1 00:00:00,090 --> 00:00:02,490 The following content is provided under a Creative 2 00:00:02,490 --> 00:00:04,030 Commons license. 3 00:00:04,030 --> 00:00:06,330 Your support will help MIT OpenCourseWare 4 00:00:06,330 --> 00:00:10,690 continue to offer high quality educational resources for free. 5 00:00:10,690 --> 00:00:13,320 To make a donation or view additional materials 6 00:00:13,320 --> 00:00:17,250 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,250 --> 00:00:19,937 at ocw.mit.edu. 8 00:00:19,937 --> 00:00:20,520 PROFESSOR: OK. 9 00:00:20,520 --> 00:00:23,360 So first we're going to review last class. 10 00:00:23,360 --> 00:00:25,200 The first question, what does it mean 11 00:00:25,200 --> 00:00:31,030 if sets A, B, C are a partition of set D? 12 00:00:31,030 --> 00:00:34,299 If you know it, just raise your hand. 13 00:00:34,299 --> 00:00:35,265 Yeah. 14 00:00:35,265 --> 00:00:38,646 AUDIENCE: A, B, and C are not in set D? 15 00:00:38,646 --> 00:00:39,612 PROFESSOR: Hmm? 16 00:00:39,612 --> 00:00:43,110 AUDIENCE: A, B, and C are not in set D? 17 00:00:43,110 --> 00:00:45,802 PROFESSOR: No 18 00:00:45,802 --> 00:00:47,617 AUDIENCE: A, B, and C make up D? 19 00:00:47,617 --> 00:00:48,200 PROFESSOR: OK. 20 00:00:48,200 --> 00:00:50,130 And what are we assuming? 21 00:00:50,130 --> 00:00:52,410 AUDIENCE: That the [INAUDIBLE]? 22 00:00:52,410 --> 00:00:55,464 Oh, that A, B, and C are set D. 23 00:00:55,464 --> 00:00:56,130 PROFESSOR: Yeah. 24 00:00:56,130 --> 00:00:57,180 So they're disjoint. 25 00:00:57,180 --> 00:00:58,055 OK? 26 00:00:58,055 --> 00:01:00,270 Did everyone hear that? 27 00:01:00,270 --> 00:01:07,160 So A, B, and C are disjoint and they make up all of D. OK. 28 00:01:07,160 --> 00:01:11,300 So how do you calculate P, probability of A given B, 29 00:01:11,300 --> 00:01:14,264 using the formula for conditional probability? 30 00:01:14,264 --> 00:01:15,680 So essentially I'm just asking you 31 00:01:15,680 --> 00:01:19,330 what is the formula of conditional probability? 32 00:01:23,630 --> 00:01:24,130 Slide. 33 00:01:30,380 --> 00:01:42,100 C disjoint-- OK. 34 00:01:42,100 --> 00:01:43,649 So, two? 35 00:01:43,649 --> 00:01:44,440 Can anyone tell me? 36 00:01:48,590 --> 00:01:55,228 So probability of A given B. How do you calculate that? 37 00:01:55,228 --> 00:01:57,713 AUDIENCE: The number of possible outcomes 38 00:01:57,713 --> 00:02:05,384 of A intersect B over-- 39 00:02:05,384 --> 00:02:08,246 PROFESSOR: Yup. 40 00:02:08,246 --> 00:02:11,280 Does everyone understand why we do this? 41 00:02:11,280 --> 00:02:15,210 So like we said with the universal set, 42 00:02:15,210 --> 00:02:17,820 we always do probability of A over the universal set, 43 00:02:17,820 --> 00:02:19,220 which is 1. 44 00:02:19,220 --> 00:02:21,120 But for a conditional probability, 45 00:02:21,120 --> 00:02:23,790 we change the universal set to B. 46 00:02:23,790 --> 00:02:26,610 So we do the event that both of these 47 00:02:26,610 --> 00:02:31,210 happen over our new set, which is B. OK? 48 00:02:31,210 --> 00:02:32,920 Does everyone understand that? 49 00:02:32,920 --> 00:02:33,420 Yeah? 50 00:02:33,420 --> 00:02:34,045 AUDIENCE: Wait. 51 00:02:34,045 --> 00:02:35,670 What does the upside down U mean? 52 00:02:35,670 --> 00:02:37,140 PROFESSOR: Oh, were you not here? 53 00:02:37,140 --> 00:02:37,640 OK. 54 00:02:37,640 --> 00:02:38,140 This intersect-- 55 00:02:38,140 --> 00:02:38,435 AUDIENCE: Sorry. 56 00:02:38,435 --> 00:02:39,570 PROFESSOR: No, that's fine. 57 00:02:39,570 --> 00:02:44,790 So if you have A and B-- 58 00:02:44,790 --> 00:02:46,772 do you know anything about set theory? 59 00:02:46,772 --> 00:02:47,730 AUDIENCE: A little bit. 60 00:02:47,730 --> 00:02:48,090 PROFESSOR: A little bit. 61 00:02:48,090 --> 00:02:48,870 OK. 62 00:02:48,870 --> 00:02:51,390 So this is event A, right? 63 00:02:51,390 --> 00:02:56,540 And this is B. And the intersect is anything that both share. 64 00:02:56,540 --> 00:02:57,460 AUDIENCE: OK. 65 00:02:57,460 --> 00:02:58,085 PROFESSOR: Yup. 66 00:03:01,320 --> 00:03:03,680 OK. 67 00:03:03,680 --> 00:03:15,690 What is the difference between P, A given B and P B given A? 68 00:03:15,690 --> 00:03:17,840 It's almost kind of self explanatory. 69 00:03:20,675 --> 00:03:22,631 AUDIENCE: So the first one is what's 70 00:03:22,631 --> 00:03:27,300 the probability of A happening if you already know B. 71 00:03:27,300 --> 00:03:29,982 And the second one is what's the probability of B happening 72 00:03:29,982 --> 00:03:31,460 if you already know A. 73 00:03:31,460 --> 00:03:33,030 PROFESSOR: And are these two same? 74 00:03:33,030 --> 00:03:34,080 Always? 75 00:03:34,080 --> 00:03:34,580 No. 76 00:03:34,580 --> 00:03:35,240 OK, yeah. 77 00:03:35,240 --> 00:03:36,698 So you'll have to calculate it out, 78 00:03:36,698 --> 00:03:39,914 because your universal set is different. 79 00:03:39,914 --> 00:03:42,360 OK. 80 00:03:42,360 --> 00:03:46,290 For if B causes A, what is the conditional probability 81 00:03:46,290 --> 00:03:49,620 that P of A given B? 82 00:03:54,970 --> 00:03:58,130 So if you know that B causes A, what's 83 00:03:58,130 --> 00:03:59,860 the probability that B is happening 84 00:03:59,860 --> 00:04:03,178 given that B happened? 85 00:04:03,178 --> 00:04:06,070 AUDIENCE: 100%? 86 00:04:06,070 --> 00:04:08,570 PROFESSOR: Yup. 87 00:04:08,570 --> 00:04:14,490 So for that one, does anyone understand why that-- 88 00:04:14,490 --> 00:04:16,410 that's the reason? 89 00:04:16,410 --> 00:04:20,820 So if B causes A, if A happened, and you know B happened, 90 00:04:20,820 --> 00:04:22,352 A has to happen. 91 00:04:22,352 --> 00:04:23,360 Right? 92 00:04:23,360 --> 00:04:31,040 But if B was a possible cause of A, then you're not-- 93 00:04:31,040 --> 00:04:33,770 and A is caused by many things, then 94 00:04:33,770 --> 00:04:35,900 this is not necessarily 1, because something else 95 00:04:35,900 --> 00:04:38,270 could cause A. Right? 96 00:04:38,270 --> 00:04:43,135 But if you know B definitely causes A, then-- 97 00:04:43,135 --> 00:04:46,100 OK? 98 00:04:46,100 --> 00:04:49,340 So in the last one, does conditional probability 99 00:04:49,340 --> 00:04:51,090 require that B causes A? 100 00:04:55,419 --> 00:04:55,960 AUDIENCE: No. 101 00:04:55,960 --> 00:04:56,543 PROFESSOR: No? 102 00:04:56,543 --> 00:04:58,951 Can you give me an example? 