1 00:00:00,000 --> 00:00:00,730 2 00:00:00,730 --> 00:00:03,890 In this problem, we'll be working with a object called 3 00:00:03,890 --> 00:00:08,180 random walk, where we have a person on the line-- or a 4 00:00:08,180 --> 00:00:10,370 tight rope, according to the problem. 5 00:00:10,370 --> 00:00:13,580 Let's start from the origin, and each time step, it would 6 00:00:13,580 --> 00:00:16,420 randomly either go forward or backward with certain 7 00:00:16,420 --> 00:00:17,405 probability. 8 00:00:17,405 --> 00:00:19,860 In our case, with probability P, the person would go 9 00:00:19,860 --> 00:00:23,530 forward, and 1 minus P going backwards. 10 00:00:23,530 --> 00:00:26,280 Now, the walk is random in the following sense-- that the 11 00:00:26,280 --> 00:00:29,780 choice going forward or backward in each step is 12 00:00:29,780 --> 00:00:31,860 random, and it's completely independent 13 00:00:31,860 --> 00:00:34,920 from all past history. 14 00:00:34,920 --> 00:00:35,740 So let's look at the problem. 15 00:00:35,740 --> 00:00:36,830 It has three parts. 16 00:00:36,830 --> 00:00:40,640 In the first part, we'd like to know what's the probability 17 00:00:40,640 --> 00:00:44,300 that after two steps the person returns to the starting 18 00:00:44,300 --> 00:00:46,730 point, which in this case is 0? 19 00:00:46,730 --> 00:00:48,700 Now, throughout this problem, I'm going to be using the 20 00:00:48,700 --> 00:00:50,820 following notation. 21 00:00:50,820 --> 00:00:55,450 F indicates the action of going forward and B indicates 22 00:00:55,450 --> 00:00:57,470 the action of going backwards. 23 00:00:57,470 --> 00:01:02,290 A sequence says F and B implies the sample that the 24 00:01:02,290 --> 00:01:05,850 person first goes forward, and then backwards. 25 00:01:05,850 --> 00:01:09,540 If I add another F, it will mean, forward, backward, 26 00:01:09,540 --> 00:01:10,690 forward again. 27 00:01:10,690 --> 00:01:11,940 OK? 28 00:01:11,940 --> 00:01:13,520 29 00:01:13,520 --> 00:01:16,910 So in order for the person to go somewhere after two steps 30 00:01:16,910 --> 00:01:21,220 and return to the origin, the following must happen. 31 00:01:21,220 --> 00:01:26,030 Either the person went forward followed by backward, or 32 00:01:26,030 --> 00:01:29,270 backward followed by forward. 33 00:01:29,270 --> 00:01:32,290 And indeed, this event-- 34 00:01:32,290 --> 00:01:35,050 namely, the union of these two possibilities-- 35 00:01:35,050 --> 00:01:39,930 defines the event of interest in our case. 36 00:01:39,930 --> 00:01:44,970 And we'd like to know what's the probability of A, which 37 00:01:44,970 --> 00:01:48,410 we'll break down into the probability of forward, 38 00:01:48,410 --> 00:01:52,640 backward, backward, forward. 39 00:01:52,640 --> 00:01:54,980 Now, forward, backward and backward, forward-- they are 40 00:01:54,980 --> 00:01:57,660 two completely different outcomes. 41 00:01:57,660 --> 00:02:01,010 And we know that because they're disjoint, this would 42 00:02:01,010 --> 00:02:05,100 just be the sum of the two probabilities-- 43 00:02:05,100 --> 00:02:08,259 plus probability of backward/forward. 44 00:02:08,259 --> 00:02:12,920 45 00:02:12,920 --> 00:02:16,090 Here's where the independence will come in. 46 00:02:16,090 --> 00:02:18,850 When we try to compute the probability of going forward 47 00:02:18,850 --> 00:02:21,280 and backward, because the action-- 48 00:02:21,280 --> 00:02:24,360 each step is completely independent from the past, we 49 00:02:24,360 --> 00:02:27,630 know this is the same as saying, in the first step, we 50 00:02:27,630 --> 00:02:31,550 have probability P of going forward, in the next step, 51 00:02:31,550 --> 00:02:34,770 probability 1 minus P of going backwards. 52 00:02:34,770 --> 00:02:38,360 We can do so-- namely, writing the probability of forward, 53 00:02:38,360 --> 00:02:42,450 backward as a product of going forward times the probability 54 00:02:42,450 --> 00:02:45,950 of going backwards, because these actions are independent. 