1 00:00:00,499 --> 00:00:02,830 The following content is provided under a Creative 2 00:00:02,830 --> 00:00:04,340 Commons license. 3 00:00:04,340 --> 00:00:06,680 Your support will help MIT OpenCourseWare 4 00:00:06,680 --> 00:00:11,050 continue to offer high quality educational resources for free. 5 00:00:11,050 --> 00:00:13,670 To make a donation or view additional materials 6 00:00:13,670 --> 00:00:17,542 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,542 --> 00:00:18,167 at ocw.mit.edu. 8 00:00:23,170 --> 00:00:25,090 PROFESSOR: Let's get started. 9 00:00:25,090 --> 00:00:26,540 Drop date is coming up. 10 00:00:26,540 --> 00:00:28,499 So that you didn't do very well on the midterm, 11 00:00:28,499 --> 00:00:30,206 and you haven't been doing your homework, 12 00:00:30,206 --> 00:00:32,070 and you haven't been going to recitation, 13 00:00:32,070 --> 00:00:35,580 I would strongly consider drop date. 14 00:00:35,580 --> 00:00:37,820 And you can see your TAs to get some idea 15 00:00:37,820 --> 00:00:40,040 of what your grades are, and what you're 16 00:00:40,040 --> 00:00:43,410 likely to do if you keep on giving the same performance 17 00:00:43,410 --> 00:00:46,400 you've had so far. 18 00:00:46,400 --> 00:00:49,740 Now, today, we're going to switch gears again and cover 19 00:00:49,740 --> 00:00:51,577 the last topic for the course. 20 00:00:51,577 --> 00:00:53,160 For the remainder of the course, we're 21 00:00:53,160 --> 00:00:55,800 going to talk about probability. 22 00:00:55,800 --> 00:00:58,880 It's one of the most important subjects in all 23 00:00:58,880 --> 00:01:00,540 of math or computer science. 24 00:01:00,540 --> 00:01:03,020 In fact, I was at a frat last night for dinner, 25 00:01:03,020 --> 00:01:05,550 and they asked me what my favorite algorithm was. 26 00:01:05,550 --> 00:01:08,340 And I couldn't really think about my favorite algorithm 27 00:01:08,340 --> 00:01:11,370 was, but in terms of one of the most important subjects for you 28 00:01:11,370 --> 00:01:15,730 to know about, after induction, it's probably probability. 29 00:01:15,730 --> 00:01:18,850 It's really important across math and computer science. 30 00:01:18,850 --> 00:01:21,812 Most of the upper level courses you take in computer science 31 00:01:21,812 --> 00:01:23,020 are going to use probability. 32 00:01:23,020 --> 00:01:25,660 They're going to expect you to know the basics. 33 00:01:25,660 --> 00:01:29,110 For example, when you're designing algorithms in 006 34 00:01:29,110 --> 00:01:33,360 or 6046, you're going to be designing in some cases 35 00:01:33,360 --> 00:01:36,990 probabilistic algorithms-- algorithms that flip coins 36 00:01:36,990 --> 00:01:39,910 or generate random numbers to get the result. 37 00:01:39,910 --> 00:01:43,410 And by using randomness, they run faster. 38 00:01:43,410 --> 00:01:45,800 If you're doing software engineering later, 39 00:01:45,800 --> 00:01:47,300 you're going to use probability when 40 00:01:47,300 --> 00:01:51,950 you analyze system performance or build hash tables. 41 00:01:51,950 --> 00:01:54,190 Probability is used in information theory, 42 00:01:54,190 --> 00:01:57,590 in coding theory, in cryptography. 43 00:01:57,590 --> 00:02:01,150 In fact, our nation's codes, their security 44 00:02:01,150 --> 00:02:04,150 depends on probabilistic analysis. 45 00:02:04,150 --> 00:02:06,580 Because the bad guys are using probabilistic 46 00:02:06,580 --> 00:02:09,199 algorithms to break the crypto-system. 47 00:02:09,199 --> 00:02:12,920 And we'll give an example of that in another week or so. 48 00:02:12,920 --> 00:02:14,820 Probability is obviously important 49 00:02:14,820 --> 00:02:18,240 if you're doing game theory or if you like to gamble, 50 00:02:18,240 --> 00:02:19,480 really critical in gambling. 51 00:02:19,480 --> 00:02:22,100 In fact, we're going to analyze a lot of different gambling 52 00:02:22,100 --> 00:02:23,350 games over the next few weeks. 53 00:02:23,350 --> 00:02:25,710 And for fun, we'll play some in class. 54 00:02:25,710 --> 00:02:28,170 In fact, we'll play one later today. 55 00:02:28,170 --> 00:02:32,010 And it's amazing how often the intuition is wrong. 56 00:02:32,010 --> 00:02:33,440 You think the answer goes one way, 57 00:02:33,440 --> 00:02:36,000 and then it's actually the other. 58 00:02:36,000 --> 00:02:39,160 Probability is important in the study of fault tolerance. 59 00:02:39,160 --> 00:02:41,090 For example, how do you figure out 60 00:02:41,090 --> 00:02:43,730 the probability there's going to be some critical failure that 61 00:02:43,730 --> 00:02:46,120 causes the space shuttle to crash? 62 00:02:46,120 --> 00:02:48,990 Or a critical failure that causes a nuclear plant 63 00:02:48,990 --> 00:02:50,350 to meltdown? 64 00:02:50,350 --> 00:02:52,250 That's done through probabilistic analysis. 65 00:02:52,250 --> 00:02:54,780 And we'll do some examples of that. 66 00:02:54,780 --> 00:02:57,370 Probability comes up in everyday life. 67 00:02:57,370 --> 00:02:59,112 We just had an election. 68 00:02:59,112 --> 00:03:00,570 And running up to the election, you 69 00:03:00,570 --> 00:03:03,780 get all these polls that say that's so-and-so 70 00:03:03,780 --> 00:03:08,060 is ahead of so-and-so by three points with a margin of error 71 00:03:08,060 --> 00:03:09,780 of five points. 72 00:03:09,780 --> 00:03:12,650 So it's a statistical dead heat, is what they say. 73 00:03:12,650 --> 00:03:14,930 Well, what the heck does that mean? 74 00:03:14,930 --> 00:03:16,990 In fact, usually they've got it screwed up. 75 00:03:16,990 --> 00:03:19,156 And we're going to spend the recitation where you're 76 00:03:19,156 --> 00:03:22,880 going to do a margin of error analysis for a poll, 77 00:03:22,880 --> 00:03:25,500 and see how many people do you need to poll to get 78 00:03:25,500 --> 00:03:28,490 a pretty accurate result? 79 00:03:28,490 --> 00:03:31,390 Comes up in the medical studies. 80 00:03:31,390 --> 00:03:34,400 Does cholesterol cause heart disease? 81 00:03:34,400 --> 00:03:37,090 Does smoking cause lung cancer? 82 00:03:37,090 --> 00:03:39,410 How effective is a certain drug? 83 00:03:39,410 --> 00:03:43,780 Say you get tested for a disease and it comes back positive. 84 00:03:43,780 --> 00:03:45,950 Well, there's false positives and false negatives. 85 00:03:45,950 --> 00:03:48,970 What are the chances you really have the disease? 86 00:03:48,970 --> 00:03:51,942 So the list goes on and on. 87 00:03:51,942 --> 00:03:53,900 In fact, we're going to see all these examples. 88 00:03:53,900 --> 00:03:55,910 We've got eight lectures and eight recitations. 89 00:03:55,910 --> 00:03:58,020 And there's just no end of interesting examples 90 00:03:58,020 --> 00:04:00,770 where probability comes into play. 91 00:04:00,770 --> 00:04:04,920 The bad news is, probability is probably 92 00:04:04,920 --> 00:04:09,400 the most misunderstood subject in all of mathematics 93 00:04:09,400 --> 00:04:11,370 and the sciences. 94 00:04:11,370 --> 00:04:15,750 In fact, Mark Twain once said, there's 95 00:04:15,750 --> 00:04:23,300 three kinds of lies-- lies, damned lies, and statistics, 96 00:04:23,300 --> 00:04:25,210 meaning probability. 97 00:04:25,210 --> 00:04:29,890 Because it's so easy to lie with statistics. 98 00:04:29,890 --> 00:04:33,750 And we're going to see lots and lots of examples of that. 99 00:04:33,750 --> 00:04:36,790 In fact, newspaper articles, you know, 100 00:04:36,790 --> 00:04:40,000 have all sorts of things that are just horribly fallacious 101 00:04:40,000 --> 00:04:41,032 based on statistics. 102 00:04:41,032 --> 00:04:41,990 And we'll see examples. 103 00:04:41,990 --> 00:04:46,690 In fact, we're going to even debunk a very famous article-- 104 00:04:46,690 --> 00:04:48,920 published research paper-- by some famous computer 105 00:04:48,920 --> 00:04:50,720 scientists at Berkeley. 106 00:04:50,720 --> 00:04:53,610 Where they got to a conclusion using a very standard method 107 00:04:53,610 --> 00:04:56,650 you see all the time-- completely fallacious. 108 00:04:56,650 --> 00:04:58,120 And we'll see that. 109 00:04:58,120 --> 00:05:00,280 So you can think of that as, I'm going to teach you 110 00:05:00,280 --> 00:05:03,080 how to lie with statistics. 111 00:05:03,080 --> 00:05:08,380 Even better, how not to be taken in by incorrect arguments using 112 00:05:08,380 --> 00:05:10,860 probability theory. 113 00:05:10,860 --> 00:05:15,390 Now, at MIT, it's probably the subject 114 00:05:15,390 --> 00:05:17,910 that most often trips up students 115 00:05:17,910 --> 00:05:20,760 in their PhD qualifying exams. 116 00:05:20,760 --> 00:05:23,450 You know, there's no better way to strike 117 00:05:23,450 --> 00:05:27,109 terror in the heart of a student then to be sitting there 118 00:05:27,109 --> 00:05:28,650 and ask them a question while they're 119 00:05:28,650 --> 00:05:32,480 at the board, what's the probability that foo happens? 120 00:05:32,480 --> 00:05:34,760 It's like panic sets in when they've got 121 00:05:34,760 --> 00:05:36,060 to do a probabilistic analysis. 122 00:05:36,060 --> 00:05:39,330 And now, MIT PhD students are good. 123 00:05:39,330 --> 00:05:42,260 You know, they can crank out the most horrendous calculation, 124 00:05:42,260 --> 00:05:45,880 no problem, come up with really clever ideas to solve problems. 125 00:05:45,880 --> 00:05:49,170 But you get into probability, it gets into trouble. 126 00:05:49,170 --> 00:05:51,070 In fact, even the other faculty sitting there 127 00:05:51,070 --> 00:05:55,050 start squirming around when this question's being posed. 128 00:05:55,050 --> 00:05:57,894 You know, and probability questions are-- 129 00:05:57,894 --> 00:05:59,810 and we'll see some examples today-- are always 130 00:05:59,810 --> 00:06:01,980 the best at faculty parties, just 131 00:06:01,980 --> 00:06:03,450 can ruin people's whole night. 132 00:06:03,450 --> 00:06:05,366 You know, what's the probability this happens? 133 00:06:05,366 --> 00:06:06,580 And it's some weird answer. 134 00:06:09,180 --> 00:06:11,130 Now, the strange thing about this-- well, 135 00:06:11,130 --> 00:06:13,820 first strange thing is that our human intuition is terrible. 136 00:06:13,820 --> 00:06:15,380 And you will see that. 137 00:06:15,380 --> 00:06:17,960 Your intuition will be wrong, just the way 138 00:06:17,960 --> 00:06:21,840 our brains are created genetically. 139 00:06:21,840 --> 00:06:25,880 But the good news is that probability can be very simple. 140 00:06:25,880 --> 00:06:29,880 If you force yourself to just go to the basic principles-- which 141 00:06:29,880 --> 00:06:32,920 we're going to cover-- and go through the basic template 142 00:06:32,920 --> 00:06:35,630 of how to solve a problem-- which we're going to cover-- 143 00:06:35,630 --> 00:06:39,370 it is easy, and you can't go wrong. 144 00:06:39,370 --> 00:06:43,660 The key is forcing yourself to just actually write it 145 00:06:43,660 --> 00:06:45,850 down, follow the basic principles, 146 00:06:45,850 --> 00:06:47,110 and get to the answer. 147 00:06:47,110 --> 00:06:51,969 As opposed to saying, oh, the answer's that and here's why. 148 00:06:51,969 --> 00:06:54,010 Because it's amazing how often you'll be wrong if 149 00:06:54,010 --> 00:06:57,620 you just try to rely on your intuition. 