103 00:04:58,951 --> 00:05:05,092 AUDIENCE: So if A is there is a white dog in the room, 104 00:05:05,092 --> 00:05:07,016 and B is there's a dog in the room. 105 00:05:07,016 --> 00:05:10,394 And you know that 45% of dogs are white. 106 00:05:10,394 --> 00:05:13,530 Then the fact that there's a dog in the room doesn't 107 00:05:13,530 --> 00:05:15,110 cause it to be a white dog, 108 00:05:15,110 --> 00:05:15,776 PROFESSOR: Yeah. 109 00:05:15,776 --> 00:05:18,260 AUDIENCE: But it does mean that there's a dog in the room. 110 00:05:18,260 --> 00:05:19,843 So it's more likely that there's going 111 00:05:19,843 --> 00:05:23,194 to be a white dog than if you don't know [INAUDIBLE] at all. 112 00:05:23,194 --> 00:05:23,860 PROFESSOR: Yeah. 113 00:05:23,860 --> 00:05:26,340 So B gives you information about A, 114 00:05:26,340 --> 00:05:30,600 but it doesn't necessarily have to cause A. OK? 115 00:05:30,600 --> 00:05:33,090 Does that make sense? 116 00:05:33,090 --> 00:05:34,994 OK. 117 00:05:34,994 --> 00:05:36,270 Any questions? 118 00:05:36,270 --> 00:05:36,770 Yeah. 119 00:05:36,770 --> 00:05:37,561 AUDIENCE: So, wait. 120 00:05:37,561 --> 00:05:38,670 I'm certainly confused. 121 00:05:38,670 --> 00:05:40,907 What is the A [INAUDIBLE] B means? 122 00:05:40,907 --> 00:05:41,490 PROFESSOR: OK. 123 00:05:41,490 --> 00:05:44,940 So from last class we learned that-- 124 00:05:44,940 --> 00:05:47,070 you know that, populated B. Right? 125 00:05:47,070 --> 00:05:48,960 You know this is, right? 126 00:05:48,960 --> 00:05:51,594 So this means given B. It's like-- 127 00:05:51,594 --> 00:05:52,260 AUDIENCE: Given. 128 00:05:52,260 --> 00:05:52,926 PROFESSOR: Yeah. 129 00:05:52,926 --> 00:05:55,260 B is knowledge that you are given, 130 00:05:55,260 --> 00:05:57,810 and then you have to figure out the probability of A 131 00:05:57,810 --> 00:05:59,910 with that new knowledge. 132 00:05:59,910 --> 00:06:00,874 AUDIENCE: OK. 133 00:06:00,874 --> 00:06:01,840 PROFESSOR: OK? 134 00:06:01,840 --> 00:06:03,936 Anything else? 135 00:06:03,936 --> 00:06:04,436 OK. 136 00:06:08,380 --> 00:06:12,870 We're going to move on to that Monty Hall problem. 137 00:06:12,870 --> 00:06:15,020 Did anyone work on it? 138 00:06:15,020 --> 00:06:16,100 You did have to, but-- 139 00:06:16,100 --> 00:06:16,661 OK. 140 00:06:16,661 --> 00:06:18,160 So what did you get for your answer? 141 00:06:18,160 --> 00:06:20,220 Do you stay or switch? 142 00:06:20,220 --> 00:06:21,260 AUDIENCE: Always switch. 143 00:06:21,260 --> 00:06:21,843 PROFESSOR: OK. 144 00:06:21,843 --> 00:06:23,782 How about? 145 00:06:23,782 --> 00:06:24,490 AUDIENCE: Switch. 146 00:06:24,490 --> 00:06:25,355 PROFESSOR: Switch? 147 00:06:25,355 --> 00:06:26,191 AUDIENCE: Switch. 148 00:06:26,191 --> 00:06:26,690 Switch. 149 00:06:26,690 --> 00:06:28,920 PROFESSOR: OK. 150 00:06:28,920 --> 00:06:31,390 Do any of you feel comfortable explaining it up here? 151 00:06:31,390 --> 00:06:32,760 How you? 152 00:06:32,760 --> 00:06:34,720 I know that you've done it once, last class. 153 00:06:34,720 --> 00:06:36,684 So you want to come up and explain it? 154 00:06:36,684 --> 00:06:38,475 AUDIENCE: I sort of know how to explain it. 155 00:06:38,475 --> 00:06:39,058 PROFESSOR: OK. 156 00:06:39,058 --> 00:06:42,355 AUDIENCE: I know that it's the [INAUDIBLE],, but-- 157 00:06:42,355 --> 00:06:43,067 PROFESSOR: OK. 158 00:06:43,067 --> 00:06:43,567 All right. 159 00:06:43,567 --> 00:06:45,529 How about you want to tell me first? 160 00:06:45,529 --> 00:06:46,070 AUDIENCE: OK. 161 00:06:46,070 --> 00:06:50,970 Well for the probability of door 1 is 2/3, 162 00:06:50,970 --> 00:06:55,870 and then, wait, and then if he shows the Door 2 then 163 00:06:55,870 --> 00:06:57,830 you know that the probability, most people 164 00:06:57,830 --> 00:07:01,440 would think that now it's cut in half, but it's actually 2/3. 165 00:07:01,440 --> 00:07:06,042 So there's more of a probability that it would be-- 166 00:07:06,042 --> 00:07:06,625 PROFESSOR: OK. 167 00:07:06,625 --> 00:07:08,885 So yours is more intuitive, actually. 168 00:07:08,885 --> 00:07:09,635 AUDIENCE: I guess. 169 00:07:09,635 --> 00:07:10,000 PROFESSOR: Yeah. 170 00:07:10,000 --> 00:07:11,810 So you don't really need to write anything. 171 00:07:11,810 --> 00:07:14,850 Did everyone hear her? 172 00:07:14,850 --> 00:07:15,350 Yes? 173 00:07:15,350 --> 00:07:15,620 No? 174 00:07:15,620 --> 00:07:16,210 AUDIENCE: No. 175 00:07:16,210 --> 00:07:16,793 PROFESSOR: No. 176 00:07:16,793 --> 00:07:17,690 OK. 177 00:07:17,690 --> 00:07:20,940 So what she said was-- 178 00:07:20,940 --> 00:07:24,090 what she's essentially doing is kind of combining the doors. 179 00:07:24,090 --> 00:07:24,590 Oh, wait. 180 00:07:24,590 --> 00:07:24,870 Sorry. 181 00:07:24,870 --> 00:07:26,411 For you guys who weren't here, do you 182 00:07:26,411 --> 00:07:28,550 know what the problem is? 183 00:07:28,550 --> 00:07:29,050 No. 184 00:07:29,050 --> 00:07:29,660 OK. 185 00:07:29,660 --> 00:07:31,790 I'll go over that first. 186 00:07:31,790 --> 00:07:33,530 Just bear with me, people who were here. 187 00:07:33,530 --> 00:07:35,060 OK. 188 00:07:35,060 --> 00:07:38,520 So the Monty Hall problem on the sheet. 189 00:07:38,520 --> 00:07:40,640 So you're at a game show. 190 00:07:40,640 --> 00:07:44,180 There's three doors, and one of these doors 191 00:07:44,180 --> 00:07:49,650 has equal probability of getting 100 million prize behind it. 192 00:07:49,650 --> 00:07:51,810 So the first step is that you pick a random door. 193 00:07:51,810 --> 00:07:54,220 So let's say you pick this one. 194 00:07:54,220 --> 00:07:56,120 And then-- but you don't open it yet, 195 00:07:56,120 --> 00:07:57,830 so you don't know what's behind it. 196 00:07:57,830 --> 00:07:59,990 And then the host picks one of these doors 197 00:07:59,990 --> 00:08:02,540 and opens it to reveal nothing. 198 00:08:02,540 --> 00:08:04,900 So let's say he picks this one. 199 00:08:04,900 --> 00:08:05,910 I can't draw. 200 00:08:05,910 --> 00:08:07,640 Whatever. 201 00:08:07,640 --> 00:08:09,020 And there's nothing, right? 202 00:08:09,020 --> 00:08:11,310 There's nothing there. 203 00:08:11,310 --> 00:08:13,486 So this is what the host picks. 204 00:08:13,486 --> 00:08:16,670 And this is what you originally picked. 