice as big. 00:44:28.830 --> 00:44:32.130 align:middle line:90% So this is our model. 00:44:32.130 --> 00:44:34.580 align:middle line:90% We can now answer questions. 00:44:34.580 --> 00:44:35.730 align:middle line:90% Let's answer the easy one. 00:44:35.730 --> 00:44:40.660 align:middle line:84% What's the probability that the outcome is exactly this point? 00:44:40.660 --> 00:44:47.500 align:middle line:84% That of course is zero because a single point has zero area. 00:44:47.500 --> 00:44:49.580 align:middle line:84% And since this probability is equal to area, 00:44:49.580 --> 00:44:51.510 align:middle line:90% that's zero probability. 00:44:51.510 --> 00:44:54.180 align:middle line:84% How about the probability that the sum 00:44:54.180 --> 00:44:58.150 align:middle line:84% of the coordinates of the point that we got is less than 00:44:58.150 --> 00:45:00.090 align:middle line:90% or equal to 1/2? 00:45:00.090 --> 00:45:01.570 align:middle line:90% How do you deal with it? 00:45:01.570 --> 00:45:04.770 align:middle line:84% Well, you look at the picture again, at your sample space, 00:45:04.770 --> 00:45:08.130 align:middle line:84% and try to describe the event that you're talking about. 00:45:08.130 --> 00:45:11.480 align:middle line:84% The sum being less than 1/2 corresponds 00:45:11.480 --> 00:45:14.960 align:middle line:84% to getting an outcome that's below this line, where 00:45:14.960 --> 00:45:19.600 align:middle line:84% this line is the line where x plus y equals to 1/2. 00:45:19.600 --> 00:45:25.860 align:middle line:84% So the intercepts of that line with the axis are 1/2 and 1/2. 00:45:25.860 --> 00:45:28.630 align:middle line:84% So you describe the event visually 00:45:28.630 --> 00:45:30.780 align:middle line:84% and then you use your probability law. 00:45:30.780 --> 00:45:32.250 align:middle line:84% The probability law that we have is 00:45:32.250 --> 00:45:36.620 align:middle line:84% that the probability of a set is equal to the area of that set. 00:45:36.620 --> 00:45:39.900 align:middle line:84% So all we need to find is the area of this triangle, which 00:45:39.900 --> 00:45:48.881 align:middle line:84% is 1/2 times 1/2 times 1/2, half, equals to 1/8. 00:45:48.881 --> 00:45:49.380 align:middle line:90% OK. 00:45:49.380 --> 00:45:52.020 align:middle line:84% Moral from these two examples is that it's always 00:45:52.020 --> 00:45:54.960 align:middle line:84% useful to have a picture and work with a picture 00:45:54.960 --> 00:45:58.750 align:middle line:84% to visualize the events that you're talking about. 00:45:58.750 --> 00:46:01.340 align:middle line:84% And once you have a probability law in your hands, 00:46:01.340 --> 00:46:03.700 align:middle line:84% then it's a matter of calculation 00:46:03.700 --> 00:46:06.540 align:middle line:84% to find the probabilities of an event of interest. 00:46:06.540 --> 00:46:09.080 align:middle line:84% The calculations we did in these two examples, of course, 00:46:09.080 --> 00:46:10.130 align:middle line:90% were very simple. 00:46:10.130 --> 00:46:13.320 align:middle line:84% Sometimes calculations may be a lot harder, 00:46:13.320 --> 00:46:15.480 align:middle line:90% but it's a different business. 00:46:15.480 --> 00:46:18.820 align:middle line:84% It's a business of calculus, for example, or being 00:46:18.820 --> 00:46:20.250 align:middle line:90% good in algebra and so on. 00:46:20.250 --> 00:46:22.950 align:middle line:84% As far as probability is concerned, 00:46:22.950 --> 00:46:24.950 align:middle line:84% it's clear what you will be doing, 00:46:24.950 --> 00:46:27.920 align:middle line:84% and then maybe you're faced with a harder algebraic part 00:46:27.920 --> 00:46:30.540 align:middle line:84% to actually carry out the calculations. 00:46:30.540 --> 00:46:32.870 align:middle line:84% The area of a triangle is easy to compute. 00:46:32.870 --> 00:46:35.740 align:middle line:84% If I had put down a very complicated shape, 00:46:35.740 --> 00:46:38.390 align:middle line:84% then you might need to solve a hard integration 00:46:38.390 --> 00:46:40.650 align:middle line:84% problem to find the area of that shape, 00:46:40.650 --> 00:46:42.960 align:middle line:84% but that's stuff that belongs to another class 00:46:42.960 --> 00:46:46.306 align:middle line:84% that you have presumably mastered by now. 00:46:46.306 --> 00:46:47.000 align:middle line:90% Good, OK. 00:46:47.000 --> 00:46:48.580 align:middle line:84% So now let me spend just a couple 00:46:48.580 --> 00:46:52.170 align:middle line:84% of minutes to return to a point that I raised before. 