150 00:06:57,620 --> 00:07:00,730 Now, we're going to start today our study of probability 151 00:07:00,730 --> 00:07:04,760 with a famous, simple game. 152 00:07:04,760 --> 00:07:08,270 And the game is known as the Monty Hall problem. 153 00:07:08,270 --> 00:07:11,500 Now, do you all know who Monty Hall was? 154 00:07:11,500 --> 00:07:12,590 You're probably too young. 155 00:07:12,590 --> 00:07:14,850 Yeah, OK, not many people know who Monte Hall was. 156 00:07:14,850 --> 00:07:19,530 But back in the 1970s, there was a very popular daytime TV 157 00:07:19,530 --> 00:07:24,850 show-- a game show-- called Let's Make a Deal. 158 00:07:24,850 --> 00:07:26,090 And it was a great show. 159 00:07:26,090 --> 00:07:27,870 It was the precursor of Wheel of Fortune-- 160 00:07:27,870 --> 00:07:30,240 if you're old enough to have seen that-- and then 161 00:07:30,240 --> 00:07:31,900 Deal or No Deal after that. 162 00:07:31,900 --> 00:07:34,500 There's always some game show that's really popular. 163 00:07:34,500 --> 00:07:37,660 Anyway, back in the '70s, it was Let's Make a Deal, 164 00:07:37,660 --> 00:07:40,132 and Monty Hall was the host. 165 00:07:40,132 --> 00:07:42,340 How many people have ever heard of Let's Make a Deal? 166 00:07:42,340 --> 00:07:44,214 Raise your hand if you've ever heard of that. 167 00:07:44,214 --> 00:07:45,630 OK, so it still has some fame. 168 00:07:45,630 --> 00:07:46,460 Good. 169 00:07:46,460 --> 00:07:47,690 All right. 170 00:07:47,690 --> 00:07:50,970 Now, Monty Hall, of course, was the precursor 171 00:07:50,970 --> 00:07:54,740 of Pat Sajak of Wheel of Fortune and Howie Mandel, 172 00:07:54,740 --> 00:07:56,410 you know, from Deal or No Deal. 173 00:07:56,410 --> 00:07:59,070 And his assistant was Carol Merrill. 174 00:07:59,070 --> 00:08:02,620 She was also famous as the beautiful assistant. 175 00:08:02,620 --> 00:08:05,430 Now, in Let's Make a Deal, they TV audience 176 00:08:05,430 --> 00:08:08,890 would come dressed up in wild costumes. 177 00:08:08,890 --> 00:08:12,690 And Monty would go into the audience 178 00:08:12,690 --> 00:08:15,240 and pick out a contestant from the audience based 179 00:08:15,240 --> 00:08:17,620 on how wild their costume was-- you know, 180 00:08:17,620 --> 00:08:19,910 somebody that would look interesting for TV. 181 00:08:19,910 --> 00:08:22,440 And what he would do is he'd buy a piece of the costume-- 182 00:08:22,440 --> 00:08:25,290 maybe their hat or shirt or something-- 183 00:08:25,290 --> 00:08:27,960 and give them $100 for a piece of their costume. 184 00:08:27,960 --> 00:08:31,370 And then he would say, hey look, you know, 185 00:08:31,370 --> 00:08:34,340 if you give me back the $100 I just gave you, 186 00:08:34,340 --> 00:08:37,230 I'm going to let you have your choice of what's 187 00:08:37,230 --> 00:08:39,669 behind door number one, door number two, or door 188 00:08:39,669 --> 00:08:40,730 number three. 189 00:08:40,730 --> 00:08:43,320 Or sometimes it would be box number one, box number two, 190 00:08:43,320 --> 00:08:45,260 and box number three. 191 00:08:45,260 --> 00:08:48,560 Now, because people have seen the game show before, 192 00:08:48,560 --> 00:08:50,620 they knew that behind one of the doors 193 00:08:50,620 --> 00:08:54,380 was a brand new car or underneath one of the boxes 194 00:08:54,380 --> 00:08:56,520 was a diamond ring. 195 00:08:56,520 --> 00:09:00,520 And behind the other doors were goats or donkeys 196 00:09:00,520 --> 00:09:01,960 or something totally useless. 197 00:09:01,960 --> 00:09:04,010 So one of the three doors has a big prize, 198 00:09:04,010 --> 00:09:07,930 the other two got nothing. 199 00:09:07,930 --> 00:09:11,790 But because anybody would pay $100 for a one-in-three chance 200 00:09:11,790 --> 00:09:12,990 at the car, they played. 201 00:09:12,990 --> 00:09:14,820 And they're on national TV. 202 00:09:14,820 --> 00:09:18,930 So they give the $100 back to Monty, and they pick a door, 203 00:09:18,930 --> 00:09:21,830 say door number one. 204 00:09:21,830 --> 00:09:24,690 Monty wouldn't open door number one right away. 205 00:09:24,690 --> 00:09:28,110 What he would do is say, OK, you're sure you like that? 206 00:09:28,110 --> 00:09:29,792 All right. 207 00:09:29,792 --> 00:09:31,250 Carol, can you please reveal what's 208 00:09:31,250 --> 00:09:33,490 behind door number three? 209 00:09:33,490 --> 00:09:35,270 So Carol opens up door number three, 210 00:09:35,270 --> 00:09:37,150 and there's one of the donkeys sitting there 211 00:09:37,150 --> 00:09:38,920 or a goat sitting there. 212 00:09:38,920 --> 00:09:40,650 Well, now there's only two doors left. 213 00:09:40,650 --> 00:09:42,500 One of them has the car. 214 00:09:42,500 --> 00:09:46,070 So he'd ask the contestant, well, you've seen the goat. 215 00:09:46,070 --> 00:09:48,510 There's a car behind one of the two doors. 216 00:09:48,510 --> 00:09:50,620 Do you want to stick with door number one? 217 00:09:50,620 --> 00:09:54,260 Or do you want to switch and get door number two? 218 00:09:54,260 --> 00:09:56,960 And then the audience is screaming, you know, switch, 219 00:09:56,960 --> 00:09:59,570 stick, you know, door number one, door number two. 220 00:09:59,570 --> 00:10:01,320 You know, the guy's wife's there screaming 221 00:10:01,320 --> 00:10:02,840 at him and everything. 222 00:10:02,840 --> 00:10:07,300 You know, he's got to pick, so he picks one. 223 00:10:07,300 --> 00:10:12,620 And you know, if he picks the car, well, his wife's thrilled. 224 00:10:12,620 --> 00:10:13,760 They live happily ever. 225 00:10:13,760 --> 00:10:15,559 He's a hero on national TV. 226 00:10:15,559 --> 00:10:18,100 And if he picks the goat, you know, he's looking at a divorce 227 00:10:18,100 --> 00:10:20,560 and his career's ruined. 228 00:10:20,560 --> 00:10:24,250 So it's pretty important that he gets the right strategy here. 229 00:10:24,250 --> 00:10:27,380 Now, the question we're going to look at today 230 00:10:27,380 --> 00:10:29,670 is whether or not the contestant should 231 00:10:29,670 --> 00:10:33,050 switch, playing this game. 232 00:10:33,050 --> 00:10:35,720 Now, before we do the analysis, we're 233 00:10:35,720 --> 00:10:38,540 going to actually play the game in class. 234 00:10:38,540 --> 00:10:41,414 I've got three boxes here. 235 00:10:41,414 --> 00:10:43,330 The game we're going to play is the same idea. 236 00:10:43,330 --> 00:10:45,300 The prizes are a little different. 237 00:10:45,300 --> 00:10:47,470 Instead of a diamond ring or a brand new car, 238 00:10:47,470 --> 00:10:51,700 we've got $10 Toski's gift certificates. 239 00:10:51,700 --> 00:10:55,370 You know, your tuition only goes so far here. 240 00:10:55,370 --> 00:10:56,790 Instead of donkeys or goats, we've 241 00:10:56,790 --> 00:10:59,871 got something much better, the ever-popular nerd pride pocket 242 00:10:59,871 --> 00:11:00,370 protector. 243 00:11:03,470 --> 00:11:06,230 And just to be clear, this is not 244 00:11:06,230 --> 00:11:08,590 the prize, the gift certificate here-- 245 00:11:08,590 --> 00:11:10,559 this is much, much better. 246 00:11:10,559 --> 00:11:11,350 [AUDIENCE LAUGHTER] 247 00:11:11,350 --> 00:11:15,890 Now, in terms of Monty Hall, we don't have him, 248 00:11:15,890 --> 00:11:17,680 but you have me. 249 00:11:17,680 --> 00:11:20,580 But I am very happy to report that we're bringing Carol 250 00:11:20,580 --> 00:11:22,530 Merrill out of retirement. 251 00:11:22,530 --> 00:11:25,630 She's coming and making a special appearance today, 252 00:11:25,630 --> 00:11:28,680 I hope, at 6042. 253 00:11:28,680 --> 00:11:30,882 Do we have Carol? 254 00:11:30,882 --> 00:11:31,840 GUEST SPEAKER: Hi, Tom. 255 00:11:31,840 --> 00:11:34,140 PROFESSOR: Oh, yes, we have Carol. 256 00:11:34,140 --> 00:11:36,186 That's scary. 257 00:11:36,186 --> 00:11:37,530 [AUDIENCE LAUGHTER] 258 00:11:37,530 --> 00:11:38,085 Oh, Jesus. 259 00:11:41,160 --> 00:11:42,190 OK. 260 00:11:42,190 --> 00:11:45,340 [APPLAUSE] 261 00:11:45,340 --> 00:11:49,290 As you can see, the years have not been kind to Carol. 262 00:11:49,290 --> 00:11:52,720 And your tuition only does goes so far. 263 00:11:52,720 --> 00:11:55,840 Now, we need three volunteers. 264 00:11:55,840 --> 00:11:57,760 We're going to play the game three times. 265 00:11:57,760 --> 00:11:59,600 All right, got a couple there. 266 00:11:59,600 --> 00:12:00,690 Why don't you come down? 267 00:12:00,690 --> 00:12:03,180 We've got three volunteers here. 268 00:12:03,180 --> 00:12:08,490 All right, we'll get the boxes set up. 269 00:12:08,490 --> 00:12:10,090 And Carol, why don't you set those up 270 00:12:10,090 --> 00:12:14,660 and load them up with-- these are the gift certificates. 271 00:12:14,660 --> 00:12:16,645 That's the pocket protector there. 272 00:12:16,645 --> 00:12:17,770 All right, now, don't look. 273 00:12:17,770 --> 00:12:19,590 You've got to get back over there so you can't see-- 274 00:12:19,590 --> 00:12:20,550 [AUDIENCE LAUGHTER] 275 00:12:20,550 --> 00:12:23,730 And no, no, no, now, you've got the film going, you know, 276 00:12:23,730 --> 00:12:25,070 here we go. 277 00:12:25,070 --> 00:12:26,910 All right. 278 00:12:26,910 --> 00:12:30,810 So Carol is going to put the prize in one of the boxes 279 00:12:30,810 --> 00:12:35,210 and the pocket protectors in the other. 280 00:12:35,210 --> 00:12:37,560 Now, I want you all to be thinking, you know, 281 00:12:37,560 --> 00:12:39,270 what's the right strategy here? 282 00:12:39,270 --> 00:12:41,020 Is it better to switch? 283 00:12:41,020 --> 00:12:43,030 Does it matter? 284 00:12:43,030 --> 00:12:44,960 What are the chances of winning? 285 00:12:44,960 --> 00:12:45,840 One in three? 286 00:12:45,840 --> 00:12:47,570 One in two? 287 00:12:47,570 --> 00:12:51,930 All right, so who's our first volunteer here? 288 00:12:51,930 --> 00:12:52,950 All right, come on over. 289 00:12:52,950 --> 00:12:54,741 We're going to write your name on the board 290 00:12:54,741 --> 00:12:57,590 because we're going to keep track of what happens here. 291 00:12:57,590 --> 00:12:58,879 What's your name? 292 00:12:58,879 --> 00:12:59,670 TERRANCE: Terrance. 293 00:12:59,670 --> 00:13:00,503 PROFESSOR: Terrance. 294 00:13:02,570 --> 00:13:03,530 All right. 295 00:13:03,530 --> 00:13:05,960 Now, first thing you've got to do, 296 00:13:05,960 --> 00:13:11,980 Terrance, is pick a box as your basic choice to start. 297 00:13:11,980 --> 00:13:12,610 TERRANCE: Two. 298 00:13:12,610 --> 00:13:16,260 PROFESSOR: Number two, OK. 299 00:13:16,260 --> 00:13:20,106 All right, you picked number two. 300 00:13:20,106 --> 00:13:23,070 Hm, interesting choice. 301 00:13:23,070 --> 00:13:26,695 Now, Carol, can you show him one of the other boxes? 302 00:13:34,344 --> 00:13:36,260 All right, which on was that you showed there? 303 00:13:36,260 --> 00:13:37,820 You revealed number three. 304 00:13:40,990 --> 00:13:43,832 And what was under number three? 305 00:13:43,832 --> 00:13:45,275 AUDIENCE: Pocket protector. 306 00:13:45,275 --> 00:13:46,718 PROFESSOR: Pocket protector. 307 00:13:46,718 --> 00:13:50,185 Oh, God, you may never live this down, Carol. 308 00:13:50,185 --> 00:13:50,685 OK. 309 00:13:53,820 --> 00:13:56,740 Now, Terrance, as you know, one of these 310 00:13:56,740 --> 00:13:59,280 has the gift certificate. 311 00:13:59,280 --> 00:14:02,300 One of them has another pocket protector. 