205 00:08:16,670 --> 00:08:20,000 And now he says, before he opens your door you still 206 00:08:20,000 --> 00:08:22,970 have the chance of switching to this door. 207 00:08:22,970 --> 00:08:25,730 So the an-- the question was, do you 208 00:08:25,730 --> 00:08:29,090 want to stay with your original door, or do you want to switch? 209 00:08:29,090 --> 00:08:31,340 Because you want to maximize your probability of 210 00:08:31,340 --> 00:08:35,240 does this one still have the prize, or is it this one. 211 00:08:35,240 --> 00:08:37,510 Does that problem make sense to everyone? 212 00:08:37,510 --> 00:08:38,010 Wait. 213 00:08:38,010 --> 00:08:39,650 OK. 214 00:08:39,650 --> 00:08:41,505 So what-- what was your name, again? 215 00:08:41,505 --> 00:08:42,296 AUDIENCE: Priyanka. 216 00:08:42,296 --> 00:08:44,450 PROFESSOR: Priyanka says that here you 217 00:08:44,450 --> 00:08:47,450 have originally one third chance, right? 218 00:08:47,450 --> 00:08:48,037 AUDIENCE: 2/3. 219 00:08:48,037 --> 00:08:49,870 PROFESSOR: Oh, no what-- your original door. 220 00:08:49,870 --> 00:08:50,578 AUDIENCE: Oh, 1/3 221 00:08:50,578 --> 00:08:51,680 PROFESSOR: Yeah. 222 00:08:51,680 --> 00:08:56,030 So when you first choose you only have 1/3 of a chance. 223 00:08:56,030 --> 00:08:58,820 And what she's doing is kind of combining these two 224 00:08:58,820 --> 00:09:01,960 doors as one, as 2/3 chance. 225 00:09:01,960 --> 00:09:05,770 And since you already know that this one is nothing, 226 00:09:05,770 --> 00:09:08,030 it doesn't really affect this probability, 227 00:09:08,030 --> 00:09:09,680 even though he opened it. 228 00:09:09,680 --> 00:09:12,590 So this door actually has 2/3 of a chance. 229 00:09:12,590 --> 00:09:15,380 2/3 of a chance to have the million dollar prize. 230 00:09:15,380 --> 00:09:17,510 So the fact that the host opened this 231 00:09:17,510 --> 00:09:21,650 does nothing to change this probability. 232 00:09:21,650 --> 00:09:26,830 So you're kind of seeing it as one door with one third chance, 233 00:09:26,830 --> 00:09:29,270 and this door a 2/3 chance. 234 00:09:29,270 --> 00:09:31,860 Right? 235 00:09:31,860 --> 00:09:34,070 That's the intuitive explanation. 236 00:09:34,070 --> 00:09:35,996 Does everyone understand this? 237 00:09:35,996 --> 00:09:37,120 It will take a little time. 238 00:09:37,120 --> 00:09:40,900 I know it took me like, a long time to understand it 239 00:09:40,900 --> 00:09:41,400 intuitively. 240 00:09:41,400 --> 00:09:44,380 So OK. 241 00:09:44,380 --> 00:09:48,740 Did anyone do it using conditional probability, 242 00:09:48,740 --> 00:09:50,770 mathematical way? 243 00:09:50,770 --> 00:09:51,274 None of you? 244 00:09:51,274 --> 00:09:52,815 AUDIENCE: Well, you can just draw out 245 00:09:52,815 --> 00:09:55,550 all the possibilities, because there are only six, right? 246 00:09:55,550 --> 00:09:56,360 PROFESSOR: Right. 247 00:09:56,360 --> 00:10:00,500 So do you want to explain it up here? 248 00:10:00,500 --> 00:10:02,240 If you don't, I can do it. 249 00:10:02,240 --> 00:10:04,170 I don't want to make you. 250 00:10:04,170 --> 00:10:04,760 No? 251 00:10:04,760 --> 00:10:05,260 OK. 252 00:10:05,260 --> 00:10:06,055 I'll do it. 253 00:10:06,055 --> 00:10:07,430 So does everyone understand this? 254 00:10:07,430 --> 00:10:09,107 Can I erase it? 255 00:10:09,107 --> 00:10:10,440 Or I should just erase this one. 256 00:10:10,440 --> 00:10:10,940 OK. 257 00:10:20,330 --> 00:10:23,410 If I need to erase better, let me know, too. 258 00:10:23,410 --> 00:10:27,010 Because I don't know what it looks like far away. 259 00:10:27,010 --> 00:10:27,510 OK. 260 00:10:31,260 --> 00:10:34,150 So using conditional probability, 261 00:10:34,150 --> 00:10:35,870 we know about the trees, right? 262 00:10:35,870 --> 00:10:38,310 And how you have certain events, and every time 263 00:10:38,310 --> 00:10:42,120 a new event happens you branch out the tree. 264 00:10:42,120 --> 00:10:42,780 OK. 265 00:10:42,780 --> 00:10:47,090 So let's say we have three doors again. 266 00:10:47,090 --> 00:10:49,480 If I'm in your way, let me know. 267 00:10:49,480 --> 00:10:52,193 OK. 268 00:10:52,193 --> 00:10:52,692 All right. 269 00:10:52,692 --> 00:10:55,660 And you don't know which on has a million dollars. 270 00:10:55,660 --> 00:11:09,890 So let's just assume you pick door A. 271 00:11:09,890 --> 00:11:12,050 So we're just going to assume in this tree 272 00:11:12,050 --> 00:11:14,316 that you picked door A. OK? 273 00:11:24,648 --> 00:11:26,620 OK. 274 00:11:26,620 --> 00:11:30,580 So your first event is, where's the prize? 275 00:11:30,580 --> 00:11:31,310 Right? 276 00:11:31,310 --> 00:11:36,910 So where is the prize? 277 00:11:36,910 --> 00:11:45,710 It has 1/3 chance, right, of being in door A, B, or C. OK. 278 00:11:45,710 --> 00:11:47,020 So this is your first step. 279 00:11:47,020 --> 00:11:49,480 Does everyone do that? 280 00:11:49,480 --> 00:11:50,320 OK. 281 00:11:50,320 --> 00:11:52,090 So the important thing to know is 282 00:11:52,090 --> 00:11:53,980 that your host knows where the door-- where 283 00:11:53,980 --> 00:11:55,688 the prize is, because if it doesn't, then 284 00:11:55,688 --> 00:11:58,330 it doesn't really add any additional information, right? 285 00:11:58,330 --> 00:12:07,420 So now we need to know the host picks which door, right? 286 00:12:07,420 --> 00:12:12,640 So if the prize is behind door A, and you pick door A, 287 00:12:12,640 --> 00:12:14,200 then he has two choices, right? 288 00:12:14,200 --> 00:12:15,670 Because both are empty. 289 00:12:15,670 --> 00:12:20,290 He can choose either B or C, right? 290 00:12:20,290 --> 00:12:21,790 And there's a half chance that he'll 291 00:12:21,790 --> 00:12:26,930 picked B and C. Does that make sense to everyone? 292 00:12:26,930 --> 00:12:30,950 So if he wants to reveal an empty door, he has two choices. 293 00:12:30,950 --> 00:12:33,600 Because the actual prize is in the door you chose. 294 00:12:33,600 --> 00:12:37,720 But if the prize is in B, and you picked A, which door can 295 00:12:37,720 --> 00:12:38,813 he open? 296 00:12:38,813 --> 00:12:39,560 AUDIENCE: C. 297 00:12:39,560 --> 00:12:41,080 PROFESSOR: C. Yeah. 298 00:12:41,080 --> 00:12:45,430 So the only one he can open is C, right? 299 00:12:45,430 --> 00:12:52,600 Same thing for C. He only can choose B. 300 00:12:52,600 --> 00:12:56,170 So does everyone see that? 301 00:12:56,170 --> 00:12:56,670 Yeah? 302 00:12:56,670 --> 00:12:57,630 OK. 303 00:12:57,630 --> 00:13:00,045 So you know with trees we multiply out the probabilities, 304 00:13:00,045 --> 00:13:00,545 right? 