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Would you like to inspect the original subtitles? These are the user uploaded subtitles that are being translated: 0 00:00:00,000 --> 00:00:02,760 PETER REDDIEN: What we're going to do 1 00:00:02,760 --> 00:00:07,770 is now try to set up a situation where 2 00:00:07,770 --> 00:00:09,960 we can get new information about probability, 3 00:00:09,960 --> 00:00:13,270 given new data or more information. 4 00:00:13,270 --> 00:00:16,740 So let's just consider some events here, 5 00:00:16,740 --> 00:00:27,190 back to our individual K. OK, so we 6 00:00:27,190 --> 00:00:33,640 could define some events here and say event 7 00:00:33,640 --> 00:00:47,260 A is that L is unaffected, and event B 8 00:00:47,260 --> 00:00:50,140 will be what we care about, that K is a carrier. 9 00:00:50,140 --> 00:01:02,270 10 00:01:02,270 --> 00:01:05,030 So what we really want to know is 11 00:01:05,030 --> 00:01:08,585 what is the probability of event B, given event A? 12 00:01:08,585 --> 00:01:12,200 That was the question I set up here. 13 00:01:12,200 --> 00:01:14,573 OK, so we want to derive an equation that 14 00:01:14,573 --> 00:01:17,240 would allow us to calculate that from these simple relationships 15 00:01:17,240 --> 00:01:20,360 we've gone through here. 16 00:01:20,360 --> 00:01:23,210 So what we're going to do is we're 17 00:01:23,210 --> 00:01:28,100 going to set this equation equal to that, because they're 18 00:01:28,100 --> 00:01:32,960 both equal to the probability of getting A and B. 19 00:01:32,960 --> 00:01:40,860 So we'll say that the probability 20 00:01:40,860 --> 00:01:47,520 of B given A times the probability of A is equal to-- 21 00:01:47,520 --> 00:02:00,966 22 00:02:00,966 --> 00:02:04,891 the probability of B given A times the probability of A, 23 00:02:04,891 --> 00:02:07,270 that's this one, is equal to the probability 24 00:02:07,270 --> 00:02:09,460 of A given B times the probability of B. 25 00:02:09,460 --> 00:02:12,640 That's this one. 26 00:02:12,640 --> 00:02:14,640 So then we can say the probability of B 27 00:02:14,640 --> 00:02:19,950 given A is equal to the probability of A given 28 00:02:19,950 --> 00:02:26,550 B times the probability of B divided 29 00:02:26,550 --> 00:02:44,080 by the probability of A. This is Bayes' theorem, which 30 00:02:44,080 --> 00:02:49,130 is one of the most widely used relations in statistics today. 31 00:02:49,130 --> 00:03:12,140 This will be the posterior probability, 32 00:03:12,140 --> 00:03:16,025 the new probability given our data, our information A. 33 00:03:16,025 --> 00:03:24,320 And this is the prior probability, the probability 34 00:03:24,320 --> 00:03:28,550 of event B before we knew this information about A. 35 00:03:28,550 --> 00:03:31,070 So using this, we can now get data 36 00:03:31,070 --> 00:03:32,630 and update our probability of event 37 00:03:32,630 --> 00:03:35,780 B. It's called conditional probability. 38 00:03:35,780 --> 00:03:38,300 39 00:03:38,300 --> 00:03:44,360 So now let's apply it to our question about scenario 40 00:03:44,360 --> 00:03:52,010 K. Or, sorry, event A and B with K being a carrier or not. 41 00:03:52,010 --> 00:03:55,910 42 00:03:55,910 --> 00:04:05,480 All right, so our probability of event B, the K is a carrier, 43 00:04:05,480 --> 00:04:08,048 was 0.5. 44 00:04:08,048 --> 00:04:09,215 We went through that before. 45 00:04:09,215 --> 00:04:12,590 46 00:04:12,590 --> 00:04:23,840 Now our probability of A given B is one half. 47 00:04:23,840 --> 00:04:28,580 48 00:04:28,580 --> 00:04:33,060 If K is a carrier, there's a one half chance of transmission. 49 00:04:33,060 --> 00:04:41,600 Now what's our probability of A, the probability 50 00:04:41,600 --> 00:04:46,550 that L is unaffected, prior to having any data. 51 00:04:46,550 --> 00:04:48,590 Well, it could be that K was a carrier, 52 00:04:48,590 --> 00:04:51,470 or it could be the K was not a carrier. 53 00:04:51,470 --> 00:04:54,260 So the probability of A will be the probability 54 00:04:54,260 --> 00:04:58,370 of A given B times the probability 55 00:04:58,370 --> 00:05:06,320 of A times the probability of B plus the probability 56 00:05:06,320 --> 00:05:15,740 of A given not B times the probability of not B. 57 00:05:15,740 --> 00:05:19,340 What we're doing here is just considering the two scenarios 58 00:05:19,340 --> 00:05:21,440 that K is a carrier, or K is not a carrier. 59 00:05:21,440 --> 00:05:25,920 In both cases, what would the probability of getting event A 60 00:05:25,920 --> 00:05:26,420 be? 61 00:05:26,420 --> 00:05:29,660 62 00:05:29,660 --> 00:05:30,605 So this is 3/4. 63 00:05:30,605 --> 00:05:35,730 64 00:05:35,730 --> 00:05:39,920 So now we can calculate our probability 65 00:05:39,920 --> 00:05:40,775 that K is a carrier. 66 00:05:40,775 --> 00:05:53,060 67 00:05:53,060 --> 00:06:00,260 One half times one half divided by 3/4 equals one third. 68 00:06:00,260 --> 00:06:05,800 69 00:06:05,800 --> 00:06:09,430 So given the information that L would be unaffected, 70 00:06:09,430 --> 00:06:12,190 now the probability that K is a carrier is no longer one half, 71 00:06:12,190 --> 00:06:13,090 it's one third. 72 00:06:13,090 --> 00:06:14,240 It has dropped. 73 00:06:14,240 --> 00:06:17,590 And you could calculate that for any type of event 74 00:06:17,590 --> 00:06:19,530 using this relation. 5318