312 00:14:02,300 --> 00:14:04,860 And now, you've got to decide, are you 313 00:14:04,860 --> 00:14:08,470 going to stick with number two, or are you going to switch? 314 00:14:08,470 --> 00:14:10,239 In fact, class, you know, Terrance 315 00:14:10,239 --> 00:14:12,030 may need some help here, figuring this out, 316 00:14:12,030 --> 00:14:13,420 so what do you recommend? 317 00:14:13,420 --> 00:14:19,180 [AUDIENCE CHATTER] 318 00:14:19,180 --> 00:14:20,630 TERRANCE: I'll switch. 319 00:14:20,630 --> 00:14:22,960 PROFESSOR: He is going to switch. 320 00:14:22,960 --> 00:14:25,865 All right, and Carol, can you reveal his prize? 321 00:14:29,670 --> 00:14:30,710 What did you get? 322 00:14:30,710 --> 00:14:33,000 Ah, he gets the prize. 323 00:14:33,000 --> 00:14:35,320 So he wins. 324 00:14:35,320 --> 00:14:36,650 Very nice, OK. 325 00:14:36,650 --> 00:14:38,200 So you get $10 at Tosci's. 326 00:14:38,200 --> 00:14:39,570 Well done. 327 00:14:39,570 --> 00:14:40,960 Who's next? 328 00:14:40,960 --> 00:14:41,905 What's your name? 329 00:14:41,905 --> 00:14:42,530 CHINUA: Chinua. 330 00:14:42,530 --> 00:14:43,280 PROFESSOR: Chinua. 331 00:14:43,280 --> 00:14:44,578 How do I spell that? 332 00:14:44,578 --> 00:14:46,375 CHINUA: Chin-U-A. 333 00:14:46,375 --> 00:14:47,500 PROFESSOR: All right, easy. 334 00:14:47,500 --> 00:14:48,460 OK. 335 00:14:48,460 --> 00:14:51,660 Now, Chinua, which-- have we got new prizes loaded in there? 336 00:14:51,660 --> 00:14:52,690 All right. 337 00:14:52,690 --> 00:14:55,230 Now, it may not be the same as last time. 338 00:14:55,230 --> 00:14:55,730 OK. 339 00:14:55,730 --> 00:14:56,830 What box do you like? 340 00:14:56,830 --> 00:14:58,060 CHINUA: Number three. 341 00:14:58,060 --> 00:14:59,337 PROFESSOR: Number three. 342 00:14:59,337 --> 00:15:00,420 Which one is number three? 343 00:15:00,420 --> 00:15:01,620 Is that the one close to me? 344 00:15:01,620 --> 00:15:02,240 Yeah, that is. 345 00:15:02,240 --> 00:15:02,740 OK. 346 00:15:02,740 --> 00:15:03,744 Number three. 347 00:15:03,744 --> 00:15:06,160 All right, Carol, can you show him one of the other boxes, 348 00:15:06,160 --> 00:15:06,660 please? 349 00:15:13,550 --> 00:15:16,820 Carol shows you number two. 350 00:15:16,820 --> 00:15:19,110 All right, Chinua, what do you think? 351 00:15:19,110 --> 00:15:22,780 Do you want to stick or switch? 352 00:15:22,780 --> 00:15:25,240 Do you want to stick with three or switch to one? 353 00:15:25,240 --> 00:15:28,203 One had it last time, right? 354 00:15:28,203 --> 00:15:32,395 [AUDIENCE CHATTER] 355 00:15:32,395 --> 00:15:33,020 CHINUA: Switch. 356 00:15:33,020 --> 00:15:35,246 PROFESSOR: He's going to switch. 357 00:15:35,246 --> 00:15:38,700 OK, so that means you are picking number one. 358 00:15:38,700 --> 00:15:42,130 Carol, can you show him what he's won? 359 00:15:42,130 --> 00:15:45,416 Another gift certificate. 360 00:15:45,416 --> 00:15:47,919 All right, now, Carol it is OK to use a different box 361 00:15:47,919 --> 00:15:48,960 for the gift certificate. 362 00:15:48,960 --> 00:15:49,800 [LAUGHTER] 363 00:15:49,800 --> 00:15:52,170 We can move them around. 364 00:15:52,170 --> 00:15:53,970 All right, who's our last contestant? 365 00:15:53,970 --> 00:15:55,100 What's your name? 366 00:15:55,100 --> 00:15:55,600 TY: Ty. 367 00:15:55,600 --> 00:15:56,736 PROFESSOR: T-Y, OK. 368 00:15:56,736 --> 00:15:59,220 That's good. 369 00:15:59,220 --> 00:16:00,970 Oh, yeah, yeah, turn around. 370 00:16:00,970 --> 00:16:01,770 No peeking here. 371 00:16:06,600 --> 00:16:07,250 OK. 372 00:16:07,250 --> 00:16:10,350 Ty, what's your first choice? 373 00:16:10,350 --> 00:16:10,850 TY: One. 374 00:16:10,850 --> 00:16:11,945 PROFESSOR: Box number one. 375 00:16:11,945 --> 00:16:15,500 Yeah, that's been the one with the gift certificate, for sure. 376 00:16:15,500 --> 00:16:20,500 Carol, can you show him another box? 377 00:16:20,500 --> 00:16:22,860 Way too many cameras in this room. 378 00:16:22,860 --> 00:16:24,982 This is not good for Carol. 379 00:16:24,982 --> 00:16:26,670 [LAUGHTER] 380 00:16:26,670 --> 00:16:30,830 OK, Carol has revealed box number two. 381 00:16:30,830 --> 00:16:31,630 [AUDIENCE CHATTER] 382 00:16:31,630 --> 00:16:33,822 Switch has worked twice in a row. 383 00:16:33,822 --> 00:16:35,030 AUDIENCE: Now you gotta stay. 384 00:16:35,030 --> 00:16:36,817 AUDIENCE: Take them both. 385 00:16:36,817 --> 00:16:37,650 PROFESSOR: Just one. 386 00:16:41,355 --> 00:16:43,688 AUDIENCE: Well, if you switch, there's two-thirds chance 387 00:16:43,688 --> 00:16:46,288 of winning, and they already won so-- 388 00:16:46,288 --> 00:16:47,720 [LAUGHTER] 389 00:16:47,720 --> 00:16:50,660 PROFESSOR: Oh, getting some theory here now. 390 00:16:50,660 --> 00:16:53,210 What do you think, Ty? 391 00:16:53,210 --> 00:16:54,520 Switch. 392 00:16:54,520 --> 00:16:55,112 All right. 393 00:16:55,112 --> 00:16:55,695 What happened? 394 00:16:58,360 --> 00:17:00,142 What do you know? 395 00:17:00,142 --> 00:17:01,600 All right, now, gentlemen, I'd like 396 00:17:01,600 --> 00:17:06,515 you to explain why-- each in turn, why did you switch? 397 00:17:06,515 --> 00:17:08,990 TERRANCE: I switched-- it seemed like the majority vote. 398 00:17:08,990 --> 00:17:10,365 PROFESSOR: The majority knew what 399 00:17:10,365 --> 00:17:13,940 they were talking about, OK. 400 00:17:13,940 --> 00:17:15,425 You have faith in these guys, huh? 401 00:17:15,425 --> 00:17:16,910 All right. 402 00:17:16,910 --> 00:17:18,976 How about you, Chinua, why did you switch? 403 00:17:18,976 --> 00:17:20,840 CHINUA: Um, I was told that it's more 404 00:17:20,840 --> 00:17:22,660 likely to be if you switch. 405 00:17:22,660 --> 00:17:24,576 PROFESSOR: And you believe what you hear, huh? 406 00:17:24,576 --> 00:17:26,007 [LAUGHTER] 407 00:17:28,392 --> 00:17:31,254 TY: I don't know who said it, but it sounded good. 408 00:17:31,254 --> 00:17:32,535 PROFESSOR: Sounded good Yeah, that two-thirds thing. 409 00:17:32,535 --> 00:17:34,160 Well, as you can see, you're guaranteed 410 00:17:34,160 --> 00:17:35,787 to win if you switch, right? 411 00:17:35,787 --> 00:17:36,286 [LAUGHTER] 412 00:17:36,286 --> 00:17:37,830 All right, Thank you very much. 413 00:17:37,830 --> 00:17:39,860 Enjoy the ice cream. 414 00:17:39,860 --> 00:17:40,460 OK. 415 00:17:40,460 --> 00:17:45,340 So how many people think it helps to switch? 416 00:17:45,340 --> 00:17:47,690 Raise your hand if you think it helps to switch. 417 00:17:47,690 --> 00:17:50,400 How many people think it does not help to switch? 418 00:17:50,400 --> 00:17:51,960 Raise your hand. 419 00:17:51,960 --> 00:17:52,530 Anybody? 420 00:17:52,530 --> 00:17:55,860 Oh, everybody likes the switch thing here. 421 00:17:55,860 --> 00:17:59,450 Does this data help you decide? 422 00:17:59,450 --> 00:17:59,960 No. 423 00:17:59,960 --> 00:18:01,918 Anything could have happened with three trials. 424 00:18:01,918 --> 00:18:03,895 In fact, we'll talk about statistical inference 425 00:18:03,895 --> 00:18:05,610 and statistical sampling later. 426 00:18:08,130 --> 00:18:10,783 If you switch, what do you think the odds of winning are? 427 00:18:10,783 --> 00:18:11,650 AUDIENCE: 100% 428 00:18:11,650 --> 00:18:12,490 PROFESSOR: 100%? 429 00:18:12,490 --> 00:18:14,630 OK, who likes 100%? 430 00:18:14,630 --> 00:18:15,810 Raise your hand. 431 00:18:15,810 --> 00:18:17,410 Well, we've got a couple. 432 00:18:17,410 --> 00:18:20,940 How many people think it's 50-50 if you switch? 433 00:18:20,940 --> 00:18:21,810 Yeah, a lot. 434 00:18:21,810 --> 00:18:26,250 How many people think it's worse than 50-50 if you switch? 435 00:18:26,250 --> 00:18:26,990 Not many. 436 00:18:26,990 --> 00:18:29,426 How many people think it's better than 50-50? 437 00:18:29,426 --> 00:18:33,250 Ah, that's I think most of you there, pretty close. 438 00:18:33,250 --> 00:18:36,090 All right, now, the answers to these questions 439 00:18:36,090 --> 00:18:38,320 are very easy to work out-- and we're going to do it. 440 00:18:38,320 --> 00:18:40,700 We're going to figure out the probability exactly-- 441 00:18:40,700 --> 00:18:42,960 if you use mathematics. 442 00:18:42,960 --> 00:18:44,500 But it's really tricky if you try 443 00:18:44,500 --> 00:18:48,490 to use intuition or hand waving arguments. 444 00:18:48,490 --> 00:18:51,610 Now, for example, this game-- Carol 445 00:18:51,610 --> 00:18:54,220 wants to leave-- yeah, Carol, let's give Carol a hand here. 446 00:18:54,220 --> 00:18:56,790 She did a wonderful job. 447 00:18:56,790 --> 00:18:57,526 Well done. 448 00:18:57,526 --> 00:18:58,804 [APPLAUSE] 449 00:19:00,020 --> 00:19:02,249 Yikes. 450 00:19:02,249 --> 00:19:04,040 I'm going to get in trouble here, too, now. 451 00:19:04,040 --> 00:19:05,590 You know? 452 00:19:05,590 --> 00:19:07,770 OK. 453 00:19:07,770 --> 00:19:11,710 For example, this game, and this particular version of it, 454 00:19:11,710 --> 00:19:16,050 was the subject of a series of articles in Parade Magazine. 455 00:19:16,050 --> 00:19:19,500 This is an old version of Parade Magazine. 456 00:19:19,500 --> 00:19:22,260 And what I'm going to do is hand out the series of articles. 457 00:19:22,260 --> 00:19:26,130 And the TAs are going to help me hand these out. 458 00:19:26,130 --> 00:19:28,580 There was three of them with a lot of write-ins here. 459 00:19:28,580 --> 00:19:32,724 I'm going to give you a bundle here. 460 00:19:32,724 --> 00:19:34,859 There you go. 461 00:19:34,859 --> 00:19:35,400 There you go. 462 00:19:35,400 --> 00:19:36,707 I'll give the rest to Oscar. 463 00:19:39,960 --> 00:19:40,460 Great. 464 00:19:54,100 --> 00:19:59,030 Now, Marilyn has a column every week in Parade. 465 00:19:59,030 --> 00:20:01,200 And people write in questions, usually 466 00:20:01,200 --> 00:20:04,700 about their love life and weird things. 467 00:20:04,700 --> 00:20:06,350 And she answers them. 468 00:20:06,350 --> 00:20:08,210 And she's well qualified to do this 469 00:20:08,210 --> 00:20:10,890 because she's listed in the Guinness Book of World Records 470 00:20:10,890 --> 00:20:16,350 as having the highest IQ of a human being ever, at 228. 471 00:20:16,350 --> 00:20:18,100 And so, you know she's in some sense 472 00:20:18,100 --> 00:20:21,600 qualified to answer any question that is written in. 473 00:20:21,600 --> 00:20:26,390 Now, in September of 1990-- and we're passing out the articles 474 00:20:26,390 --> 00:20:32,880 now and it shows the first article-- 475 00:20:32,880 --> 00:20:36,700 this guy named Craig Whitaker writes in and asks 476 00:20:36,700 --> 00:20:40,469 Marilyn if you should switch in this game. 477 00:20:40,469 --> 00:20:41,510 So you can all read that. 478 00:20:41,510 --> 00:20:43,830 It says, suppose you're on a game show and you're given 479 00:20:43,830 --> 00:20:45,500 the choice of three doors. 480 00:20:45,500 --> 00:20:47,120 Behind one door is a car. 481 00:20:47,120 --> 00:20:49,180 Behind the others, goats. 482 00:20:49,180 --> 00:20:52,470 You pick a door, say number one, and the host, 483 00:20:52,470 --> 00:20:54,120 who knows what's behind the doors, 484 00:20:54,120 --> 00:20:58,910 opens another one, say door three, which has a goat. 485 00:20:58,910 --> 00:21:02,480 He says to you, do you want to pick door number two? 486 00:21:02,480 --> 00:21:05,055 Is it to your advantage to switch? 487 00:21:07,720 --> 00:21:10,670 Now, Marilyn writes back saying, very plainly, 488 00:21:10,670 --> 00:21:12,070 yes, you should switch. 