305 00:13:00,545 --> 00:13:08,970 So you know this has, what, 1/6, 1/6, 1/3 1/3. 306 00:13:08,970 --> 00:13:11,274 OK? 307 00:13:11,274 --> 00:13:12,690 So that's just the problem set up. 308 00:13:12,690 --> 00:13:14,910 If you want to answer the question, 309 00:13:14,910 --> 00:13:20,300 you need to figure out, what if I stay in door A, 310 00:13:20,300 --> 00:13:21,270 and what if I switch? 311 00:13:26,380 --> 00:13:29,820 So if you stay here, you win, right? 312 00:13:29,820 --> 00:13:31,230 So you get money. 313 00:13:31,230 --> 00:13:34,080 If you stay, you win. 314 00:13:34,080 --> 00:13:35,070 Right? 315 00:13:35,070 --> 00:13:38,040 And if you switch, you lose. 316 00:13:38,040 --> 00:13:42,780 So here you lose, right? 317 00:13:42,780 --> 00:13:47,390 Because it's actually in B. And here you 318 00:13:47,390 --> 00:13:50,260 lose because the prize is actually in C, right? 319 00:13:50,260 --> 00:13:53,240 And if you switch, you get it. 320 00:13:53,240 --> 00:13:54,281 Does this make sense? 321 00:13:54,281 --> 00:13:55,530 I know I'm going kind of fast. 322 00:13:55,530 --> 00:13:59,190 So Are there any questions so far? 323 00:14:06,847 --> 00:14:07,600 OK. 324 00:14:07,600 --> 00:14:09,900 So you can figure out the probability 325 00:14:09,900 --> 00:14:12,240 of winning if you stay. 326 00:14:12,240 --> 00:14:17,920 So you add 1/6 plus 1/6, is 1/3. 327 00:14:17,920 --> 00:14:20,010 Right? 328 00:14:20,010 --> 00:14:25,540 And then the probability of winning if you switch is 2/3. 329 00:14:32,840 --> 00:14:38,020 So this proves mathematically that switching will win-- 330 00:14:38,020 --> 00:14:40,390 will get you the better probability of winning. 331 00:14:45,550 --> 00:14:47,630 Is there any part that confuses anybody? 332 00:14:47,630 --> 00:14:48,350 Yeah? 333 00:14:48,350 --> 00:14:51,816 AUDIENCE: Can you go over the stay and switch part? 334 00:14:51,816 --> 00:14:52,760 PROFESSOR: OK. 335 00:14:52,760 --> 00:14:57,050 So this is all assuming that along each way 336 00:14:57,050 --> 00:14:58,970 that the prize is in A, right? 337 00:14:58,970 --> 00:15:02,560 So if the prize is in A-- 338 00:15:02,560 --> 00:15:04,302 that's all right here, right-- 339 00:15:04,302 --> 00:15:08,650 if the prize is in A, you stay in A, 340 00:15:08,650 --> 00:15:10,450 you're going to win, right? 341 00:15:10,450 --> 00:15:12,070 But if you switch, and you're assuming 342 00:15:12,070 --> 00:15:14,200 that the prize is in door A, then you're 343 00:15:14,200 --> 00:15:16,990 not going to win, right? 344 00:15:16,990 --> 00:15:21,020 But for this row, you're assuming that prize is in B, 345 00:15:21,020 --> 00:15:24,332 and if you switch, you have to win. 346 00:15:24,332 --> 00:15:28,351 And symmetrically this is the same way, too. 347 00:15:28,351 --> 00:15:29,516 All right. 348 00:15:29,516 --> 00:15:30,320 That make sense? 349 00:15:30,320 --> 00:15:33,145 Anything else? 350 00:15:33,145 --> 00:15:35,140 OK. 351 00:15:35,140 --> 00:15:39,000 So that just proves, sometimes conditional probability 352 00:15:39,000 --> 00:15:40,080 is good for you. 353 00:15:40,080 --> 00:15:41,410 OK. 354 00:15:41,410 --> 00:15:45,810 So we're going to move on to this class's stuff, which 355 00:15:45,810 --> 00:15:46,580 is Bayes' rule. 356 00:15:51,730 --> 00:15:52,300 Oh, crap. 357 00:15:52,300 --> 00:15:53,830 Did anyone need this? 358 00:15:53,830 --> 00:15:55,310 I can leave it up. 359 00:15:55,310 --> 00:15:56,530 No? 360 00:15:56,530 --> 00:15:58,420 OK. 361 00:15:58,420 --> 00:16:00,340 If you still need it after class, let me know. 362 00:16:00,340 --> 00:16:01,050 I'm sorry. 363 00:16:01,050 --> 00:16:04,188 I'll ask before I do that. 364 00:16:04,188 --> 00:16:04,688 OK. 365 00:16:09,940 --> 00:16:10,440 OK. 366 00:16:10,440 --> 00:16:12,356 So I'm not going to write out the first thing. 367 00:16:12,356 --> 00:16:15,520 But Bayes' rule is basically finding out 368 00:16:15,520 --> 00:16:17,840 your reverse probability. 369 00:16:17,840 --> 00:16:20,920 So remember I was asking you what's 370 00:16:20,920 --> 00:16:27,449 the difference between this, and this. 371 00:16:27,449 --> 00:16:27,949 Right? 372 00:16:32,350 --> 00:16:38,530 So if you look at the slide that I handed you out. 373 00:16:38,530 --> 00:16:41,020 The first slide that says Bayes' rule. 374 00:16:41,020 --> 00:16:44,320 If we use the radar example that we showed from before. 375 00:16:48,190 --> 00:16:51,010 When we did do the problem we figured out the probability 376 00:16:51,010 --> 00:16:54,340 that the radar registers, given that the plane is present. 377 00:16:54,340 --> 00:16:55,490 Right? 378 00:16:55,490 --> 00:16:58,960 But now we want to know if the plane is present, 379 00:16:58,960 --> 00:17:01,270 given that the radar registers. 380 00:17:01,270 --> 00:17:03,600 Does everyone see the difference in that? 381 00:17:03,600 --> 00:17:07,720 So the first one is kind of saying 382 00:17:07,720 --> 00:17:10,690 how accurate the radar is, but the second one 383 00:17:10,690 --> 00:17:13,180 is what you really want to know, is how much 384 00:17:13,180 --> 00:17:15,040 you can rely on the radar. 385 00:17:15,040 --> 00:17:19,960 Because the first one is more of a mechanical thing, 386 00:17:19,960 --> 00:17:23,589 but the second is actually using the radar. 387 00:17:23,589 --> 00:17:25,520 Does that make sense to everyone? 388 00:17:25,520 --> 00:17:26,200 OK. 389 00:17:26,200 --> 00:17:30,870 So I'll write it up here. 390 00:17:30,870 --> 00:17:36,110 P is our event that the plane is present. 391 00:17:40,420 --> 00:17:40,920 All right. 392 00:17:40,920 --> 00:17:41,970 If you can't read my handwriting, 393 00:17:41,970 --> 00:17:43,110 let me know that, too. 394 00:17:46,086 --> 00:17:46,586 Radar. 395 00:17:51,877 --> 00:17:53,330 OK. 396 00:17:53,330 --> 00:17:56,470 So what we want to know is the probability 397 00:17:56,470 --> 00:18:01,150 that the plane is there given that the radar registers. 398 00:18:01,150 --> 00:18:04,410 OK, so for you guys who weren't here, and just for you 399 00:18:04,410 --> 00:18:09,090 other guys, too, I'll write out the chart again. 400 00:18:09,090 --> 00:18:11,635 The tree chart. 