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These are the user uploaded subtitles that are being translated: 0 00:00:00,000 --> 00:00:01,042 PETER REDDIEN: All right. 1 00:00:01,042 --> 00:00:04,200 So let's turn now to some probability to go through this. 2 00:00:04,200 --> 00:00:07,510 3 00:00:07,510 --> 00:00:08,010 OK. 4 00:00:08,010 --> 00:00:10,540 5 00:00:10,540 --> 00:00:17,980 So this will be part three of the lecture called 6 00:00:17,980 --> 00:00:18,760 Bayes' theorem. 7 00:00:18,760 --> 00:00:26,740 8 00:00:26,740 --> 00:00:28,660 OK. 9 00:00:28,660 --> 00:00:31,900 So let's have some sample space here. 10 00:00:31,900 --> 00:00:40,190 11 00:00:40,190 --> 00:00:45,360 OK, which is going to be a collection of all 12 00:00:45,360 --> 00:00:49,050 possible discrete events or discrete outcomes. 13 00:00:49,050 --> 00:01:11,910 14 00:01:11,910 --> 00:01:12,430 OK. 15 00:01:12,430 --> 00:01:13,972 So that's our sample space collection 16 00:01:13,972 --> 00:01:16,000 of all possible discrete outcomes, 17 00:01:16,000 --> 00:01:22,780 and then we'll have some events, where 18 00:01:22,780 --> 00:01:25,600 an event is a collection of points in this sample space. 19 00:01:25,600 --> 00:01:44,960 20 00:01:44,960 --> 00:01:45,460 OK. 21 00:01:45,460 --> 00:01:47,293 So let's go through this with some examples. 22 00:01:47,293 --> 00:01:54,700 I'll give you an event, event A. OK. 23 00:01:54,700 --> 00:01:57,430 We'll imagine we're rolling dice. 24 00:01:57,430 --> 00:01:59,980 And we've rolled the dice. 25 00:01:59,980 --> 00:02:06,190 And we get a 1, a 2, a 4, or a 6. 26 00:02:06,190 --> 00:02:10,570 27 00:02:10,570 --> 00:02:13,550 A 1, 2, a 4, or a 6. 28 00:02:13,550 --> 00:02:14,050 OK. 29 00:02:14,050 --> 00:02:18,460 So we could put those outcomes possible outcomes in our sample 30 00:02:18,460 --> 00:02:23,020 space here, 1, 2, 4, and 6, and then 31 00:02:23,020 --> 00:02:26,470 I'll just circle those and say that's event 32 00:02:26,470 --> 00:02:32,760 A. That one of those discrete outcomes happens. 33 00:02:32,760 --> 00:02:35,088 So that'll be A. 34 00:02:35,088 --> 00:02:49,870 And now I'll give you an event B, where we get a 3 or a 6. 35 00:02:49,870 --> 00:02:56,620 36 00:02:56,620 --> 00:02:57,120 OK. 37 00:02:57,120 --> 00:03:00,390 So we get a 3, or a 6. 38 00:03:00,390 --> 00:03:06,930 So that's our event B. And we could also 39 00:03:06,930 --> 00:03:12,210 have obtained a 5 on the dice as an outcome, 40 00:03:12,210 --> 00:03:17,030 but that was neither event A or event B. OK. 41 00:03:17,030 --> 00:03:20,000 42 00:03:20,000 --> 00:03:22,400 So now we can ask some things about the probability 43 00:03:22,400 --> 00:03:24,330 of these events. 44 00:03:24,330 --> 00:03:26,850 So let's start easy here. 45 00:03:26,850 --> 00:03:30,350 So we just say the probability of event A happening. 46 00:03:30,350 --> 00:03:32,940 47 00:03:32,940 --> 00:03:35,100 So there's six possible outcomes. 48 00:03:35,100 --> 00:03:37,170 Event A has four of those outcomes. 49 00:03:37,170 --> 00:03:38,795 So the probability of event A is 4/6. 50 00:03:38,795 --> 00:03:42,470 51 00:03:42,470 --> 00:03:45,645 Probability of event B 2/6. 52 00:03:45,645 --> 00:03:52,030 53 00:03:52,030 --> 00:03:52,530 OK. 54 00:03:52,530 --> 00:03:54,900 So now we could say, what's the probability 55 00:03:54,900 --> 00:03:57,870 of event A or event B? 56 00:03:57,870 --> 00:04:05,760 57 00:04:05,760 --> 00:04:06,780 OK. 58 00:04:06,780 --> 00:04:09,570 Now you recall in the past we've talked a little bit 59 00:04:09,570 --> 00:04:15,090 about probabilities of two events happening, where 60 00:04:15,090 --> 00:04:16,470 it could be one or the other. 61 00:04:16,470 --> 00:04:19,779 If they are mutually exclusive independent events, 62 00:04:19,779 --> 00:04:21,779 then you just add the probabilities of those two 63 00:04:21,779 --> 00:04:23,400 things happening together. 