489 00:21:12,070 --> 00:21:14,880 The first door had a one-in-three chance of winning. 490 00:21:14,880 --> 00:21:18,530 The second door has a two-in-three chance. 491 00:21:18,530 --> 00:21:21,320 Here's a good way to visualize what's going on. 492 00:21:21,320 --> 00:21:24,540 Suppose there are a million doors. 493 00:21:24,540 --> 00:21:26,780 You pick door number one. 494 00:21:26,780 --> 00:21:29,480 Then the host, who knows what's behind the doors, 495 00:21:29,480 --> 00:21:36,710 reveals every other door but one and 777,777. 496 00:21:36,710 --> 00:21:39,900 Well, it would sort of look like 777,777 497 00:21:39,900 --> 00:21:43,205 is the one with the car, and you'd switch pretty fast. 498 00:21:43,205 --> 00:21:44,620 All right that's her argument. 499 00:21:44,620 --> 00:21:47,990 That is not exactly a proof, OK, that the answer 500 00:21:47,990 --> 00:21:52,450 is you should switch or that the probability is 2/3. 501 00:21:52,450 --> 00:21:53,270 All right? 502 00:21:53,270 --> 00:21:55,140 Now, because she's giving another problem 503 00:21:55,140 --> 00:21:58,070 with a million doors, but you've got a problem with three doors. 504 00:21:58,070 --> 00:22:01,530 And as we'll see in recitation tomorrow, the number of doors 505 00:22:01,530 --> 00:22:03,980 actually matters in terms of some strategies here. 506 00:22:06,540 --> 00:22:09,760 So people weren't convinced by this. 507 00:22:09,760 --> 00:22:11,510 And so, if you look to the next page, 508 00:22:11,510 --> 00:22:12,930 you see people writing in. 509 00:22:12,930 --> 00:22:15,340 And there's a lot a write-ins. 510 00:22:15,340 --> 00:22:18,830 The first one's by Robert Sachs, who's actually-- 511 00:22:18,830 --> 00:22:21,360 I looked him up on Google, and he's 512 00:22:21,360 --> 00:22:25,450 on the faculty at George Mason University. 513 00:22:25,450 --> 00:22:27,390 I also noticed that in his history, 514 00:22:27,390 --> 00:22:29,400 he got his BA from Harvard. 515 00:22:29,400 --> 00:22:32,130 So that should tell you something about the quality 516 00:22:32,130 --> 00:22:34,400 of his response here. 517 00:22:34,400 --> 00:22:38,330 He says, since you seem to enjoy coming straight to the point, 518 00:22:38,330 --> 00:22:39,560 I'll do the same. 519 00:22:39,560 --> 00:22:40,890 You blew it. 520 00:22:40,890 --> 00:22:42,170 Let me explain. 521 00:22:42,170 --> 00:22:44,120 If one door is shown to be a loser, 522 00:22:44,120 --> 00:22:47,120 that information changes the probability 523 00:22:47,120 --> 00:22:49,940 of either remaining choice, neither of which 524 00:22:49,940 --> 00:22:53,210 has any reason to be more likely. 525 00:22:53,210 --> 00:22:56,340 And so, the answer is one half of having the car. 526 00:22:56,340 --> 00:22:58,830 Then he says, as a professional mathematician, 527 00:22:58,830 --> 00:23:00,830 I'm very concerned with the general public's 528 00:23:00,830 --> 00:23:02,760 lack of mathematical skills. 529 00:23:02,760 --> 00:23:04,545 Please help by confessing your error. 530 00:23:04,545 --> 00:23:07,090 And in the future, be more careful. 531 00:23:07,090 --> 00:23:10,810 Just a little gratuitous extra statement there. 532 00:23:10,810 --> 00:23:14,270 Next we've got a PhD from Florida, Scott Smith, who says, 533 00:23:14,270 --> 00:23:16,600 you blew it, and you blew it big. 534 00:23:16,600 --> 00:23:18,210 Since you seem to have difficulty 535 00:23:18,210 --> 00:23:22,350 grasping the basic principles at work here, I'll explain. 536 00:23:22,350 --> 00:23:24,440 After the host reveals a goat, you now 537 00:23:24,440 --> 00:23:26,890 have a one-in-two chance of being correct. 538 00:23:26,890 --> 00:23:29,530 Whether you change your selection or not, 539 00:23:29,530 --> 00:23:31,150 the odds are the same. 540 00:23:31,150 --> 00:23:34,140 There is enough mathematical illiteracy in this country, 541 00:23:34,140 --> 00:23:37,560 and we don't need the world's highest IQ propagating more. 542 00:23:37,560 --> 00:23:39,199 Shame. 543 00:23:39,199 --> 00:23:41,240 Well, now, this guy is right about that one point 544 00:23:41,240 --> 00:23:44,370 about mathematical illiteracy in the country. 545 00:23:44,370 --> 00:23:47,290 The next guy, Barry Pasternack, you know, 546 00:23:47,290 --> 00:23:50,360 he lists himself as California Faculty Association. 547 00:23:50,360 --> 00:23:55,050 He actually was president of the California Faculty Association. 548 00:23:55,050 --> 00:23:57,870 He says, your answer to the question is an error, 549 00:23:57,870 --> 00:24:01,120 but if it's any constellation, many of my academic colleagues 550 00:24:01,120 --> 00:24:04,080 have also been stumped by this problem. 551 00:24:04,080 --> 00:24:04,580 OK. 552 00:24:04,580 --> 00:24:07,360 So next page, Marilyn writes back. 553 00:24:07,360 --> 00:24:09,510 And she says, you know, good heavens, 554 00:24:09,510 --> 00:24:11,572 with so much learned opposition, I'll 555 00:24:11,572 --> 00:24:14,030 bet this is going to keep math classes all over the country 556 00:24:14,030 --> 00:24:14,960 busy on Monday. 557 00:24:14,960 --> 00:24:15,760 And it did. 558 00:24:15,760 --> 00:24:18,910 It was a huge uproar at the time about this. 559 00:24:18,910 --> 00:24:22,230 So now she tries to explain it some 560 00:24:22,230 --> 00:24:26,430 more with shells and stuff under the shells and so forth, 561 00:24:26,430 --> 00:24:29,040 trying to argue it. 562 00:24:29,040 --> 00:24:32,750 And then she says, just try the experiment six times 563 00:24:32,750 --> 00:24:35,116 or something, and you'll see. 564 00:24:35,116 --> 00:24:38,300 You know, but, it's not really convincing. 565 00:24:38,300 --> 00:24:39,640 It's not a proof. 566 00:24:39,640 --> 00:24:42,340 She's sort of waving hands at it to give you some idea of why 567 00:24:42,340 --> 00:24:43,686 she thinks she's right. 568 00:24:43,686 --> 00:24:45,310 But when I look at the hand waving here 569 00:24:45,310 --> 00:24:48,080 versus the hand waving on those faculty, 570 00:24:48,080 --> 00:24:50,460 well, it's all hand waving. 571 00:24:50,460 --> 00:24:50,960 All right. 572 00:24:50,960 --> 00:24:53,740 So you go to the next page, and now the letters 573 00:24:53,740 --> 00:24:55,910 are getting a little nastier. 574 00:24:55,910 --> 00:24:58,670 We got another University of Florida guy. 575 00:24:58,670 --> 00:25:01,450 He says, may I suggest you obtain and refer 576 00:25:01,450 --> 00:25:03,860 to a standard textbook on probability 577 00:25:03,860 --> 00:25:07,720 before you try to answer a question of this type again. 578 00:25:07,720 --> 00:25:10,380 Then Robert Smith, Georgia State, 579 00:25:10,380 --> 00:25:12,280 I am sure you will receive many letters 580 00:25:12,280 --> 00:25:15,780 on this topic from high school and college students. 581 00:25:15,780 --> 00:25:17,970 Perhaps you should keep a few addresses for help 582 00:25:17,970 --> 00:25:20,380 with future columns. 583 00:25:20,380 --> 00:25:22,190 These are great. 584 00:25:22,190 --> 00:25:26,525 Now this next guy, Ray Bobo, now, he's a real guy. 585 00:25:26,525 --> 00:25:27,400 I looked him up, too. 586 00:25:27,400 --> 00:25:28,150 It's not fake. 587 00:25:28,150 --> 00:25:29,820 That's his real name. 588 00:25:29,820 --> 00:25:32,190 And I've got to say, if my name was Bobo, 589 00:25:32,190 --> 00:25:35,630 I'd think twice before writing into Parade Magazine, 590 00:25:35,630 --> 00:25:39,070 you know, especially if I'm wrong. 591 00:25:39,070 --> 00:25:41,080 Anyway, here's Bobo. 592 00:25:41,080 --> 00:25:44,260 You are utterly incorrect about the game show question. 593 00:25:44,260 --> 00:25:47,170 And I hope this controversy will call some public attention 594 00:25:47,170 --> 00:25:50,415 to the serious national crisis in mathematical education. 595 00:25:50,415 --> 00:25:52,394 Well, that's fair. 596 00:25:52,394 --> 00:25:53,810 If you could admit your error, you 597 00:25:53,810 --> 00:25:55,310 will have contributed constructively 598 00:25:55,310 --> 00:25:58,091 towards a solution of a deplorable situation. 599 00:25:58,091 --> 00:25:59,090 Now, here's [INAUDIBLE]. 600 00:25:59,090 --> 00:26:01,940 How many irate mathematicians are needed 601 00:26:01,940 --> 00:26:04,380 to get you to change your mind? 602 00:26:04,380 --> 00:26:07,794 Well that's proof by intimidation, right? 603 00:26:07,794 --> 00:26:08,600 [LAUGHTER] 604 00:26:08,600 --> 00:26:11,420 You know, it's not, here's the reason why. 605 00:26:11,420 --> 00:26:13,490 It's just, so many irate mathematicians said 606 00:26:13,490 --> 00:26:16,510 it, they must, must be right. 607 00:26:16,510 --> 00:26:17,550 Now the next guy. 608 00:26:17,550 --> 00:26:19,340 I am in shock that after being corrected 609 00:26:19,340 --> 00:26:21,110 by at least three mathematicians, 610 00:26:21,110 --> 00:26:23,300 you still did not see your mistake. 611 00:26:23,300 --> 00:26:24,880 As if three mathematicians, that's 612 00:26:24,880 --> 00:26:28,340 sort of the criteria for correctness. 613 00:26:28,340 --> 00:26:31,190 Oh, this guy is bad. 614 00:26:31,190 --> 00:26:34,970 Maybe women look at math problems differently than men. 615 00:26:34,970 --> 00:26:36,550 Ooh. 616 00:26:36,550 --> 00:26:39,220 Well, maybe it's a good thing in this case. 617 00:26:39,220 --> 00:26:41,740 I don't know. 618 00:26:41,740 --> 00:26:46,690 Oh, then we have Glen Calkins, you are the goat. 619 00:26:46,690 --> 00:26:50,190 Finally, this guy, US Army Research Institute, 620 00:26:50,190 --> 00:26:51,550 a little scary. 621 00:26:51,550 --> 00:26:54,230 you made a mistake, but look at the positive side. 622 00:26:54,230 --> 00:26:56,840 If all those PhD's were wrong, the country 623 00:26:56,840 --> 00:26:58,620 would be in some very serious trouble. 624 00:27:01,481 --> 00:27:01,980 All right. 625 00:27:01,980 --> 00:27:05,620 Now Marilyn writes an even longer response 626 00:27:05,620 --> 00:27:08,970 because she's getting zillions of letters, 90% voting 627 00:27:08,970 --> 00:27:11,660 against her, that she's wrong. 628 00:27:11,660 --> 00:27:15,130 She begins now, suggests a nationwide experiment, 629 00:27:15,130 --> 00:27:18,490 so that, nationwide, you'd do the sample a million times 630 00:27:18,490 --> 00:27:21,414 and get the answer, which really wouldn't prove it either. 631 00:27:21,414 --> 00:27:22,580 Then we go to the last page. 632 00:27:22,580 --> 00:27:23,630 There's one last letter. 633 00:27:23,630 --> 00:27:27,420 And this is what finished the controversy in the press. 634 00:27:27,420 --> 00:27:29,350 Comes from MIT. 635 00:27:29,350 --> 00:27:30,590 You are indeed correct. 636 00:27:30,590 --> 00:27:33,110 My colleagues at work had a ball with this problem, 637 00:27:33,110 --> 00:27:36,280 and I dare say that most of them, including me at first, 638 00:27:36,280 --> 00:27:38,480 thought you were wrong. 639 00:27:38,480 --> 00:27:41,870 So maybe that $50,000 a year somebody's paying for education 640 00:27:41,870 --> 00:27:47,260 is worth something, because MIT came in and agreed with her. 641 00:27:47,260 --> 00:27:48,490 OK. 642 00:27:48,490 --> 00:27:52,650 It turns out that Marilyn was correct in her statement 643 00:27:52,650 --> 00:27:54,180 that you should switch. 644 00:27:54,180 --> 00:27:57,630 And if you switch, you have a 2/3 chance of winning. 