WEBVTT 00:00:00.210 --> 00:00:03.360 In this lecture, we introduce and develop the concept of 00:00:03.360 --> 00:00:05.940 independence between events. 00:00:05.940 --> 00:00:08.630 The general idea is the following. 00:00:08.630 --> 00:00:12.190 If I tell you that a certain event A has occurred, this 00:00:12.190 --> 00:00:15.400 will generally change the probability of some other 00:00:15.400 --> 00:00:18.760 event B. Probabilities will have to be replaced by 00:00:18.760 --> 00:00:21.210 conditional probabilities. 00:00:21.210 --> 00:00:24.170 But if the conditional probability turns out to be 00:00:24.170 --> 00:00:27.890 the same as the unconditional probability, then the 00:00:27.890 --> 00:00:31.840 occurrence of event A does not carry any useful information 00:00:31.840 --> 00:00:34.530 on whether event B will occur. 00:00:34.530 --> 00:00:37.970 In such a case, we say that events A and B are 00:00:37.970 --> 00:00:40.140 independent. 00:00:40.140 --> 00:00:43.320 We will develop some intuition about the meaning of 00:00:43.320 --> 00:00:47.780 independence of two events and introduce an extension, the 00:00:47.780 --> 00:00:50.890 concept of conditional independence. 00:00:50.890 --> 00:00:54.180 We will then proceed to define the independence of a 00:00:54.180 --> 00:00:57.470 collection of more than two events. 00:00:57.470 --> 00:01:00.620 If, for any two of the events in the collection we have 00:01:00.620 --> 00:01:04.610 independence between them, we will say that we have pairwise 00:01:04.610 --> 00:01:05.910 independence. 00:01:05.910 --> 00:01:08.750 But we will see that independence of the entire 00:01:08.750 --> 00:01:11.230 collection is something different. 00:01:11.230 --> 00:01:14.470 It involves additional conditions. 00:01:14.470 --> 00:01:17.910 Finally, we will close with an application in reliability 00:01:17.910 --> 00:01:21.650 analysis and with a nice puzzle that will serve as a 00:01:21.650 --> 00:01:25.130 word of caution about putting together probabilistic models.