1 00:00:00,210 --> 00:00:03,360 In this lecture, we introduce and develop the concept of 2 00:00:03,360 --> 00:00:05,940 independence between events. 3 00:00:05,940 --> 00:00:08,630 The general idea is the following. 4 00:00:08,630 --> 00:00:12,190 If I tell you that a certain event A has occurred, this 5 00:00:12,190 --> 00:00:15,400 will generally change the probability of some other 6 00:00:15,400 --> 00:00:18,760 event B. Probabilities will have to be replaced by 7 00:00:18,760 --> 00:00:21,210 conditional probabilities. 8 00:00:21,210 --> 00:00:24,170 But if the conditional probability turns out to be 9 00:00:24,170 --> 00:00:27,890 the same as the unconditional probability, then the 10 00:00:27,890 --> 00:00:31,840 occurrence of event A does not carry any useful information 11 00:00:31,840 --> 00:00:34,530 on whether event B will occur. 12 00:00:34,530 --> 00:00:37,970 In such a case, we say that events A and B are 13 00:00:37,970 --> 00:00:40,140 independent. 14 00:00:40,140 --> 00:00:43,320 We will develop some intuition about the meaning of 15 00:00:43,320 --> 00:00:47,780 independence of two events and introduce an extension, the 16 00:00:47,780 --> 00:00:50,890 concept of conditional independence. 17 00:00:50,890 --> 00:00:54,180 We will then proceed to define the independence of a 18 00:00:54,180 --> 00:00:57,470 collection of more than two events. 19 00:00:57,470 --> 00:01:00,620 If, for any two of the events in the collection we have 20 00:01:00,620 --> 00:01:04,610 independence between them, we will say that we have pairwise 21 00:01:04,610 --> 00:01:05,910 independence. 22 00:01:05,910 --> 00:01:08,750 But we will see that independence of the entire 23 00:01:08,750 --> 00:01:11,230 collection is something different. 24 00:01:11,230 --> 00:01:14,470 It involves additional conditions. 25 00:01:14,470 --> 00:01:17,910 Finally, we will close with an application in reliability 26 00:01:17,910 --> 00:01:21,650 analysis and with a nice puzzle that will serve as a 27 00:01:21,650 --> 00:01:25,130 word of caution about putting together probabilistic models.