645 00:27:57,630 --> 00:28:01,050 So if you don't switch, you've got a 1/3 chance of winning, 646 00:28:01,050 --> 00:28:03,750 provided that Monty is guaranteed 647 00:28:03,750 --> 00:28:05,285 to open a door with goat. 648 00:28:05,285 --> 00:28:07,780 You know and that's the assumption here. 649 00:28:07,780 --> 00:28:10,720 And the proof is simple. 650 00:28:10,720 --> 00:28:14,270 Although Marilyn's reasoning wasn't so convincing, 651 00:28:14,270 --> 00:28:15,590 I didn't think. 652 00:28:15,590 --> 00:28:18,500 And all those PhDs that wrote in with those stupid letters, 653 00:28:18,500 --> 00:28:20,220 they're probably intelligent people. 654 00:28:20,220 --> 00:28:22,480 After all, they got advanced degrees in mathematics, 655 00:28:22,480 --> 00:28:25,270 and most of them were faculty teaching mathematics. 656 00:28:25,270 --> 00:28:26,880 A little scary. 657 00:28:26,880 --> 00:28:30,640 But they just weren't following the basic principles. 658 00:28:30,640 --> 00:28:32,600 They were following their intuition. 659 00:28:32,600 --> 00:28:34,220 They didn't go through the basic steps 660 00:28:34,220 --> 00:28:37,046 to figure out the probability. 661 00:28:37,046 --> 00:28:38,420 So that's what we're going to do. 662 00:28:38,420 --> 00:28:39,602 We're going to go through the basic steps 663 00:28:39,602 --> 00:28:41,140 now and see how to solve it. 664 00:28:41,140 --> 00:28:43,210 And we will use these steps to solve pretty much 665 00:28:43,210 --> 00:28:45,320 every problem in probability. 666 00:28:45,320 --> 00:28:46,840 OK? 667 00:28:46,840 --> 00:28:52,860 So the key, the first step, is to look at the sample 668 00:28:52,860 --> 00:28:56,512 space of possible outcomes. 669 00:28:56,512 --> 00:28:58,710 Let me define sample space and outcome. 670 00:29:03,080 --> 00:29:13,550 The sample space for an experiment, or a probability 671 00:29:13,550 --> 00:29:22,000 game-- you can treat this Monty Hall game as an experiment-- 672 00:29:22,000 --> 00:29:28,990 is simply the set of all possible outcomes, 673 00:29:28,990 --> 00:29:30,790 things that could happen. 674 00:29:33,750 --> 00:29:36,660 In fact, let me define an outcome, 675 00:29:36,660 --> 00:29:40,040 also known as an atomic event or a sample point. 676 00:29:43,580 --> 00:29:50,790 An outcome, also known as a sample 677 00:29:50,790 --> 00:29:59,470 point-- we'll use those terms interchangeably-- 678 00:29:59,470 --> 00:30:21,320 consists of all the information about the experiment, 679 00:30:21,320 --> 00:30:24,160 including-- and that's after it's 680 00:30:24,160 --> 00:30:42,940 been performed-- including the values of all random choices. 681 00:30:55,780 --> 00:31:02,330 For example, say we want to know the probability of winning 682 00:31:02,330 --> 00:31:05,486 in Monty Hall if you switch. 683 00:31:05,486 --> 00:31:07,780 All right, let's figure that out. 684 00:31:07,780 --> 00:31:13,230 So in that case, we're going to define an outcome of the Monty 685 00:31:13,230 --> 00:31:30,740 Hall game or a sample point-- and I'm assuming in this case, 686 00:31:30,740 --> 00:31:47,434 we know you're going to switch and the contestant switches-- 687 00:31:47,434 --> 00:31:48,475 consists of three things. 688 00:31:52,710 --> 00:32:00,780 First, the box with the prize, where did Carroll put it? 689 00:32:00,780 --> 00:32:03,735 Second, the box chosen by the contestant. 690 00:32:10,420 --> 00:32:12,960 And third, the box that was revealed. 691 00:32:18,160 --> 00:32:19,720 Once you know those three things, 692 00:32:19,720 --> 00:32:23,150 you know everything about what happened. 693 00:32:23,150 --> 00:32:24,880 All right? 694 00:32:24,880 --> 00:32:27,450 For example, let's look at a typical sample point. 695 00:32:30,330 --> 00:32:40,330 Say I took sample point, or outcome, 2,1,3. 696 00:32:40,330 --> 00:32:53,490 This is the outcome where the prize is the box 2. 697 00:32:58,290 --> 00:33:09,130 The player picks first box 1. 698 00:33:09,130 --> 00:33:16,150 And Carol reveals box 3. 699 00:33:19,600 --> 00:33:22,140 OK, now in this case, in this outcome, 700 00:33:22,140 --> 00:33:23,205 did the contestant win? 701 00:33:28,570 --> 00:33:30,700 Yes, because we're assuming-- for henceforth, 702 00:33:30,700 --> 00:33:31,550 for most of the rest of the day, we're 703 00:33:31,550 --> 00:33:33,230 assuming the switch case, because we're 704 00:33:33,230 --> 00:33:34,360 going to analyze that. 705 00:33:34,360 --> 00:33:35,860 So that's fixed, they switch. 706 00:33:35,860 --> 00:33:38,470 They come in knowing they're going to switch. 707 00:33:38,470 --> 00:33:40,900 So the prize is in 2, all right? 708 00:33:40,900 --> 00:33:41,650 The prize is here. 709 00:33:44,590 --> 00:33:46,560 The player started here. 710 00:33:46,560 --> 00:33:48,770 Carol revealed 3. 711 00:33:48,770 --> 00:33:52,370 And when they switched, they got the prize. 712 00:33:52,370 --> 00:33:56,310 So in this outcome, the player won. 713 00:33:56,310 --> 00:33:57,950 OK. 714 00:33:57,950 --> 00:34:02,570 Now, does every 3-tuple here correspond to a sample point, 715 00:34:02,570 --> 00:34:05,601 to our possible outcome? 716 00:34:05,601 --> 00:34:06,100 No. 717 00:34:06,100 --> 00:34:08,909 What's an example of a sample point with numbers 1, 2, 718 00:34:08,909 --> 00:34:09,980 and 3 that's not. 719 00:34:12,489 --> 00:34:12,989 Yeah? 720 00:34:16,239 --> 00:34:17,199 Very good. 721 00:34:17,199 --> 00:34:27,010 Yeah, so for example, 1, 2, 1 is not a sample point, 722 00:34:27,010 --> 00:34:31,500 is not in the set, the sample space, 723 00:34:31,500 --> 00:34:37,310 because we can't reveal the box with the prize. 724 00:34:37,310 --> 00:34:42,060 Similarly, 2, 1, 1 is not a sample point, 725 00:34:42,060 --> 00:34:46,810 because we can't reveal the box that the player picked. 726 00:34:46,810 --> 00:34:47,351 Yeah? 727 00:34:47,351 --> 00:34:49,976 AUDIENCE : I think the first two pieces of data actually define 728 00:34:49,976 --> 00:34:54,732 the third one, because once, or-- actually, no, 729 00:34:54,732 --> 00:34:57,210 if they're the same. 730 00:34:57,210 --> 00:34:59,490 PROFESSOR: Right, so if the player 731 00:34:59,490 --> 00:35:02,321 picked the box with the prize, which 732 00:35:02,321 --> 00:35:04,110 never happened in our case. 733 00:35:04,110 --> 00:35:12,460 For example, 1, 1 2 and 1, 1, 3 are both possible 734 00:35:12,460 --> 00:35:16,230 because you could put the prize here, the player could pick it, 735 00:35:16,230 --> 00:35:19,960 now Carol has a choice of revealing 2 or 3. 736 00:35:19,960 --> 00:35:22,000 OK? 737 00:35:22,000 --> 00:35:22,790 So these are OK. 738 00:35:28,810 --> 00:35:34,030 All right, so let's figure out a nice way of characterizing 739 00:35:34,030 --> 00:35:37,400 all outcomes in the sample space. 740 00:35:37,400 --> 00:35:40,920 Now, to do this, we're going to set up a tree. 741 00:35:40,920 --> 00:35:42,980 And we're going to use a thing called the tree 742 00:35:42,980 --> 00:35:54,679 method to construct the sample space, 743 00:35:54,679 --> 00:35:55,720 getting all the outcomes. 744 00:36:07,860 --> 00:36:08,360 All right. 745 00:36:08,360 --> 00:36:12,300 So the first thing we do is figure out where the prize is. 746 00:36:17,060 --> 00:36:19,980 And we'll represent that as sort of like a decision tree. 747 00:36:19,980 --> 00:36:22,060 There's three choices. 748 00:36:22,060 --> 00:36:26,320 It's in box 1, box 2, or box 3. 749 00:36:26,320 --> 00:36:29,971 Then the next thing is the box that the player picked. 750 00:36:34,210 --> 00:36:36,640 And we'll continue our decision tree here. 751 00:36:36,640 --> 00:36:39,200 Each of these now has three branches. 752 00:36:43,910 --> 00:36:45,778 And they're labeled 1, 2, or 3. 753 00:36:50,520 --> 00:36:55,030 And then, the last step is which box is revealed. 754 00:37:00,190 --> 00:37:03,620 Now, in this case, if the prize is in one 755 00:37:03,620 --> 00:37:08,680 and the player chose one, there's two possibilities. 756 00:37:08,680 --> 00:37:12,090 We could reveal 2 or reveal 3. 757 00:37:12,090 --> 00:37:18,830 So we get 1, 2, 3 here and 1, 2, 3 here. 758 00:37:18,830 --> 00:37:20,880 What about here? 759 00:37:20,880 --> 00:37:26,490 Prize is in box 1, box 2 is revealed-- sorry, 760 00:37:26,490 --> 00:37:27,730 player picked box 2. 761 00:37:27,730 --> 00:37:31,850 Prize is in 1, player picks 2, how many choices for which 762 00:37:31,850 --> 00:37:34,940 box is revealed? 763 00:37:34,940 --> 00:37:35,660 One. 764 00:37:35,660 --> 00:37:39,390 And which box is going to be revealed? 765 00:37:39,390 --> 00:37:41,870 3. 766 00:37:41,870 --> 00:37:44,900 All right, this is sample point 1, 2, 3. 767 00:37:44,900 --> 00:37:50,120 And here we've got 1, 3, 2 is the only choice. 768 00:37:53,150 --> 00:37:54,660 All right. 769 00:37:54,660 --> 00:37:56,625 And let's do-- maybe I'll, we ought 770 00:37:56,625 --> 00:37:58,140 to fill the whole thing out here. 771 00:37:58,140 --> 00:38:08,060 2, 1, 3-- I'm going to run out of space, but-- 2, 2, now this 772 00:38:08,060 --> 00:38:13,110 splits into 1 and 3. 773 00:38:13,110 --> 00:38:13,790 All right? 774 00:38:13,790 --> 00:38:23,120 So we have here 2, 2, 1, 2, 2, 3. 775 00:38:23,120 --> 00:38:26,420 Now let me do the last one, 2, 3, 1 here. 776 00:38:29,390 --> 00:38:33,070 And now I'll bring these up over here. 777 00:38:33,070 --> 00:38:36,135 I've got 3, 1, 2. 778 00:38:41,880 --> 00:38:44,770 Then I've got 3, 2, 1. 779 00:38:48,210 --> 00:38:53,780 And finally, I've got 3, 3 with two choices, 1 and 2. 780 00:38:59,650 --> 00:39:00,150 All right? 781 00:39:00,150 --> 00:39:04,670 That is now the construction of the entire sample space, 782 00:39:04,670 --> 00:39:07,100 all the possible outcomes. 783 00:39:07,100 --> 00:39:08,850 How many sample points are there here? 784 00:39:08,850 --> 00:39:10,030 How many possible outcomes? 785 00:39:14,000 --> 00:39:14,690 AUDIENCE : 12 786 00:39:14,690 --> 00:39:16,190 PROFESSOR: 12. 787 00:39:16,190 --> 00:39:22,100 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, OK. 788 00:39:22,100 --> 00:39:26,570 Let's see which ones correspond to victory for the player. 789 00:39:26,570 --> 00:39:28,740 Let's label each one as a win for the player 790 00:39:28,740 --> 00:39:31,150 or a loss for the player. 791 00:39:31,150 --> 00:39:34,150 What's this one? 792 00:39:34,150 --> 00:39:34,650 Yeah. 793 00:39:34,650 --> 00:39:37,040 Prize is in 1, the player chose 1 794 00:39:37,040 --> 00:39:40,690 where the prize was, and then switched away from it. 795 00:39:40,690 --> 00:39:41,300 Loss. 796 00:39:41,300 --> 00:39:43,650 What about this one? 797 00:39:43,650 --> 00:39:45,140 Loss, same reason. 798 00:39:45,140 --> 00:39:48,180 What about this one? 799 00:39:48,180 --> 00:39:48,760 That's a win. 800 00:39:48,760 --> 00:39:51,560 The prize is here, the player went there and switched, 801 00:39:51,560 --> 00:39:53,520 so they wound up there. 802 00:39:53,520 --> 00:39:55,390 Win. 803 00:39:55,390 --> 00:39:58,050 And these are all wins for the same reason. 804 00:39:58,050 --> 00:40:01,280 Whoop, not this one. 805 00:40:01,280 --> 00:40:04,080 2, 1, 1, the player got it right to start with, 806 00:40:04,080 --> 00:40:05,170 and then switched away. 807 00:40:05,170 --> 00:40:06,830 That's a loss. 808 00:40:06,830 --> 00:40:08,470 Loss. 809 00:40:08,470 --> 00:40:12,320 Win, win, win. 810 00:40:12,320 --> 00:40:14,955 And here the player got it right, they switched away, 811 00:40:14,955 --> 00:40:16,350 it's a lose. 812 00:40:16,350 --> 00:40:18,680 All right, so we've labeled every outcome 813 00:40:18,680 --> 00:40:23,420 as to whether the player who's switching wins or loses. 