00:46:52.170 --> 00:46:56.270 align:middle line:84% I was saying that the axiom that we had about additivity 00:46:56.270 --> 00:46:58.730 align:middle line:90% might not quite be enough. 00:46:58.730 --> 00:47:01.730 align:middle line:84% Let's illustrate what I mean by the following example. 00:47:01.730 --> 00:47:04.790 align:middle line:84% Think of the experiment where you keep flipping a coin 00:47:04.790 --> 00:47:08.120 align:middle line:84% and you wait until you obtain heads for the first time. 00:47:08.120 --> 00:47:11.390 align:middle line:84% What's the sample space of this experiment? 00:47:11.390 --> 00:47:12.970 align:middle line:84% It might happen the first flip, it 00:47:12.970 --> 00:47:14.700 align:middle line:90% might happen in the tenth flip. 00:47:14.700 --> 00:47:18.490 align:middle line:84% Heads for the first time might occur in the millionth flip. 00:47:18.490 --> 00:47:20.370 align:middle line:84% So the outcome of this experiment 00:47:20.370 --> 00:47:22.020 align:middle line:84% is going to be an integer and there's 00:47:22.020 --> 00:47:23.820 align:middle line:90% no bound to that integer. 00:47:23.820 --> 00:47:26.780 align:middle line:84% You might have to wait very much until that happens. 00:47:26.780 --> 00:47:28.860 align:middle line:84% So the natural sample space is the set 00:47:28.860 --> 00:47:30.950 align:middle line:90% of all possible integers. 00:47:30.950 --> 00:47:34.690 align:middle line:84% Somebody tells you some information 00:47:34.690 --> 00:47:36.250 align:middle line:90% about the probability law. 00:47:36.250 --> 00:47:39.250 align:middle line:84% The probability that you have to wait for n flips 00:47:39.250 --> 00:47:41.130 align:middle line:90% is equal to two to the minus n. 00:47:41.130 --> 00:47:42.850 align:middle line:90% Where did this come from? 00:47:42.850 --> 00:47:44.220 align:middle line:90% That's a separate story. 00:47:44.220 --> 00:47:45.730 align:middle line:90% Where did it come from? 00:47:45.730 --> 00:47:49.680 align:middle line:84% Somebody tells this to us, and those probabilities 00:47:49.680 --> 00:47:52.150 align:middle line:84% are plotted here as a function of n. 00:47:52.150 --> 00:47:54.770 align:middle line:84% And you're asked to find the probability that the outcome is 00:47:54.770 --> 00:47:56.660 align:middle line:90% an even number. 00:47:56.660 --> 00:47:59.920 align:middle line:84% How do you go about calculating that probability? 00:47:59.920 --> 00:48:01.830 align:middle line:84% So the probability of being an even number 00:48:01.830 --> 00:48:05.300 align:middle line:84% is the probability of the subset that consists 00:48:05.300 --> 00:48:08.380 align:middle line:90% of just the even numbers. 00:48:08.380 --> 00:48:11.040 align:middle line:84% So it would be a subset of this kind, that 00:48:11.040 --> 00:48:13.760 align:middle line:90% includes two, four, and so on. 00:48:13.760 --> 00:48:17.080 align:middle line:84% So any reasonable person would say, 00:48:17.080 --> 00:48:19.960 align:middle line:84% well the probability of obtaining an outcome that's 00:48:19.960 --> 00:48:23.160 align:middle line:84% either two or four or six and so on 00:48:23.160 --> 00:48:25.770 align:middle line:84% is equal to the probability of obtaining a two, 00:48:25.770 --> 00:48:28.180 align:middle line:84% plus the probability of obtaining a four, 00:48:28.180 --> 00:48:31.130 align:middle line:84% plus the probability of obtaining a six, and so on. 00:48:31.130 --> 00:48:33.640 align:middle line:84% These probabilities are given to us. 00:48:33.640 --> 00:48:35.990 align:middle line:90% So here I have to do my algebra. 00:48:35.990 --> 00:48:40.840 align:middle line:84% I add this geometric series and I get an answer of 1/3. 00:48:40.840 --> 00:48:43.430 align:middle line:84% That's what any reasonable person would do. 00:48:43.430 --> 00:48:46.460 align:middle line:84% But the person who only knows the axioms 00:48:46.460 --> 00:48:51.880 align:middle line:84% that they posted just a little earlier may get stuck. 