1 00:00:00,000 --> 00:00:02,630 PROFESSOR: Independent events are events that have nothing 2 00:00:02,630 --> 00:00:04,080 to do with each other. 3 00:00:04,080 --> 00:00:07,900 And needless to say, it's a lot easier to work with them 4 00:00:07,900 --> 00:00:09,400 because you don't have to figure out 5 00:00:09,400 --> 00:00:11,810 some weird interaction between two events that 6 00:00:11,810 --> 00:00:13,480 do have something to do with each other. 7 00:00:13,480 --> 00:00:16,200 Typical case where independent events come up 8 00:00:16,200 --> 00:00:19,160 is, for example, you toss a coin five times, 9 00:00:19,160 --> 00:00:21,520 and then you're about to toss a coin the sixth time. 10 00:00:21,520 --> 00:00:23,070 Is there any reason to think that 11 00:00:23,070 --> 00:00:25,250 what the coins did the first five times is going 12 00:00:25,250 --> 00:00:27,000 to have any influence on the flip 13 00:00:27,000 --> 00:00:30,720 of the coin for the sixth time? 14 00:00:30,720 --> 00:00:33,100 Well, it's reasonable to assume not, 15 00:00:33,100 --> 00:00:36,780 which is to say that the outcome of the sixth toss 16 00:00:36,780 --> 00:00:40,690 is arguably independent of the outcome of all 17 00:00:40,690 --> 00:00:42,450 the previous tosses. 18 00:00:42,450 --> 00:00:43,170 OK. 19 00:00:43,170 --> 00:00:46,570 Let's look at a formal definition 20 00:00:46,570 --> 00:00:48,660 now in this short video of just what 21 00:00:48,660 --> 00:00:51,520 is the technical definition of independent events. 22 00:00:51,520 --> 00:00:54,741 So what we said is that they are independent if they have 23 00:00:54,741 --> 00:00:55,990 nothing to do with each other. 24 00:00:55,990 --> 00:00:59,820 What that means is that if I tell you that B happened, 25 00:00:59,820 --> 00:01:03,220 it doesn't have any effect on the probability of A. That is, 26 00:01:03,220 --> 00:01:06,350 the probability of A, given that B happened, 27 00:01:06,350 --> 00:01:09,130 doesn't change the probability of A at all. 28 00:01:09,130 --> 00:01:10,196 That's it. 29 00:01:10,196 --> 00:01:11,320 Now this is one definition. 30 00:01:11,320 --> 00:01:13,153 Maybe this is the more intuitive definition. 31 00:01:13,153 --> 00:01:16,440 But another definition that's equivalent and is standard 32 00:01:16,440 --> 00:01:19,260 is that two events are equivalent if the product 33 00:01:19,260 --> 00:01:22,252 of their probabilities is equal to the probability 34 00:01:22,252 --> 00:01:24,210 that they both happen, that is, the probability 35 00:01:24,210 --> 00:01:25,760 of their intersection. 36 00:01:25,760 --> 00:01:27,890 Now definition one and definition two 37 00:01:27,890 --> 00:01:30,800 are trivial equivalent, just using the definition 38 00:01:30,800 --> 00:01:32,239 of conditional probability. 39 00:01:32,239 --> 00:01:34,280 And if you don't see that, this would be a moment 40 00:01:34,280 --> 00:01:38,100 to stop, get a pencil and paper, and write down 41 00:01:38,100 --> 00:01:43,440 the three-line proof of the equivalence of these two 42 00:01:43,440 --> 00:01:44,220 equalities. 43 00:01:44,220 --> 00:01:46,260 In fact, the three-line proof has this as the first line 44 00:01:46,260 --> 00:01:47,426 and that as the second line. 45 00:01:47,426 --> 00:01:49,800 So you could argue it's really just the middle line 46 00:01:49,800 --> 00:01:51,120 that you're adding. 47 00:01:51,120 --> 00:01:53,070 OK. 48 00:01:53,070 --> 00:01:56,560 Definition two has the slight advantage that it always works, 49 00:01:56,560 --> 00:01:58,720 whereas definition one implicitly 50 00:01:58,720 --> 00:02:03,390 requires that the divisor-- remember probability of A given 51 00:02:03,390 --> 00:02:06,390 B is defined as the probability of the intersection divided 52 00:02:06,390 --> 00:02:09,789 by the probability B. It's only defined if the probability of B 53 00:02:09,789 --> 00:02:10,820 is positive. 54 00:02:10,820 --> 00:02:12,930 Whereas the second definition always works, 55 00:02:12,930 --> 00:02:15,450 so we don't have to put a proviso 56 00:02:15,450 --> 00:02:17,827 in about the probability of B being non-zero. 57 00:02:17,827 --> 00:02:19,535 So that's the definition of independence. 58 00:02:22,637 --> 00:02:24,970 Looking at this definition, what you can see immediately 59 00:02:24,970 --> 00:02:27,087 is that it's completely symmetric in A and B. 60 00:02:27,087 --> 00:02:28,545 Since multiplication is commutative 61 00:02:28,545 --> 00:02:31,735 and intersection is commutative, which is A 62 00:02:31,735 --> 00:02:33,320 and which is B doesn't matter. 63 00:02:33,320 --> 00:02:36,820 And what that implies then is that A is independent of B 64 00:02:36,820 --> 00:02:41,370 if and only if B is independent of A. 65 00:02:41,370 --> 00:02:45,660 Now another fact that holds is that if the probability of B 66 00:02:45,660 --> 00:02:49,010 happens to be zero, then vacuously B 67 00:02:49,010 --> 00:02:52,550 is independent of everything-- even itself. 68 00:02:52,550 --> 00:02:56,880 Which isn't important, but is a small technicality that's worth 69 00:02:56,880 --> 00:03:00,450 remembering by that definition. 70 00:03:00,450 --> 00:03:03,470 Now again, the intuitive idea that A and B have nothing 71 00:03:03,470 --> 00:03:07,100 to do with each other is that A is independent of B means 72 00:03:07,100 --> 00:03:11,600 that A is independent of whether or not B occurs. 73 00:03:11,600 --> 00:03:13,630 That is to say, if A is independent of B, 74 00:03:13,630 --> 00:03:16,400 it ought to be independent of the complement of B. 75 00:03:16,400 --> 00:03:18,600 And that's a lemma that's also easily proved. 76 00:03:18,600 --> 00:03:20,940 A is independent of B if and only 77 00:03:20,940 --> 00:03:23,890 if A is independent of the complement of B. 78 00:03:23,890 --> 00:03:26,050 It's a simple proof using the difference rule. 79 00:03:26,050 --> 00:03:28,370 And again, I encourage you to stop 80 00:03:28,370 --> 00:03:31,090 with a piece of paper and a pencil 81 00:03:31,090 --> 00:03:35,235 and convince yourself that that follows with a one-line proof.