814 00:40:23,420 --> 00:40:27,940 How many sample points lead to victory? 815 00:40:27,940 --> 00:40:29,780 6. 816 00:40:29,780 --> 00:40:34,590 1, 2, 3, 4, 5, 6 out of 12. 817 00:40:34,590 --> 00:40:36,420 Uh-oh. 818 00:40:36,420 --> 00:40:37,290 Was Marilyn wrong? 819 00:40:37,290 --> 00:40:38,590 Is it really Bobo was right? 820 00:40:38,590 --> 00:40:40,890 It's 50-50? 821 00:40:40,890 --> 00:40:44,814 Half the sample points lead to victory. 822 00:40:44,814 --> 00:40:46,910 Hm, what's missing here? 823 00:40:46,910 --> 00:40:47,932 Yeah? 824 00:40:47,932 --> 00:40:50,360 AUDIENCE : The first two are identical, 825 00:40:50,360 --> 00:40:52,360 the box with the prize and the box chosen first. 826 00:40:52,360 --> 00:40:55,804 Each of those two sample points coming from that 827 00:40:55,804 --> 00:40:59,876 are half as likely as the single sample points. 828 00:40:59,876 --> 00:41:00,876 PROFESSOR: That's right. 829 00:41:00,876 --> 00:41:04,060 What we're missing here is the likelihood 830 00:41:04,060 --> 00:41:08,160 of an outcome, the probability that this is what happens. 831 00:41:08,160 --> 00:41:09,846 We're missing one more thing. 832 00:41:09,846 --> 00:41:11,470 And this is really critical when you're 833 00:41:11,470 --> 00:41:13,630 making your sample spaces. 834 00:41:13,630 --> 00:41:16,550 You need to make sure you get the probabilities assigned 835 00:41:16,550 --> 00:41:17,180 to each event. 836 00:41:17,180 --> 00:41:21,650 They're not necessarily equally likely. 837 00:41:21,650 --> 00:41:25,060 In fact, what we need to do is to construct the probability 838 00:41:25,060 --> 00:41:26,400 space. 839 00:41:26,400 --> 00:41:27,440 So lets define that. 840 00:41:33,870 --> 00:41:44,300 Your probability space consists of a sample space-- which 841 00:41:44,300 --> 00:41:46,670 we've already talked about. 842 00:41:46,670 --> 00:42:02,140 That's just all these outcomes-- and a probability function 843 00:42:02,140 --> 00:42:06,460 that maps-- and we call this function, usually, PR. 844 00:42:06,460 --> 00:42:11,500 We'll have, sometimes it'll be f or something like that-- 845 00:42:11,500 --> 00:42:21,325 but PR maps the sample space, S, to the real numbers 846 00:42:21,325 --> 00:42:30,220 such that first, for every outcome in the sample space, 847 00:42:30,220 --> 00:42:37,590 the probability is between 0 and 1 of that outcome. 848 00:42:37,590 --> 00:42:40,570 And 2, if you sum up the probabilities 849 00:42:40,570 --> 00:42:42,870 of all the outcomes, you get 1. 850 00:42:42,870 --> 00:42:50,100 So the sum over all outcomes of the probability of the outcome 851 00:42:50,100 --> 00:42:52,541 is 1. 852 00:42:52,541 --> 00:42:53,040 All right? 853 00:42:53,040 --> 00:42:55,750 So the sample space is all the outcomes. 854 00:42:55,750 --> 00:42:58,850 The probability space adds in-- has this extra thing-- 855 00:42:58,850 --> 00:43:01,410 the probability of each outcome. 856 00:43:01,410 --> 00:43:02,880 And really, it could be anything, 857 00:43:02,880 --> 00:43:05,290 as long is the probability of every outcome 858 00:43:05,290 --> 00:43:09,140 is between 0 and 1 and they all add up to 1. 859 00:43:09,140 --> 00:43:09,710 That's it. 860 00:43:12,190 --> 00:43:12,690 All right. 861 00:43:12,690 --> 00:43:15,550 Now, there's a natural interpretation 862 00:43:15,550 --> 00:43:18,090 of what the probability function is. 863 00:43:18,090 --> 00:43:20,812 And that is the probability of a sample point, 864 00:43:20,812 --> 00:43:22,770 well, should be the probability of its outcome. 865 00:43:28,967 --> 00:43:31,300 And this is sort of where you get the to Marilyn saying, 866 00:43:31,300 --> 00:43:34,000 hey, try it a million times and you'll see 2/3 of the time, 867 00:43:34,000 --> 00:43:38,600 roughly, you get the right, that you win. 868 00:43:38,600 --> 00:43:42,990 So for every outcome, the probability 869 00:43:42,990 --> 00:43:46,020 of that outcome, that real number, 870 00:43:46,020 --> 00:43:54,000 should be the probability that when you run the experiment, 871 00:43:54,000 --> 00:43:55,045 w is what happens. 872 00:43:59,270 --> 00:44:00,580 w will be the outcome. 873 00:44:07,090 --> 00:44:08,690 In other words, if probability of w 874 00:44:08,690 --> 00:44:12,620 were a half, what that means is you 875 00:44:12,620 --> 00:44:15,670 should get that half the time, roughly speaking. 876 00:44:15,670 --> 00:44:17,915 We'll get into really details of that later. 877 00:44:17,915 --> 00:44:20,810 In other words, you toss a fair coin, you flip it. 878 00:44:20,810 --> 00:44:23,260 Well, you might say, a probability half of a heads. 879 00:44:23,260 --> 00:44:23,760 All right? 880 00:44:23,760 --> 00:44:28,620 And so, this function, then, the probability of heads is 1/2. 881 00:44:28,620 --> 00:44:32,540 Now, assigning the right probabilities to the outcomes 882 00:44:32,540 --> 00:44:36,029 is 95% of the work. 883 00:44:36,029 --> 00:44:37,570 In fact, if you screw that up, you're 884 00:44:37,570 --> 00:44:40,517 doomed because you can't get the right answer. 885 00:44:40,517 --> 00:44:42,600 So for example, if we just said, OK, each of these 886 00:44:42,600 --> 00:44:46,571 is probability 1/12, we'd get Bobo's answer 887 00:44:46,571 --> 00:44:47,820 as the probability of winning. 888 00:44:47,820 --> 00:44:51,009 So that's not too good. 889 00:44:51,009 --> 00:44:53,050 Now, to get the right probabilities for the Monty 890 00:44:53,050 --> 00:44:56,000 Hall problem, we have to make some assumptions. 891 00:44:58,690 --> 00:45:00,254 So let's write those down. 892 00:45:24,140 --> 00:45:26,710 So the first assumption we're going to make 893 00:45:26,710 --> 00:45:30,170 is that Carol put the prize in a random box. 894 00:45:30,170 --> 00:45:33,740 They're all equally likely. 895 00:45:33,740 --> 00:45:44,470 So the prize is in each box with probability 1/3. 896 00:45:47,040 --> 00:45:49,380 The next assumption we're going to make 897 00:45:49,380 --> 00:45:54,200 is that the player didn't see which box has the prize. 898 00:45:54,200 --> 00:45:54,700 OK? 899 00:45:54,700 --> 00:45:58,110 In other words, and more specifically, 900 00:45:58,110 --> 00:46:08,350 no matter where the prize is, the player 901 00:46:08,350 --> 00:46:11,380 picks each box with probability of 1/3. 902 00:46:11,380 --> 00:46:13,735 So the player's making, effectively, a random choice. 903 00:46:27,449 --> 00:46:30,482 All right, and there's one more assumption we're going to make. 904 00:46:30,482 --> 00:46:31,940 Anybody have any ideas about what's 905 00:46:31,940 --> 00:46:34,786 left to make as an assumption here? 906 00:46:34,786 --> 00:46:35,286 Yeah? 907 00:46:37,724 --> 00:46:39,890 AUDIENCE : I guess, if Carol's the one who's opening 908 00:46:39,890 --> 00:46:42,290 the boxes, she knows where all the prizes are. 909 00:46:42,290 --> 00:46:43,110 PROFESSOR: Yeah. 910 00:46:43,110 --> 00:46:43,740 OK. 911 00:46:43,740 --> 00:46:46,240 So what assumption should I make? 912 00:46:50,980 --> 00:46:55,077 Yeah, that she picks a box at random if she has a choice. 913 00:46:55,077 --> 00:46:57,160 All right, so we're going to make that assumption. 914 00:46:57,160 --> 00:47:04,220 So that no matter where the prize 915 00:47:04,220 --> 00:47:19,740 is, if Carol has a choice, she picks 916 00:47:19,740 --> 00:47:21,614 each box with probability 1/2. 917 00:47:35,960 --> 00:47:38,430 Now, why is this third one important? 918 00:47:38,430 --> 00:47:41,115 Why might that really matter? 919 00:47:41,115 --> 00:47:43,490 Can anybody think of a reason why that could be relevant? 920 00:47:46,358 --> 00:47:50,182 What if Carol had some strategy in mind 921 00:47:50,182 --> 00:47:52,572 that the contestant knew about? 922 00:47:52,572 --> 00:47:54,896 Can you think of why that might be useful. 923 00:47:54,896 --> 00:47:57,794 AUDIENCE : If the contestant were able to produce 924 00:47:57,794 --> 00:48:00,853 information based off of that, then there's a higher chance 925 00:48:00,853 --> 00:48:03,107 a contestant could figure out which one to choose. 926 00:48:03,107 --> 00:48:04,073 PROFESSOR: Yeah. 927 00:48:04,073 --> 00:48:09,010 In fact, several years ago, when I was lecturing, 928 00:48:09,010 --> 00:48:11,770 I said, in fact it could make a difference. 929 00:48:11,770 --> 00:48:14,220 That there is a way that Carol can convey information 930 00:48:14,220 --> 00:48:17,930 to the player by of two she's choosing, which one to pick. 931 00:48:17,930 --> 00:48:20,150 Like, if she has a choice of 2 or 3, 932 00:48:20,150 --> 00:48:22,790 say she always picks bigger one, 3. 933 00:48:22,790 --> 00:48:24,920 Well, that might tell you information about, 934 00:48:24,920 --> 00:48:27,370 she didn't pick 2, so maybe it has to have, 935 00:48:27,370 --> 00:48:30,460 or she-- well, I don't know. 936 00:48:30,460 --> 00:48:32,970 In fact, that's wrong, and it doesn't 937 00:48:32,970 --> 00:48:35,239 matter in the case of 2 boxes, this assumption. 938 00:48:35,239 --> 00:48:37,030 We're going to make it, because we're going 939 00:48:37,030 --> 00:48:38,900 to do the analysis with it. 940 00:48:38,900 --> 00:48:41,640 But for three boxes, you can show that it really doesn't 941 00:48:41,640 --> 00:48:43,650 matter for this assumption. 942 00:48:43,650 --> 00:48:45,820 But if you go to four boxes, it does. 943 00:48:45,820 --> 00:48:47,510 And so in recitation tomorrow, you're 944 00:48:47,510 --> 00:48:50,250 going to consider the four box version of this game. 945 00:48:50,250 --> 00:48:53,810 And if Carol has a strategy that the player knows about, 946 00:48:53,810 --> 00:48:57,480 you could improve the player's chances of winning. 947 00:48:57,480 --> 00:48:59,730 But not for three, it turns out. 948 00:48:59,730 --> 00:49:02,690 In any case, when you're setting up a problem, 949 00:49:02,690 --> 00:49:04,940 you've got to set up your assumptions. 950 00:49:04,940 --> 00:49:07,530 Otherwise, you don't get anywhere. 951 00:49:07,530 --> 00:49:10,900 These are sort of like axioms, same kind of thing. 952 00:49:10,900 --> 00:49:12,150 It's fine to make assumptions. 953 00:49:12,150 --> 00:49:15,304 Just state what they are. 954 00:49:15,304 --> 00:49:17,470 Now, of course, you also want them to be reasonable. 955 00:49:23,300 --> 00:49:23,800 All right. 956 00:49:23,800 --> 00:49:27,530 So maybe we're going to go back now and figure out 957 00:49:27,530 --> 00:49:31,100 how to get the probabilities of the sample 958 00:49:31,100 --> 00:49:34,690 points, or the outcomes, for Monty Hall. 959 00:49:34,690 --> 00:49:36,762 Let's go do that. 960 00:49:36,762 --> 00:49:38,886 And maybe I'll draw them in a different color here. 961 00:49:43,280 --> 00:49:45,790 What I'm going to do is label every edge 962 00:49:45,790 --> 00:49:50,990 with a probability associated with taking that step. 963 00:49:50,990 --> 00:49:55,260 So if the prize is placed in each box with a probability 964 00:49:55,260 --> 00:50:00,400 of 1/3, that says this happens with probability 1/3, that's 965 00:50:00,400 --> 00:50:02,370 1/3, and that's 1/3. 