401 00:18:11,635 --> 00:18:12,635 I'll just do it up here. 402 00:18:45,240 --> 00:18:49,090 So if you guys first remember, if the plane was present, 403 00:18:49,090 --> 00:18:55,180 we have a 0.05 probability that the plane is present. 404 00:18:55,180 --> 00:19:02,224 Which means a 0.95 probability that it's not present. 405 00:19:02,224 --> 00:19:06,490 And the next thing we have was whether the radar picked up 406 00:19:06,490 --> 00:19:07,780 on it, right? 407 00:19:07,780 --> 00:19:09,053 So whether it registered. 408 00:19:21,378 --> 00:19:22,370 OK. 409 00:19:22,370 --> 00:19:26,030 So we were given, last time, the probabilities of this. 410 00:19:26,030 --> 00:19:28,880 Given whether the plane was there or not. 411 00:19:28,880 --> 00:19:33,950 So the radar registers 0.99 at the time 412 00:19:33,950 --> 00:19:39,480 the plane is there, which means 0.01 if it's there 413 00:19:39,480 --> 00:19:41,070 but it doesn't register. 414 00:19:41,070 --> 00:19:44,630 And then if it registers anyway, even if it's not there, 415 00:19:44,630 --> 00:19:52,500 that's a 0.1 chance, which means a 0.90 chance here. 416 00:19:52,500 --> 00:19:54,850 For you guys who weren't here, do 417 00:19:54,850 --> 00:19:56,100 understand the problem setup? 418 00:19:56,100 --> 00:19:59,890 This is all you need to know to understand the next step. 419 00:19:59,890 --> 00:20:01,670 OK. 420 00:20:01,670 --> 00:20:12,760 And using the multiplication rule you can get 0.0455, 421 00:20:12,760 --> 00:20:27,144 0.0005, 0.0950, and 0.8550. 422 00:20:27,144 --> 00:20:29,100 OK? 423 00:20:29,100 --> 00:20:32,060 Does everyone understand what these numbers are referring to? 424 00:20:32,060 --> 00:20:35,690 So this is the probability that this part of the branch 425 00:20:35,690 --> 00:20:36,190 happened. 426 00:20:36,190 --> 00:20:38,800 The plane is there and the radar says yes. 427 00:20:38,800 --> 00:20:40,896 Et cetera, et cetera, et cetera. 428 00:20:40,896 --> 00:20:41,670 OK? 429 00:20:41,670 --> 00:20:41,820 Yeah? 430 00:20:41,820 --> 00:20:44,361 AUDIENCE: So that's achieved by multiplying the two together, 431 00:20:44,361 --> 00:20:46,620 right? 432 00:20:46,620 --> 00:20:49,480 PROFESSOR: Which you can do for a sequence of events like this. 433 00:20:52,360 --> 00:20:53,320 All right. 434 00:21:08,732 --> 00:21:09,740 OK. 435 00:21:09,740 --> 00:21:17,430 So if you see 0.99 is the probability 436 00:21:17,430 --> 00:21:22,860 that the radar registers given that the plane is there. 437 00:21:22,860 --> 00:21:24,800 Right? 438 00:21:24,800 --> 00:21:28,980 But we actually want to know this, the reverse of it. 439 00:21:28,980 --> 00:21:32,880 So you've seen the definition of probability. 440 00:21:32,880 --> 00:21:40,289 You have probability of P given R equals-- 441 00:21:40,289 --> 00:21:41,080 can anyone tell me? 442 00:21:46,569 --> 00:21:51,430 AUDIENCE: The intersect R. over the probability of R. 443 00:21:51,430 --> 00:21:52,422 PROFESSOR: Yup. 444 00:21:52,422 --> 00:21:54,410 OK. 445 00:21:54,410 --> 00:22:03,220 So first we can find probability of R. So given this thing, 446 00:22:03,220 --> 00:22:04,990 can anyone tell me what the probability 447 00:22:04,990 --> 00:22:08,880 that the radar registers is? 448 00:22:08,880 --> 00:22:11,310 Or have an idea of how you can figure out 449 00:22:11,310 --> 00:22:12,630 what this probability is? 450 00:22:17,888 --> 00:22:18,850 OK. 451 00:22:18,850 --> 00:22:21,685 So probability of the radar registry-- 452 00:22:21,685 --> 00:22:23,680 I should have done this over there, but-- 453 00:22:23,680 --> 00:22:26,730 you have it here and here, right? 454 00:22:26,730 --> 00:22:28,320 So what you need to do-- 455 00:22:28,320 --> 00:22:34,070 so that's this branch plus this branch. 456 00:22:34,070 --> 00:22:34,570 Right? 457 00:22:38,150 --> 00:22:43,140 So you add this probability plus this probability. 458 00:22:43,140 --> 00:22:56,530 So you have 0.0495 plus 0.0950. 459 00:22:56,530 --> 00:22:58,684 And what's the probability of this? 460 00:22:58,684 --> 00:23:00,100 Can anyone see that in the branch? 461 00:23:03,480 --> 00:23:05,710 The probability that the plane is there and the radar 462 00:23:05,710 --> 00:23:09,070 registers. 463 00:23:09,070 --> 00:23:11,870 AUDIENCE: 0.0495. 464 00:23:11,870 --> 00:23:13,531 PROFESSOR: So it's only this branch. 465 00:23:13,531 --> 00:23:14,030 Right? 466 00:23:14,030 --> 00:23:15,120 What she said. 467 00:23:15,120 --> 00:23:24,910 So it's 0.0495, and you're left with 0.3426, 468 00:23:24,910 --> 00:23:29,150 which is about a 34% chance. 469 00:23:29,150 --> 00:23:31,310 Does everyone understand how we got 470 00:23:31,310 --> 00:23:34,730 from what we were given, the probability of the radar 471 00:23:34,730 --> 00:23:38,600 registering given that the plane is present, to the probability 472 00:23:38,600 --> 00:23:42,180 that the plane is present given that the radar says it's there. 473 00:23:44,750 --> 00:23:47,610 Does everyone understand? 474 00:23:47,610 --> 00:23:51,291 So this probability, even though the radar is pretty accurate, 475 00:23:51,291 --> 00:23:51,790 right? 476 00:23:51,790 --> 00:23:54,360 It's 0.99% chance. 477 00:23:54,360 --> 00:23:57,060 You still have a 34% chance that the radar-- 478 00:23:57,060 --> 00:24:01,560 that the plane is actually there given that the radar says yes. 479 00:24:01,560 --> 00:24:05,130 So even though it seems like a very accurate radar, 480 00:24:05,130 --> 00:24:07,330 this probability is not really what you want. 481 00:24:07,330 --> 00:24:10,110 You want to be sure that the plane is there 482 00:24:10,110 --> 00:24:12,150 if the radar says yes. 483 00:24:12,150 --> 00:24:15,540 So in an ideal world you have 100%. 484 00:24:15,540 --> 00:24:17,190 Right? 485 00:24:17,190 --> 00:24:20,080 So the thing that's throwing this off is-- 486 00:24:20,080 --> 00:24:23,850 the radar off, is probably this part. 487 00:24:23,850 --> 00:24:27,550 You don't want the radar to say yes if it's not actually there. 488 00:24:27,550 --> 00:24:31,530 So if you have an ideal radar, this would be zero. 489 00:24:31,530 --> 00:24:32,192 Right? 490 00:24:32,192 --> 00:24:33,650 AUDIENCE: Can I go to the restroom. 491 00:24:33,650 --> 00:24:35,340 PROFESSOR: Yes. 492 00:24:35,340 --> 00:24:36,710 If you need to pee, go. 493 00:24:36,710 --> 00:24:37,210 Yes? 494 00:24:37,210 --> 00:24:40,089 AUDIENCE: So [INAUDIBLE] one of these problems is you 495 00:24:40,089 --> 00:24:42,454 kind of find out what the probability of P 496 00:24:42,454 --> 00:24:46,070 is in this example if R is true. 497 00:24:46,070 --> 00:24:46,870 All right. 