55 00:02:45,950 --> 00:02:49,310 And similarly, for the second one, we have going backwards 56 00:02:49,310 --> 00:02:52,980 first, times going forward the second time. 57 00:02:52,980 --> 00:02:59,440 Adding these two up, we have 2 times P times 1 minus P. And 58 00:02:59,440 --> 00:03:01,850 that will be the answer to the first part of the problem. 59 00:03:01,850 --> 00:03:04,440 60 00:03:04,440 --> 00:03:06,540 In the second part of the problem, we're interested in 61 00:03:06,540 --> 00:03:11,370 the probability that after three steps, the person ends 62 00:03:11,370 --> 00:03:14,410 up in position 1, or one step forward 63 00:03:14,410 --> 00:03:16,750 compared to where he started. 64 00:03:16,750 --> 00:03:19,610 Now, the only possibilities here are that among the three 65 00:03:19,610 --> 00:03:23,440 steps, exactly two steps are forward, and one step is 66 00:03:23,440 --> 00:03:26,500 backwards, because otherwise there's no way the person will 67 00:03:26,500 --> 00:03:29,050 end up in position 1. 68 00:03:29,050 --> 00:03:32,980 To do so, there, again, are three possibilities in which 69 00:03:32,980 --> 00:03:36,670 we go forward, forward, backward, or forward, 70 00:03:36,670 --> 00:03:41,690 backward, forward, or backward, forward, forward. 71 00:03:41,690 --> 00:03:44,850 And that exhausts all the possibilities that the person 72 00:03:44,850 --> 00:03:47,490 can end up in position 1 after three steps. 73 00:03:47,490 --> 00:03:50,340 And we'll define the collection of all these 74 00:03:50,340 --> 00:03:56,440 outcomes as event C. The probability of event C-- 75 00:03:56,440 --> 00:03:57,660 same as before-- 76 00:03:57,660 --> 00:03:59,930 is simply the sum of the probability of 77 00:03:59,930 --> 00:04:02,560 each individual outcome. 78 00:04:02,560 --> 00:04:05,640 Now, based on the independence assumption that we used 79 00:04:05,640 --> 00:04:09,770 before, each outcome here has the same probability, which is 80 00:04:09,770 --> 00:04:15,330 equal to P squared times 1 minus P. The P squared comes 81 00:04:15,330 --> 00:04:18,680 from the fact that two forward steps are taken, and 1 minus 82 00:04:18,680 --> 00:04:23,440 P, the probability of that one backwards step. 83 00:04:23,440 --> 00:04:27,910 And since there are three of them, we multiply 3 in front, 84 00:04:27,910 --> 00:04:30,030 and that will give us the probability. 85 00:04:30,030 --> 00:04:32,490 In the last part of the problem, we're asked to 86 00:04:32,490 --> 00:04:36,480 compute that, conditional on event C already took place, 87 00:04:36,480 --> 00:04:40,250 what is the probability that the first step he took was a 88 00:04:40,250 --> 00:04:41,580 forward step? 89 00:04:41,580 --> 00:04:43,510 Without going into the details, let's take a look at 90 00:04:43,510 --> 00:04:47,840 the C, in which we have three elements, and only the first 91 00:04:47,840 --> 00:04:52,440 two elements correspond to a forward step 92 00:04:52,440 --> 00:04:54,500 in the first step. 93 00:04:54,500 --> 00:04:58,490 So we can define event D as simply 94 00:04:58,490 --> 00:05:00,100 the first two outcomes-- 95 00:05:00,100 --> 00:05:02,580 forward, forward, backward, and 96 00:05:02,580 --> 00:05:06,170 forward, backward, forward. 97 00:05:06,170 --> 00:05:09,910 Now, the probability we're interested in is simply 98 00:05:09,910 --> 00:05:15,540 probability of D conditional on C. We'd write it out using 99 00:05:15,540 --> 00:05:17,950 the law of conditional probability-- 100 00:05:17,950 --> 00:05:22,810 D intersection C conditional on C. Now, because D is a 101 00:05:22,810 --> 00:05:27,270 subset of C, we have probability of D divided by 102 00:05:27,270 --> 00:05:29,510 the probability of C. 103 00:05:29,510 --> 00:05:33,380 Again, because all samples here have the same 104 00:05:33,380 --> 00:05:37,290 probability, all we need to do is to count the number of 105 00:05:37,290 --> 00:05:40,190 samples here, which is 2, and divide by the number of 106 00:05:40,190 --> 00:05:43,130 samples here, which is 3. 107 00:05:43,130 --> 00:05:47,020 So we end up with 2 over 3. 108 00:05:47,020 --> 00:05:48,940 And that concludes the problem. 109 00:05:48,940 --> 00:05:50,190 See you next time. 110 00:05:50,190 --> 00:05:51,367