966 00:50:05,000 --> 00:50:06,570 Same thing here. 967 00:50:06,570 --> 00:50:09,410 Assumption two says that no matter where 968 00:50:09,410 --> 00:50:14,820 the prize is, say it's in box 1, the player chooses each box 969 00:50:14,820 --> 00:50:16,680 with a probably of 1/3. 970 00:50:16,680 --> 00:50:19,110 So given that the prize is in box 1, 971 00:50:19,110 --> 00:50:24,330 the player chooses 1 with 1/3, 2 with 1/3, 3 with 1/3. 972 00:50:24,330 --> 00:50:27,120 Same if the prize is in box 2. 973 00:50:27,120 --> 00:50:29,120 So I label every edge here with the probability. 974 00:50:31,780 --> 00:50:34,800 All right. 975 00:50:34,800 --> 00:50:36,850 Now I go to this level. 976 00:50:36,850 --> 00:50:40,248 And I'm, at this node, I'm in the state where prize 977 00:50:40,248 --> 00:50:43,960 is in box 1, player selected box 1, 978 00:50:43,960 --> 00:50:48,420 so Carol has a choice of 2 or 3 to reveal. 979 00:50:48,420 --> 00:50:53,110 And she's going to reveal each one with probability 1/2. 980 00:50:53,110 --> 00:50:55,250 So I label 1/2 on each of these, because that's 981 00:50:55,250 --> 00:50:57,980 what we've assumed. 982 00:50:57,980 --> 00:51:00,370 What probability do I put on this edge here? 983 00:51:03,280 --> 00:51:05,570 Probability 1, because there's no choice. 984 00:51:05,570 --> 00:51:07,020 Carol has to do it. 985 00:51:07,020 --> 00:51:08,430 The same here. 986 00:51:08,430 --> 00:51:17,520 And I get 1/2, 1/2, 1, 1, 1, 1/2, 1/2. 987 00:51:17,520 --> 00:51:21,670 All right, so I've labeled every edge with its probability. 988 00:51:21,670 --> 00:51:24,380 Pretty simple, because it was all assumed. 989 00:51:24,380 --> 00:51:27,196 Now I've got to get a probability for each outcome. 990 00:51:27,196 --> 00:51:28,820 And we do that with the following rule. 991 00:51:36,480 --> 00:51:49,680 The probability of a sample point 992 00:51:49,680 --> 00:52:08,210 is the product of the probabilities 993 00:52:08,210 --> 00:52:10,250 on the path leading to the outcome, 994 00:52:10,250 --> 00:52:24,675 the path in the tree leading to the outcome or the sample 995 00:52:24,675 --> 00:52:25,175 point. 996 00:52:32,360 --> 00:52:32,970 OK. 997 00:52:32,970 --> 00:52:38,080 So what this says is just really, really simple. 998 00:52:38,080 --> 00:52:43,410 The probability of this sample point is 1/3 times 1/3 times 999 00:52:43,410 --> 00:52:46,620 1/2, 1/18. 1000 00:52:46,620 --> 00:52:52,470 The probability of this sample point is 1/3 times 1/3 times 1001 00:52:52,470 --> 00:52:55,489 1/2, 1/18. 1002 00:52:55,489 --> 00:52:58,030 What's the probability of this sample point going to be here? 1003 00:53:00,570 --> 00:53:01,130 1/9. 1004 00:53:01,130 --> 00:53:06,200 I've got 1/3 times 1/3 times, well, this g was 1, is 1/9. 1005 00:53:10,030 --> 00:53:16,630 The next one here is 1/3 times 1/3 times 1, 1/9. 1006 00:53:16,630 --> 00:53:18,310 Same thing here. 1007 00:53:18,310 --> 00:53:22,720 1/3 times 1/3 times 1 is 1/9. 1008 00:53:22,720 --> 00:53:27,920 Then I get 1/3 times 1/3 times 1/2 for 1/18. 1009 00:53:27,920 --> 00:53:30,940 And I can keep on going. 1010 00:53:30,940 --> 00:53:32,970 This would be 1/9. 1011 00:53:32,970 --> 00:53:35,400 This one has a 1, so it's a 1/9. 1012 00:53:38,380 --> 00:53:42,116 An 1/18 and 1/18. 1013 00:53:45,100 --> 00:53:45,660 OK. 1014 00:53:45,660 --> 00:53:46,840 Now, that's the rule. 1015 00:53:46,840 --> 00:53:48,900 And you'll always do this. 1016 00:53:48,900 --> 00:53:51,680 You can sort of see why it works. 1017 00:53:51,680 --> 00:53:54,170 You'll understand after another week why, 1018 00:53:54,170 --> 00:53:55,740 for sure, why it works. 1019 00:53:55,740 --> 00:54:00,080 But intuitively, you know, to get to this outcome, 1020 00:54:00,080 --> 00:54:02,170 these are the choices we had to make. 1021 00:54:02,170 --> 00:54:04,820 First, the box had to be-- sorry, 1022 00:54:04,820 --> 00:54:06,760 the prize had to be in box 1. 1023 00:54:06,760 --> 00:54:09,220 There's a one-in-three chance of that happening. 1024 00:54:09,220 --> 00:54:13,190 Now, in those one-out-of-three chances, 1025 00:54:13,190 --> 00:54:16,250 the chance the player also picked box 1, 1026 00:54:16,250 --> 00:54:19,990 1/3 of the time of 1/3 of the time that happens. 1027 00:54:19,990 --> 00:54:20,950 All right? 1028 00:54:20,950 --> 00:54:22,360 1/3 of the time, you're here. 1029 00:54:22,360 --> 00:54:26,200 And of that, 1/3 of those times, you get to here. 1030 00:54:26,200 --> 00:54:28,094 That's 1/3 times 1/3, or 1/9. 1031 00:54:28,094 --> 00:54:32,590 And then, 1/2 of those times, Carol happen to reveal box 2. 1032 00:54:32,590 --> 00:54:35,260 That's 1/18. 1033 00:54:35,260 --> 00:54:37,760 So do you see why that might make sense, 1034 00:54:37,760 --> 00:54:39,580 why you're multiplying these? 1035 00:54:39,580 --> 00:54:42,400 Because at every step, you have a chance to diverge. 1036 00:54:42,400 --> 00:54:45,150 And the probability you take the step 1037 00:54:45,150 --> 00:54:47,850 that we're following here-- say, in this case, 1038 00:54:47,850 --> 00:54:51,670 it was 1/2-- that cuts down the chance you land here 1039 00:54:51,670 --> 00:54:55,320 by that probability, 1/2. 1040 00:54:55,320 --> 00:54:58,770 Any questions about that? 1041 00:54:58,770 --> 00:54:59,270 All right. 1042 00:54:59,270 --> 00:55:00,686 For now, you can take it on faith. 1043 00:55:00,686 --> 00:55:04,159 We'll see why after we do a few more of these 1044 00:55:04,159 --> 00:55:05,200 over the next week or so. 1045 00:55:07,760 --> 00:55:09,280 Any questions there? 1046 00:55:09,280 --> 00:55:12,100 Is that OK? 1047 00:55:12,100 --> 00:55:12,600 OK. 1048 00:55:12,600 --> 00:55:14,487 Oh, yeah, all right? 1049 00:55:14,487 --> 00:55:15,425 AUDIENCE : [INAUDIBLE] 1050 00:55:19,180 --> 00:55:20,880 PROFESSOR: On this one? 1051 00:55:20,880 --> 00:55:22,160 Only one choice. 1052 00:55:22,160 --> 00:55:24,061 So it happens with probability 1. 1053 00:55:27,067 --> 00:55:27,900 Any other questions? 1054 00:55:31,459 --> 00:55:32,860 OK. 1055 00:55:32,860 --> 00:55:33,970 All right. 1056 00:55:33,970 --> 00:55:38,780 So now we can figure out the probability of winning. 1057 00:55:38,780 --> 00:55:42,790 Well, the probability of winning is we have 1/9, 1/9, 1058 00:55:42,790 --> 00:55:46,801 there's 3 of them, 4 of them, 5, 6. 1059 00:55:46,801 --> 00:55:47,300 All right? 1060 00:55:47,300 --> 00:55:52,131 So the probability of winning here is 6 times 1/9. 1061 00:56:16,190 --> 00:56:18,730 Because you just sum up the probabilities of the winning 1062 00:56:18,730 --> 00:56:20,240 sample points. 1063 00:56:20,240 --> 00:56:21,310 There's six of winning. 1064 00:56:21,310 --> 00:56:23,230 They're each 1/9. 1065 00:56:23,230 --> 00:56:26,250 And that's 2/3. 1066 00:56:26,250 --> 00:56:26,750 OK? 1067 00:56:26,750 --> 00:56:29,110 So Marilyn was right. 1068 00:56:29,110 --> 00:56:31,660 There's a 2/3 chance of winning if you switch. 1069 00:56:31,660 --> 00:56:33,900 And now, this is a formal argument 1070 00:56:33,900 --> 00:56:35,630 that gives you that answer. 1071 00:56:35,630 --> 00:56:37,980 The nice thing is it works for pretty much every problem 1072 00:56:37,980 --> 00:56:39,880 you ever think about. 1073 00:56:39,880 --> 00:56:41,430 Now, in general, if you want to know 1074 00:56:41,430 --> 00:56:44,050 the probability of any event, you 1075 00:56:44,050 --> 00:56:45,540 can follow this same method. 1076 00:56:45,540 --> 00:56:47,180 Let me define what an event is. 1077 00:56:53,085 --> 00:56:58,880 An event is simply a subset of outcomes, 1078 00:56:58,880 --> 00:57:00,136 a subset of the sample space. 1079 00:57:10,040 --> 00:57:14,530 For example, we could define the event of losing. 1080 00:57:14,530 --> 00:57:20,840 Let's call EL, E sub-L, to be the event that player loses, 1081 00:57:20,840 --> 00:57:26,950 the player loses, with the switch strategy. 1082 00:57:35,390 --> 00:57:35,890 All right? 1083 00:57:35,890 --> 00:57:38,110 Now, to compute the probability of an event, 1084 00:57:38,110 --> 00:57:44,110 we just add the probabilities of the outcomes in that set. 1085 00:57:44,110 --> 00:57:47,445 So let's get that down. 1086 00:58:05,840 --> 00:58:19,577 All right, so formally, the probability 1087 00:58:19,577 --> 00:58:36,805 that an event occurs is simply the sum, we'll call an event, 1088 00:58:36,805 --> 00:58:39,590 e, let's say, event, e. 1089 00:58:39,590 --> 00:58:41,380 It's a subset of the sample space. 1090 00:58:41,380 --> 00:58:44,710 The probability that e occurs is the sum 1091 00:58:44,710 --> 00:58:49,920 over the sample points in e of the probability of those sample 1092 00:58:49,920 --> 00:58:51,650 points. 1093 00:58:51,650 --> 00:58:56,280 So for example, the probability of the event, EL, 1094 00:58:56,280 --> 00:59:00,850 where the player loses, well, there's 1095 00:59:00,850 --> 00:59:03,129 six sample points where the player loses, 1096 00:59:03,129 --> 00:59:04,420 each with the probability 1/18. 1097 00:59:07,760 --> 00:59:12,040 So it's 1/3 is the probability the player loses on the switch 1098 00:59:12,040 --> 00:59:13,140 strategy. 1099 00:59:13,140 --> 00:59:16,250 And that's not surprising because you either win 1100 00:59:16,250 --> 00:59:16,880 or you lose. 1101 00:59:16,880 --> 00:59:18,970 The probabilities have to add to 1. 1102 00:59:18,970 --> 00:59:22,820 So if won with probability 2/3, you lose with probability 1/3. 1103 00:59:25,570 --> 00:59:27,690 OK? 1104 00:59:27,690 --> 00:59:28,190 All right. 1105 00:59:28,190 --> 00:59:31,860 So Marilyn was right, much better to switch. 1106 00:59:31,860 --> 00:59:34,810 We'll do a bunch more examples with Monty Hall tomorrow. 1107 00:59:34,810 --> 00:59:39,630 Are there any other questions on Monty Hall before I change 1108 00:59:39,630 --> 00:59:40,590 gears? 1109 00:59:40,590 --> 00:59:42,572 Yeah? 1110 00:59:42,572 --> 00:59:44,480 AUDIENCE: So you showed that when you switch, 1111 00:59:44,480 --> 00:59:47,705 you are more likely to win than to lose. 1112 00:59:47,705 --> 00:59:49,580 But you didn't show that that's not also true 1113 00:59:49,580 --> 00:59:51,020 when you don't switch. 1114 00:59:51,020 --> 00:59:52,409 PROFESSOR: Ah good question. 1115 00:59:52,409 --> 00:59:54,450 What happens if you're playing the stay strategy. 1116 00:59:58,900 --> 01:00:03,060 Right, maybe you win 2/3 of the time when you stay. 1117 01:00:03,060 --> 01:00:04,190 Maybe. 1118 01:00:04,190 --> 01:00:05,590 Maybe not. 1119 01:00:05,590 --> 01:00:06,090 Right? 1120 01:00:06,090 --> 01:00:12,994 Because if I win when I switch, what 1121 01:00:12,994 --> 01:00:14,410 would have happened if I'd stayed? 1122 01:00:17,680 --> 01:00:19,570 I'd have lost. 1123 01:00:19,570 --> 01:00:21,080 Because if I win when I switch, it 1124 01:00:21,080 --> 01:00:22,380 means the prize is over there. 1125 01:00:22,380 --> 01:00:24,990 So if I stay, I'm going to lose. 1126 01:00:24,990 --> 01:00:27,680 Which means that if I do the stay strategy, 1127 01:00:27,680 --> 01:00:29,220 I'm going to lose 2/3 of the time. 1128 01:00:31,744 --> 01:00:32,410 So you're right. 1129 01:00:32,410 --> 01:00:33,750 I didn't cover that. 1130 01:00:33,750 --> 01:00:34,420 But we can. 1131 01:00:34,420 --> 01:00:38,160 It's not too hard in this case because the probability 1132 01:00:38,160 --> 01:00:41,540 of winning on a switch equals the probability 1133 01:00:41,540 --> 01:00:42,490 of losing on a stick. 