498 00:24:46,870 --> 00:24:47,495 PROFESSOR: Yup. 499 00:24:50,550 --> 00:24:51,990 Does this make sense to everyone? 500 00:24:51,990 --> 00:24:53,030 OK. 501 00:24:53,030 --> 00:24:54,720 So the next slide. 502 00:24:57,700 --> 00:25:00,150 There's an example of this, this military application. 503 00:25:00,150 --> 00:25:02,080 I'm actually working at Lincoln Labs, 504 00:25:02,080 --> 00:25:03,540 if you don't know about it. 505 00:25:03,540 --> 00:25:06,100 And what they do a lot of military defense like this. 506 00:25:06,100 --> 00:25:10,540 So if you were trying to register if a plane was there, 507 00:25:10,540 --> 00:25:12,880 that plane could be an enemy aircraft, 508 00:25:12,880 --> 00:25:14,560 or it could be a commercial aircraft. 509 00:25:14,560 --> 00:25:17,749 And you want the radar to make sure it knows what it's seeing. 510 00:25:17,749 --> 00:25:20,290 Because you don't want to waste a missile and shoot something 511 00:25:20,290 --> 00:25:23,350 that's not actually what you're seeing. 512 00:25:23,350 --> 00:25:24,220 Right? 513 00:25:24,220 --> 00:25:25,870 Yeah. 514 00:25:25,870 --> 00:25:28,840 So ideally in the military they have really good radars. 515 00:25:28,840 --> 00:25:31,340 It won't be this kind of probability. 516 00:25:31,340 --> 00:25:31,840 OK. 517 00:25:31,840 --> 00:25:36,646 Does that help figure out why we need this kind of stuff? 518 00:25:36,646 --> 00:25:38,620 OK. 519 00:25:38,620 --> 00:25:41,320 Oh, yeah, and this is an example of Bayesian probability 520 00:25:41,320 --> 00:25:44,200 that we mentioned before how Bayesian probability is 521 00:25:44,200 --> 00:25:47,950 a measure of how much you believe something will happen. 522 00:25:47,950 --> 00:25:49,341 So this is like that. 523 00:25:49,341 --> 00:25:49,840 Right? 524 00:25:49,840 --> 00:25:51,465 You can't repeat it over and over again 525 00:25:51,465 --> 00:25:55,270 like a coin, exact same experiment all by itself. 526 00:25:55,270 --> 00:25:59,776 This is dependent on whether the plane is there or not. 527 00:25:59,776 --> 00:26:00,610 OK. 528 00:26:00,610 --> 00:26:03,200 Any questions? 529 00:26:03,200 --> 00:26:03,700 OK. 530 00:26:06,440 --> 00:26:08,690 In order for us to figure out Bayes' rule, 531 00:26:08,690 --> 00:26:10,410 we have to make several assumptions. 532 00:26:14,210 --> 00:26:15,710 And you can see it on the slide. 533 00:26:18,660 --> 00:26:22,100 We have the-- 534 00:26:22,100 --> 00:26:22,880 OK. 535 00:26:22,880 --> 00:26:27,170 So we have probability of A is what we know. 536 00:26:27,170 --> 00:26:29,520 Is whether the radar registers or not. 537 00:26:29,520 --> 00:26:31,220 So we have all that information. 538 00:26:31,220 --> 00:26:32,150 Right? 539 00:26:32,150 --> 00:26:35,600 If you don't know this information. 540 00:26:35,600 --> 00:26:37,970 Like, say you don't know how accurate the radar is, 541 00:26:37,970 --> 00:26:41,551 then you can't ever figure out the reverse. 542 00:26:41,551 --> 00:26:42,050 Right? 543 00:26:42,050 --> 00:26:44,540 So you have to know everything about R before you 544 00:26:44,540 --> 00:26:48,084 can figure out P of R. 545 00:26:48,084 --> 00:26:59,720 So that's just saying that if you have a bunch of things 546 00:26:59,720 --> 00:27:06,750 like, this is event A1, this is event A2, and this is event A3. 547 00:27:06,750 --> 00:27:09,856 And you have probability of B happening. 548 00:27:09,856 --> 00:27:12,860 But you don't really know what probability of B is, 549 00:27:12,860 --> 00:27:15,980 but you know the probability of A2 550 00:27:15,980 --> 00:27:22,570 and B, probability of A1 and B, etc. 551 00:27:22,570 --> 00:27:28,170 Then in some way or another you can get probability of B. 552 00:27:28,170 --> 00:27:32,016 And I write that out better. 553 00:27:32,016 --> 00:27:35,670 So this is used-- 554 00:27:35,670 --> 00:27:39,660 Bayes' rule uses the total probability theorem, 555 00:27:39,660 --> 00:27:43,540 which I have written out there. 556 00:27:43,540 --> 00:27:45,360 I'll write it out again. 557 00:27:45,360 --> 00:27:54,330 Probability of B equals probability of A1 and B 558 00:27:54,330 --> 00:28:06,480 happening, plus everything all the way up to An and B. 559 00:28:06,480 --> 00:28:11,160 So for the radar example we only had two A's. 560 00:28:11,160 --> 00:28:11,895 Right? 561 00:28:11,895 --> 00:28:17,430 A is the probability that the radar registers, 562 00:28:17,430 --> 00:28:21,140 and A2 is whether-- 563 00:28:21,140 --> 00:28:23,620 wait, hold on. 564 00:28:23,620 --> 00:28:26,190 Sorry. 565 00:28:26,190 --> 00:28:27,440 Hold on. 566 00:28:27,440 --> 00:28:27,940 OK. 567 00:28:34,650 --> 00:28:35,150 It is there. 568 00:28:35,150 --> 00:28:36,040 OK. 569 00:28:36,040 --> 00:28:38,440 So this was actually our probability 570 00:28:38,440 --> 00:28:47,120 that the radar registers given that the plane is there, 571 00:28:47,120 --> 00:28:56,230 union with the radar registers plus the probability that-- 572 00:28:56,230 --> 00:28:59,920 I'm sorry, this P is really confusing-- 573 00:28:59,920 --> 00:29:03,900 plus the probability that the plane isn't there. 574 00:29:03,900 --> 00:29:09,110 Union with R. Can you guys see that? 575 00:29:09,110 --> 00:29:13,010 So you can do that for multiple things, but for this case, 576 00:29:13,010 --> 00:29:14,500 we only had two. 577 00:29:14,500 --> 00:29:16,560 So probability that the plane was there, 578 00:29:16,560 --> 00:29:19,960 probability that it wasn't there. 579 00:29:19,960 --> 00:29:23,900 So that's this part, and this part. 580 00:29:27,720 --> 00:29:33,380 And the way you can see this with a tree 581 00:29:33,380 --> 00:29:44,460 is that you have A1 happening, A2 happening, An happening. 582 00:29:44,460 --> 00:29:48,150 And then you know if B happens or if B does not happen. 583 00:29:50,760 --> 00:29:53,745 You know if B happens or B doesn't happen. 584 00:30:00,600 --> 00:30:02,390 So that's kind of what we did, right? 585 00:30:02,390 --> 00:30:04,290 We know the plane is there or not, 586 00:30:04,290 --> 00:30:06,670 and then we know in each case whether the radar registers 587 00:30:06,670 --> 00:30:08,000 or not. 588 00:30:08,000 --> 00:30:11,645 And then to get probability of B you just 589 00:30:11,645 --> 00:30:17,330 do this branch, this branch, and this branch. 