00:48:51.880 --> 00:48:53.610 align:middle line:84% They would get stuck at this point. 00:48:53.610 --> 00:48:55.700 align:middle line:90% How do we justify this? 00:48:55.700 --> 00:48:59.000 align:middle line:90% 00:48:59.000 --> 00:49:03.740 align:middle line:84% We had this property for the union of disjoint sets 00:49:03.740 --> 00:49:05.780 align:middle line:84% and the corresponding property that tells us 00:49:05.780 --> 00:49:10.950 align:middle line:84% that the total probability of finitely many things, outcomes, 00:49:10.950 --> 00:49:13.740 align:middle line:84% is the sum of their individual probabilities. 00:49:13.740 --> 00:49:17.940 align:middle line:84% But here we're using it on an infinite collection. 00:49:17.940 --> 00:49:21.400 align:middle line:84% The probability of infinitely many points 00:49:21.400 --> 00:49:24.350 align:middle line:84% is equal to the sum of the probabilities 00:49:24.350 --> 00:49:26.070 align:middle line:90% of each one of these. 00:49:26.070 --> 00:49:30.840 align:middle line:84% To justify this step we need to introduce one additional rule, 00:49:30.840 --> 00:49:35.060 align:middle line:84% an additional axiom, that tells us that this step is actually 00:49:35.060 --> 00:49:36.160 align:middle line:90% legitimate. 00:49:36.160 --> 00:49:38.850 align:middle line:84% And this is the countable additivity axiom, 00:49:38.850 --> 00:49:42.560 align:middle line:84% which is a little stronger, or quite a bit stronger, 00:49:42.560 --> 00:49:45.140 align:middle line:84% than the additivity axiom we had before. 00:49:45.140 --> 00:49:48.830 align:middle line:84% It tells us that if we have a sequence of sets 00:49:48.830 --> 00:49:53.760 align:middle line:84% that are disjoint and we want to find their total probability, 00:49:53.760 --> 00:49:58.230 align:middle line:84% then we are allowed to add their individual probabilities. 00:49:58.230 --> 00:50:01.000 align:middle line:84% So the picture might be such as follows. 00:50:01.000 --> 00:50:07.420 align:middle line:84% We have a sequence of sets, A1, A2, A3, and so on. 00:50:07.420 --> 00:50:09.950 align:middle line:84% I guess in order to fit them inside the sample space, 00:50:09.950 --> 00:50:13.920 align:middle line:84% the sets need to get smaller and smaller perhaps. 00:50:13.920 --> 00:50:15.340 align:middle line:90% They are disjoint. 00:50:15.340 --> 00:50:17.330 align:middle line:90% We have a sequence of such sets. 00:50:17.330 --> 00:50:20.180 align:middle line:84% The total probability of falling anywhere 00:50:20.180 --> 00:50:23.140 align:middle line:84% inside one of those sets is the sum 00:50:23.140 --> 00:50:25.740 align:middle line:84% of their individual probabilities. 00:50:25.740 --> 00:50:28.510 align:middle line:84% A key subtlety that's involved here 00:50:28.510 --> 00:50:33.710 align:middle line:84% is that we're talking about a sequence of events. 00:50:33.710 --> 00:50:36.330 align:middle line:84% By "sequence" we mean that these events 00:50:36.330 --> 00:50:38.450 align:middle line:90% can be arranged in order. 00:50:38.450 --> 00:50:41.660 align:middle line:84% I can tell you the first event, the second event, 00:50:41.660 --> 00:50:43.530 align:middle line:90% the third event, and so on. 00:50:43.530 --> 00:50:45.860 align:middle line:84% So if you have such a collection of events 00:50:45.860 --> 00:50:50.060 align:middle line:84% that can be ordered as first, second, third, and so on, 00:50:50.060 --> 00:50:53.200 align:middle line:84% then you can add their probabilities 00:50:53.200 --> 00:50:55.790 align:middle line:84% to find the probability of their union. 00:50:55.790 --> 00:50:57.910 align:middle line:84% So this point is actually a little more subtle 00:50:57.910 --> 00:50:59.670 align:middle line:84% than you might appreciate at this point, 00:50:59.670 --> 00:51:02.480 align:middle line:84% and I'm going to return to it at the beginning 00:51:02.480 --> 00:51:04.010 align:middle line:90% of the next lecture. 00:51:04.010 --> 00:51:07.160 align:middle line:84% For now, enjoy the first week of classes 00:51:07.160 --> 00:51:09.380 align:middle line:90% and have a good weekend. 00:51:09.380 --> 00:51:11.230 align:middle line:90% Thank you.