64 00:04:23,400 --> 00:04:26,950 But here, we see that they are not mutually exclusive. 65 00:04:26,950 --> 00:04:32,400 So if we just added the probability of A 66 00:04:32,400 --> 00:04:35,130 to the probability of B, we would 67 00:04:35,130 --> 00:04:38,850 be double counting the probability of this outcome 6. 68 00:04:38,850 --> 00:04:46,168 So we now need to subtract the probability of A and B 69 00:04:46,168 --> 00:04:48,585 to account for this fact that we would be double counting. 70 00:04:48,585 --> 00:04:54,220 71 00:04:54,220 --> 00:05:08,360 So this is 4/6 plus 2/6 minus 1/6 equals 5/6. 72 00:05:08,360 --> 00:05:10,770 73 00:05:10,770 --> 00:05:12,770 And when you just look at it in the sample space 74 00:05:12,770 --> 00:05:16,520 is sort of pretty obvious that five out of these six outcomes 75 00:05:16,520 --> 00:05:19,830 would be A or B. OK. 76 00:05:19,830 --> 00:05:22,340 77 00:05:22,340 --> 00:05:26,960 Now we could calculate a conditional probability, where 78 00:05:26,960 --> 00:05:33,710 we say what is the probability of A given B. 79 00:05:33,710 --> 00:05:36,035 And I'll denote this with this vertical line here. 80 00:05:36,035 --> 00:05:40,220 81 00:05:40,220 --> 00:05:47,840 For this vertical line indicates given, 82 00:05:47,840 --> 00:05:53,780 where in other words, if we know that event B has happened, 83 00:05:53,780 --> 00:05:57,257 what is the probability that A happened? 84 00:05:57,257 --> 00:05:58,840 Can anyone tell me what that would be? 85 00:05:58,840 --> 00:06:01,330 Yeah. 86 00:06:01,330 --> 00:06:05,230 So if we event B happened, we know we either got a 3 or a 6, 87 00:06:05,230 --> 00:06:07,105 there's a 50% chance it was a 6. 88 00:06:07,105 --> 00:06:11,600 89 00:06:11,600 --> 00:06:12,100 OK. 90 00:06:12,100 --> 00:06:17,200 91 00:06:17,200 --> 00:06:22,120 Go to probability of event B given A. What would that be? 92 00:06:22,120 --> 00:06:25,650 93 00:06:25,650 --> 00:06:26,150 1/4. 94 00:06:26,150 --> 00:06:30,580 95 00:06:30,580 --> 00:06:34,500 If we know that A happened, you got a 1, 2, 4, 6. 96 00:06:34,500 --> 00:06:36,500 One of those four options would have been the 6. 97 00:06:36,500 --> 00:06:42,100 So 1/4 chance we got event B given we got event A. OK. 98 00:06:42,100 --> 00:06:51,083 99 00:06:51,083 --> 00:06:52,750 Now we could ask, what's the probability 100 00:06:52,750 --> 00:07:03,520 of event A and event B. And this will 101 00:07:03,520 --> 00:07:05,860 be equal to the probability of event 102 00:07:05,860 --> 00:07:16,000 A given B times the probability of event B. 103 00:07:16,000 --> 00:07:20,210 104 00:07:20,210 --> 00:07:26,240 So if B happened, then we had a 1/2 chance of getting A. 105 00:07:26,240 --> 00:07:28,032 But there is only a 2/6 chance of B 106 00:07:28,032 --> 00:07:29,240 happening in the first place. 107 00:07:29,240 --> 00:07:34,500 108 00:07:34,500 --> 00:07:38,900 So that's the probability of event A and B happening. 109 00:07:38,900 --> 00:07:43,730 You could also do this as the probability of event B given 110 00:07:43,730 --> 00:07:55,880 A times the probability of A. So probability B given a was 1/4, 111 00:07:55,880 --> 00:07:57,440 and the probability of A was 4/6. 112 00:07:57,440 --> 00:08:01,910 113 00:08:01,910 --> 00:08:03,860 OK. 114 00:08:03,860 --> 00:08:05,864 Equals 1/6. 115 00:08:05,864 --> 00:08:09,248 You could just see it by looking at our sample space there. 116 00:08:09,248 --> 00:08:11,540 You say, the probability of getting A and B is only one 117 00:08:11,540 --> 00:08:14,030 of those six events. 118 00:08:14,030 --> 00:08:15,740 So that's how it's sort of intuitive 119 00:08:15,740 --> 00:08:17,980 what I just went through. 7665

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