590 00:30:17,330 --> 00:30:29,089 So then you can get probability of B. So does that make sense? 591 00:30:29,089 --> 00:30:30,630 It's a little confusing, let me know. 592 00:30:34,770 --> 00:30:38,340 So even though we don't know probability of B off hand, 593 00:30:38,340 --> 00:30:40,824 but you know everything that can lead to B, 594 00:30:40,824 --> 00:30:42,240 and all the probabilities, you can 595 00:30:42,240 --> 00:30:54,230 get B. Does that makes sense? 596 00:30:54,230 --> 00:30:54,990 Any questions? 597 00:30:58,410 --> 00:30:59,640 OK. 598 00:30:59,640 --> 00:31:03,530 So Bayes' rule is basically figuring out, 599 00:31:03,530 --> 00:31:06,000 you've seen the total probability theorem. 600 00:31:06,000 --> 00:31:07,607 You're reverse. 601 00:31:07,607 --> 00:31:09,065 So in the end, you have probability 602 00:31:09,065 --> 00:31:16,830 of Ai, whatever it is, for us it's A1, given 603 00:31:16,830 --> 00:31:24,130 B equals probability of A-- 604 00:31:36,390 --> 00:31:43,020 and I use this to figure out the bottom half. 605 00:31:46,160 --> 00:31:48,140 To get this. 606 00:31:48,140 --> 00:31:51,950 This is just a generic way of doing what we did earlier. 607 00:31:56,060 --> 00:31:58,942 Does everyone see that? 608 00:31:58,942 --> 00:32:00,355 OK. 609 00:32:00,355 --> 00:32:02,581 Can I erase this? 610 00:32:02,581 --> 00:32:03,080 Yeah? 611 00:32:03,080 --> 00:32:03,580 OK. 612 00:32:09,390 --> 00:32:12,360 So those are real examples of you seeing Bayes' rule. 613 00:32:12,360 --> 00:32:14,770 It's very useful. 614 00:32:14,770 --> 00:32:17,525 We use it a lot in artificial intelligence. 615 00:32:17,525 --> 00:32:21,300 In that class I use it a lot. 616 00:32:21,300 --> 00:32:24,070 I'm sure they use it a lot for other applications, too. 617 00:32:24,070 --> 00:32:25,770 So that's a very important concept. 618 00:32:25,770 --> 00:32:27,370 Make sure you get that straight. 619 00:32:31,870 --> 00:32:34,370 OK. 620 00:32:34,370 --> 00:32:38,800 So we're going to use conditional probability 621 00:32:38,800 --> 00:32:43,550 to derive what independence means. 622 00:32:43,550 --> 00:32:49,210 So if I told you that the probability of A given B 623 00:32:49,210 --> 00:32:53,290 is actually equal to the probability of A, 624 00:32:53,290 --> 00:32:54,622 what does that indicate? 625 00:32:57,400 --> 00:33:00,400 That's kind of like saying, if you have B, 626 00:33:00,400 --> 00:33:01,820 it doesn't really matter. 627 00:33:01,820 --> 00:33:05,740 You still have the probability of A. 628 00:33:05,740 --> 00:33:08,680 So B doesn't affect A in any other way. 629 00:33:08,680 --> 00:33:10,720 They're independent. 630 00:33:10,720 --> 00:33:12,720 And that also works for the reverse, 631 00:33:12,720 --> 00:33:15,990 because these are just numbers. 632 00:33:15,990 --> 00:33:17,320 Right? 633 00:33:17,320 --> 00:33:22,950 So if A is independent of B, B is independent of A. OK? 634 00:33:25,850 --> 00:33:30,710 So that's how you define independence. 635 00:33:30,710 --> 00:33:33,140 So in order to figure out if something is independent 636 00:33:33,140 --> 00:33:36,090 or not, we're going to use conditional probability. 637 00:33:36,090 --> 00:33:47,390 So you have probability of A equals 2, A given B. 638 00:33:47,390 --> 00:33:51,170 And B doesn't matter, but we're going to use this, anyway. 639 00:33:51,170 --> 00:33:54,380 So given the definition of conditional probability, 640 00:33:54,380 --> 00:34:03,930 we have this is A union B over PV. 641 00:34:08,699 --> 00:34:23,219 And you can kind of erase that and put in [INAUDIBLE] A. 642 00:34:23,219 --> 00:34:28,940 So if this is true, that means this has to be true. 643 00:34:28,940 --> 00:34:32,020 And since we don't like it when we divide by zero, 644 00:34:32,020 --> 00:34:33,730 we're just going to move this up. 645 00:34:33,730 --> 00:34:43,320 So you get this right. 646 00:34:43,320 --> 00:34:47,515 So this is how you test for independence. 647 00:34:47,515 --> 00:34:48,850 OK? 648 00:34:48,850 --> 00:34:53,190 So if I told you that the probability of A given B 649 00:34:53,190 --> 00:34:56,310 still equals the probability of A, it means B doesn't matter, 650 00:34:56,310 --> 00:34:57,870 which means this has to hold true. 651 00:35:01,380 --> 00:35:03,275 Does everyone see how I got that? 652 00:35:03,275 --> 00:35:04,180 How that makes sense? 653 00:35:04,180 --> 00:35:04,680 OK. 654 00:35:09,170 --> 00:35:11,790 This is a question mark, if you don't know. 655 00:35:11,790 --> 00:35:14,060 So if you don't know, you try it out. 656 00:35:14,060 --> 00:35:16,374 If it's equal, means independent. 657 00:35:20,250 --> 00:35:26,180 So if you have two disjoint events, are they independent? 658 00:35:26,180 --> 00:35:36,482 You have A, B. Are these two independent? 659 00:35:36,482 --> 00:35:39,224 AUDIENCE: So that means if you have one then you definitely 660 00:35:39,224 --> 00:35:40,315 can't have the other? 661 00:35:40,315 --> 00:35:43,170 PROFESSOR: Right. 662 00:35:43,170 --> 00:35:45,650 So that's intuitive, although a lot of people 663 00:35:45,650 --> 00:35:48,530 think that these two are independent. 664 00:35:48,530 --> 00:35:49,940 Or not independent. 665 00:35:49,940 --> 00:35:50,580 Or independent. 666 00:35:50,580 --> 00:35:51,080 Sorry. 667 00:35:51,080 --> 00:35:52,850 But they actually aren't independent. 668 00:35:52,850 --> 00:35:56,510 So if you do the math, the probability of each of these 669 00:35:56,510 --> 00:36:02,130 has to be greater than zero, but their intersection is zero. 670 00:36:02,130 --> 00:36:06,262 So if you have probability, you have your test. 671 00:36:06,262 --> 00:36:08,466 And this holds true with independent. 672 00:36:13,600 --> 00:36:15,930 A union B? 673 00:36:15,930 --> 00:36:17,230 Zero. 674 00:36:17,230 --> 00:36:18,910 Because they are disjoint. 675 00:36:18,910 --> 00:36:21,520 But you can't really have an event unless you have greater 676 00:36:21,520 --> 00:36:23,270 than zero probability. 677 00:36:23,270 --> 00:36:27,240 So this is a greater than zero thing. 678 00:36:27,240 --> 00:36:29,640 This is a greater than zero. 679 00:36:29,640 --> 00:36:33,070 And they obviously can't equal zero. 680 00:36:33,070 --> 00:36:34,414 So it's not independent. 681 00:36:37,260 --> 00:36:43,230 It's like saying, I flipped a coin and I got heads. 682 00:36:43,230 --> 00:36:45,390 That definitely means you can't have tails. 683 00:36:45,390 --> 00:36:48,730 So even though they're disjoint, they're 684 00:36:48,730 --> 00:36:50,730 dependent on each other, because if one happens, 685 00:36:50,730 --> 00:36:53,695 the other one can't for sure. 686 00:36:53,695 --> 00:36:55,250 Does that make sense to everyone? 687 00:36:55,250 --> 00:36:57,310 This trips a lot of people up. 688 00:36:57,310 --> 00:36:59,840 OK. 689 00:36:59,840 --> 00:37:01,572 We can't move this. 690 00:37:01,572 --> 00:37:04,420 I'll just work on this. 691 00:37:04,420 --> 00:37:08,410 So we're going to use independence 692 00:37:08,410 --> 00:37:12,020 to prove that successive rules are independent of each other. 693 00:37:12,020 --> 00:37:15,430 A lot of times they just tell you that rules are independent 694 00:37:15,430 --> 00:37:16,320 of each other. 695 00:37:16,320 --> 00:37:18,860 But we're going to use this to prove it. 696 00:37:18,860 --> 00:37:24,450 So instead of a six sided die, we have a four sided die. 697 00:37:24,450 --> 00:37:30,610 And the sides, the numbers on the sides are 1, 2, 3, and 4. 698 00:37:30,610 --> 00:37:32,320 So instead of 1-6 we have 1-4. 699 00:37:35,950 --> 00:37:38,530 And the answer we're trying to get 700 00:37:38,530 --> 00:37:41,511 is, are successive rolls independent? 701 00:37:41,511 --> 00:37:42,010 So. 702 00:38:04,640 --> 00:38:05,650 OK? 703 00:38:05,650 --> 00:38:07,670 And on the sheet I have for you, I 704 00:38:07,670 --> 00:38:10,900 did write out the sample space just in case. 705 00:38:10,900 --> 00:38:13,502 So you guys can see. 706 00:38:13,502 --> 00:38:14,960 Each one of those is a combination. 707 00:38:14,960 --> 00:38:18,950 So the first one, 1-1, is your first rolls is a 1, 708 00:38:18,950 --> 00:38:22,350 your second roll is a 1, et cetera, et cetera. 709 00:38:22,350 --> 00:38:23,000 OK? 710 00:38:23,000 --> 00:38:35,435 So for our events we're going to do A equals first roll is i. 711 00:38:38,650 --> 00:38:41,060 And because this is a very generalized answer, 712 00:38:41,060 --> 00:38:45,740 we're not going to make it first roll is 1, or 2, or 3, or 4. 713 00:38:45,740 --> 00:38:48,910 i can be any of these numbers. 714 00:38:48,910 --> 00:38:49,410 Right? 715 00:38:52,860 --> 00:38:55,080 So does that event make sense? 716 00:38:55,080 --> 00:38:55,580 OK? 717 00:38:55,580 --> 00:38:58,940 We don't want to specify too much. 718 00:38:58,940 --> 00:39:02,440 And B is the same thing, except it's 719 00:39:02,440 --> 00:39:08,000 the second roll is, say, j. 720 00:39:08,000 --> 00:39:11,611 And j has to be in here, too. 721 00:39:11,611 --> 00:39:12,110 OK? 722 00:39:12,110 --> 00:39:14,465 So that's how we're going to define our problem. 723 00:39:17,880 --> 00:39:19,100 So we have our test again. 724 00:39:23,630 --> 00:39:36,870 First I'm going to do P, A and B. So 725 00:39:36,870 --> 00:39:39,630 what this means is that the probability that the first rule 726 00:39:39,630 --> 00:39:43,320 is i and the first roll is j. 727 00:39:43,320 --> 00:39:47,100 And that has to be 1/16, because there 728 00:39:47,100 --> 00:39:49,280 are 16 different combinations. 729 00:39:49,280 --> 00:39:53,010 And given an i, given a j, you only can have one. 730 00:39:55,680 --> 00:39:58,760 Does this make sense? 731 00:39:58,760 --> 00:40:04,455 So if we say I is 1, j is 4, there's only 1/16 chance. 732 00:40:07,770 --> 00:40:12,490 And the next step is the left side. 733 00:40:12,490 --> 00:40:16,260 So I want to figure out, what's the probability of A with i. 734 00:40:19,170 --> 00:40:25,530 So if we assume that i equals 1 again, how many different ways, 735 00:40:25,530 --> 00:40:29,190 in that sample space, is the first roll an i? 736 00:40:29,190 --> 00:40:34,200 Well you have 1-1, 1-2, 1-3, 1-4. 737 00:40:34,200 --> 00:40:37,826 So it's four different combinations. 738 00:40:37,826 --> 00:40:38,700 Does that make sense? 739 00:40:38,700 --> 00:40:39,970 Should I write this out? 740 00:40:39,970 --> 00:40:42,292 So you have 1-1, 1-2-- 741 00:40:47,528 --> 00:40:55,310 this is assuming i equals 1. 742 00:40:55,310 --> 00:40:57,340 But because I want to generalize it, 743 00:40:57,340 --> 00:40:59,340 that's why I'm just saying it's i. 744 00:40:59,340 --> 00:41:01,700 But you can really just fill in any number, 745 00:41:01,700 --> 00:41:04,020 and it would be the same probability. 746 00:41:22,370 --> 00:41:24,660 So you just have B equals j on it's 747 00:41:24,660 --> 00:41:28,190 own, keep in mind that this is the second roll, right? 748 00:41:28,190 --> 00:41:33,890 So if we assume again that j equals 4, 749 00:41:33,890 --> 00:41:39,640 it's still the same probability, because now you have 1-4-- 750 00:41:39,640 --> 00:41:41,300 thank you, by the way-- 751 00:41:41,300 --> 00:41:48,290 1-4, 2-4, 3-4, 4-4. 752 00:41:50,250 --> 00:41:51,000 Everyone see that? 753 00:41:51,000 --> 00:41:53,240 So the second-- this is the probability 754 00:41:53,240 --> 00:41:56,492 that the second roll is a 4. 755 00:41:56,492 --> 00:41:57,950 But I actually generalized it to j. 756 00:41:57,950 --> 00:41:59,780 I'm just doing j equals 4. 757 00:41:59,780 --> 00:42:02,000 So you can see that better. 758 00:42:02,000 --> 00:42:05,810 So that's 4 out of 16. 759 00:42:10,340 --> 00:42:13,700 i and j can be any number, again. 760 00:42:13,700 --> 00:42:17,440 So if we do that test for independence, 761 00:42:17,440 --> 00:42:25,230 we have probability of A, probability Bj 762 00:42:25,230 --> 00:42:37,502 equals probability of Ai union B. This is 1/16. 763 00:42:37,502 --> 00:42:38,460 Does everyone see that? 764 00:42:41,300 --> 00:42:47,490 So if we assume A is 1 and B is j, 765 00:42:47,490 --> 00:42:52,160 you only can have 1/4 out of the 16 combinations. 766 00:42:55,630 --> 00:42:58,210 So that's 1/16. 767 00:42:58,210 --> 00:43:00,220 And then this probability is 4/16. 768 00:43:02,820 --> 00:43:03,455 4/16. 769 00:43:06,962 --> 00:43:09,510 1, 4. 770 00:43:09,510 --> 00:43:11,436 1, 4. 771 00:43:18,820 --> 00:43:19,730 They're equal. 772 00:43:19,730 --> 00:43:20,770 Right? 773 00:43:20,770 --> 00:43:24,270 So that means they are independent. 774 00:43:24,270 --> 00:43:26,170 Did everyone see how I did this? 775 00:43:26,170 --> 00:43:27,720 OK.