1134 01:00:45,760 --> 01:00:52,280 So the probability that you win with switch 1135 01:00:52,280 --> 01:00:58,720 equals the probability you lose with a stick strategy. 1136 01:01:02,560 --> 01:01:03,080 All right? 1137 01:01:03,080 --> 01:01:06,080 And they're both 2/3, because they're the same 1138 01:01:06,080 --> 01:01:07,950 and this is 2/3. 1139 01:01:07,950 --> 01:01:09,530 So in fact, what's, then, the chance 1140 01:01:09,530 --> 01:01:10,780 you win with a stick strategy? 1141 01:01:13,350 --> 01:01:14,952 1/3. 1142 01:01:14,952 --> 01:01:17,410 And that's sort of you know, where Marilyn was coming from. 1143 01:01:17,410 --> 01:01:19,677 Because she said, well, if you pick the box right 1144 01:01:19,677 --> 01:01:21,260 the first time, which is what you need 1145 01:01:21,260 --> 01:01:23,870 to get to win with a stick strategy, 1146 01:01:23,870 --> 01:01:26,810 the chance you pick the right box is one-in-three. 1147 01:01:26,810 --> 01:01:29,060 And showing, you know, a goat over here didn't matter. 1148 01:01:29,060 --> 01:01:32,200 It's still one-in-three. 1149 01:01:32,200 --> 01:01:34,330 Now that's gets a-- that's intuition, 1150 01:01:34,330 --> 01:01:38,260 but it can get you trouble, because say 1151 01:01:38,260 --> 01:01:41,450 you picked this box and Carol showed you both of these boxes, 1152 01:01:41,450 --> 01:01:43,519 and they had pocket protectors. 1153 01:01:43,519 --> 01:01:45,310 Then what do you know about the probability 1154 01:01:45,310 --> 01:01:47,750 you won with a stick? 1155 01:01:47,750 --> 01:01:48,710 It's 1. 1156 01:01:48,710 --> 01:01:50,900 Ooh, so things did change there. 1157 01:01:50,900 --> 01:01:51,400 All right? 1158 01:01:51,400 --> 01:01:54,375 So you have to be a little careful about how you reason, 1159 01:01:54,375 --> 01:01:55,740 OK? 1160 01:01:55,740 --> 01:01:58,300 You've got to do this. 1161 01:01:58,300 --> 01:02:03,630 Stay with that and you'll always be fine for your strategy. 1162 01:02:03,630 --> 01:02:07,160 So really the key is to resist the temptation 1163 01:02:07,160 --> 01:02:12,224 to just sort of use your intuition to think about it. 1164 01:02:12,224 --> 01:02:12,890 All right. 1165 01:02:12,890 --> 01:02:14,880 Now, this is a pretty simple example. 1166 01:02:17,530 --> 01:02:19,270 So now I'm going to do one that's 1167 01:02:19,270 --> 01:02:24,200 a little more interesting and a little more devious in a sense. 1168 01:02:24,200 --> 01:02:25,540 So I need a volunteer. 1169 01:02:25,540 --> 01:02:27,610 We're going to do a gambling game. 1170 01:02:27,610 --> 01:02:29,700 So I need a volunteer. 1171 01:02:29,700 --> 01:02:30,570 All right. 1172 01:02:30,570 --> 01:02:32,330 Now, he's raised his hands before I even 1173 01:02:32,330 --> 01:02:34,690 said what the criteria are. 1174 01:02:34,690 --> 01:02:37,065 First, you gotta be confident you understand probability. 1175 01:02:40,554 --> 01:02:41,470 Well, you can fake it. 1176 01:02:41,470 --> 01:02:43,170 But second, and more importantly, you've 1177 01:02:43,170 --> 01:02:46,200 got to have some money in your product or you've got 1178 01:02:46,200 --> 01:02:48,560 to borrow some on the way down. 1179 01:02:48,560 --> 01:02:50,185 You got a few bucks there? 1180 01:02:50,185 --> 01:02:50,769 AUDIENCE: Yes. 1181 01:02:50,769 --> 01:02:52,226 PROFESSOR: All right, come on down. 1182 01:02:52,226 --> 01:02:54,115 That's really all that matters, you know. 1183 01:03:03,735 --> 01:03:08,370 OK, now, in this game, there's three dice. 1184 01:03:08,370 --> 01:03:10,700 Well, that's 'cause you're smart. 1185 01:03:10,700 --> 01:03:13,610 OK, so let me show you the dice in this game. 1186 01:03:20,980 --> 01:03:22,935 And these, of course, are not normal dice. 1187 01:03:38,071 --> 01:03:38,570 All right? 1188 01:03:38,570 --> 01:03:43,980 So dice A has a 2, a 6, and a 7, and they're 1189 01:03:43,980 --> 01:03:45,390 the same on the reverse side. 1190 01:03:45,390 --> 01:03:47,630 So it's a six-sided die, and it's fair. 1191 01:03:47,630 --> 01:03:49,990 When you roll it, the probability A 1192 01:03:49,990 --> 01:03:51,970 comes up 2 is going to be 1/3. 1193 01:03:51,970 --> 01:03:52,490 All right? 1194 01:03:52,490 --> 01:03:55,640 Same for 6 and for 7. 1195 01:03:55,640 --> 01:03:58,850 Ah, we're just getting started. 1196 01:03:58,850 --> 01:04:07,552 Dice B has a 1, 5, and a 9 and dice C has a 3, 4, and an 8. 1197 01:04:07,552 --> 01:04:09,260 So I got-- they're all different numbers, 1198 01:04:09,260 --> 01:04:11,760 the numbers from 1 to 9. 1199 01:04:11,760 --> 01:04:15,190 So just so we're really clear, the probability 1200 01:04:15,190 --> 01:04:20,100 that A comes up 3 is 1/3. 1201 01:04:20,100 --> 01:04:28,394 The same for the probability A comes up 6 is 1/3 and so on. 1202 01:04:28,394 --> 01:04:30,405 Ah, yes I do mean 2. 1203 01:04:30,405 --> 01:04:31,110 Good. 1204 01:04:31,110 --> 01:04:31,955 All right, good, you're paying attention. 1205 01:04:31,955 --> 01:04:32,480 That's good. 1206 01:04:32,480 --> 01:04:35,021 There's some hope. 1207 01:04:35,021 --> 01:04:35,520 All right? 1208 01:04:35,520 --> 01:04:36,990 Now, here's the game. 1209 01:04:36,990 --> 01:04:38,610 We're each going to pick a die. 1210 01:04:38,610 --> 01:04:40,760 And we're going to roll our die. 1211 01:04:40,760 --> 01:04:41,890 And the higher die wins. 1212 01:04:41,890 --> 01:04:43,140 We're going to roll them once. 1213 01:04:43,140 --> 01:04:44,600 The higher one wins. 1214 01:04:44,600 --> 01:04:47,420 And the loser pays the winner $1. 1215 01:04:47,420 --> 01:04:50,560 And no tears, OK? 1216 01:04:50,560 --> 01:04:51,250 You got that? 1217 01:04:51,250 --> 01:04:51,750 That's good. 1218 01:04:51,750 --> 01:04:53,916 He's already trying to figure out which one to pick. 1219 01:04:53,916 --> 01:04:57,290 So let's put your dollar on the table. 1220 01:04:57,290 --> 01:04:59,064 Here we go. 1221 01:04:59,064 --> 01:04:59,980 AUDIENCE: I have a $5. 1222 01:04:59,980 --> 01:05:01,260 Do you have change? 1223 01:05:01,260 --> 01:05:03,460 PROFESSOR: I might have change. 1224 01:05:03,460 --> 01:05:04,150 All right. 1225 01:05:04,150 --> 01:05:06,630 Let's see here. 1226 01:05:06,630 --> 01:05:07,750 Yes, I do have change. 1227 01:05:07,750 --> 01:05:08,150 AUDIENCE: Oh, I do have a dollar. 1228 01:05:08,150 --> 01:05:09,358 PROFESSOR: You have a dollar. 1229 01:05:09,358 --> 01:05:09,962 All right. 1230 01:05:09,962 --> 01:05:11,020 Well, for now anyway. 1231 01:05:11,020 --> 01:05:14,707 Let's put the dollar there on the table. 1232 01:05:14,707 --> 01:05:16,290 AUDIENCE: It's going to be a $20 next. 1233 01:05:16,290 --> 01:05:18,165 PROFESSOR: We're just getting warmed up here. 1234 01:05:18,165 --> 01:05:20,207 OK, now, there is-- what's your name, by the way? 1235 01:05:20,207 --> 01:05:20,831 AUDIENCE: Will. 1236 01:05:20,831 --> 01:05:21,660 PROFESSOR: Will. 1237 01:05:21,660 --> 01:05:23,890 OK, nice to meet you, Will. 1238 01:05:23,890 --> 01:05:24,790 There we go. 1239 01:05:24,790 --> 01:05:25,290 All right. 1240 01:05:25,290 --> 01:05:27,830 Now, there is one problem here. 1241 01:05:27,830 --> 01:05:30,595 I don't really have those dice. 1242 01:05:30,595 --> 01:05:32,720 So we're going to have to do a mathematical version 1243 01:05:32,720 --> 01:05:33,529 of this bet. 1244 01:05:33,529 --> 01:05:35,320 AUDIENCE: You're always going to win, then. 1245 01:05:35,320 --> 01:05:37,590 PROFESSOR: So what we're going to do 1246 01:05:37,590 --> 01:05:40,710 is pretend that we did a roll, head-to-head, 1247 01:05:40,710 --> 01:05:43,770 and whichever one is more likely to win, 1248 01:05:43,770 --> 01:05:47,460 we're going to assume the roll came out that way, all right? 1249 01:05:47,460 --> 01:05:51,330 Now, I know you've been studying those three guys there, 1250 01:05:51,330 --> 01:05:53,070 those three dice. 1251 01:05:53,070 --> 01:05:56,620 I'm going to be fair and let you pick first which one you like. 1252 01:05:56,620 --> 01:06:00,910 Now, as you look at this, you know, you guys 1253 01:06:00,910 --> 01:06:04,140 can help him out here if you can figure out which one's best. 1254 01:06:04,140 --> 01:06:06,540 This has got a couple good sized numbers. 1255 01:06:06,540 --> 01:06:08,610 You know, two of these beat these. 1256 01:06:08,610 --> 01:06:11,600 This has the biggest number, but it also has the smallest. 1257 01:06:11,600 --> 01:06:14,960 This has two small guys in it. 1258 01:06:14,960 --> 01:06:17,427 You now and you want to pick the one that's likely to-- 1259 01:06:17,427 --> 01:06:19,260 AUDIENCE: They all add up to the same thing. 1260 01:06:19,260 --> 01:06:21,920 PROFESSOR: They do all add up to the same thing, that's true. 1261 01:06:21,920 --> 01:06:22,420 Yeah. 1262 01:06:25,054 --> 01:06:25,970 Which one do you want? 1263 01:06:38,606 --> 01:06:40,070 Which one do you want here Will? 1264 01:06:42,529 --> 01:06:43,654 What's your best one there? 1265 01:06:47,610 --> 01:06:49,418 A does look pretty good. 1266 01:06:49,418 --> 01:06:52,286 You've got two solid numbers there. 1267 01:06:52,286 --> 01:06:53,242 It does. 1268 01:06:53,242 --> 01:06:57,560 A looks like it beats up on C pretty good. 1269 01:06:57,560 --> 01:06:59,060 AUDIENCE: I don't know. 1270 01:06:59,060 --> 01:07:00,554 I think I'm going to lose a dollar. 1271 01:07:00,554 --> 01:07:02,045 PROFESSOR: That's probably true. 1272 01:07:02,045 --> 01:07:03,533 [LAUGHTER] 1273 01:07:04,033 --> 01:07:05,524 [AUDIENCE CHATTER] 1274 01:07:14,967 --> 01:07:15,509 I don't know. 1275 01:07:15,509 --> 01:07:16,591 A looks pretty good to us. 1276 01:07:16,591 --> 01:07:17,773 Which one do you like here? 1277 01:07:17,773 --> 01:07:19,192 We'll try them out. 1278 01:07:19,192 --> 01:07:21,560 We'll play more games here. 1279 01:07:21,560 --> 01:07:29,134 A. All right, Will likes A. All right. 1280 01:07:29,134 --> 01:07:32,169 And I'm going to pick-- I'm going 1281 01:07:32,169 --> 01:07:34,710 to go with C. I'm going to give you a fighting chance, right? 1282 01:07:34,710 --> 01:07:37,190 Because we figured A beat C for sure, right? 1283 01:07:37,190 --> 01:07:39,450 Because you've got a couple of 6 and 7 beat up 3 and 4 1284 01:07:39,450 --> 01:07:41,050 pretty good. 1285 01:07:41,050 --> 01:07:44,597 Yeah, I'm going to pick C. All right. 1286 01:07:44,597 --> 01:07:46,430 And now we're going to do-- figure out which 1287 01:07:46,430 --> 01:07:49,050 one's more likely to win. 1288 01:07:49,050 --> 01:07:51,411 And we're going to make the tree diagram. 1289 01:07:51,411 --> 01:07:51,910 All right? 1290 01:07:51,910 --> 01:07:58,265 So here's A. You've got 2, 6, and 7 are your choices. 1291 01:07:58,265 --> 01:07:59,140 And they're each 1/3. 1292 01:08:01,930 --> 01:08:05,311 And then, for mine, I've got 3, 4, and 8. 1293 01:08:10,720 --> 01:08:13,590 3, 4, and 8. 1294 01:08:13,590 --> 01:08:15,130 3, 4, and 8. 1295 01:08:15,130 --> 01:08:16,475 And they're all 1/3 chance. 1296 01:08:19,925 --> 01:08:23,430 All right, so I make my tree here. 1297 01:08:23,430 --> 01:08:28,495 I got squished at the bottom, but they're all 1/3. 1298 01:08:28,495 --> 01:08:30,939 All right, now we're going to see who wins, 1299 01:08:30,939 --> 01:08:34,640 C or A. There's nine outcomes. 1300 01:08:34,640 --> 01:08:38,034 Who wins this outcome? 1301 01:08:38,034 --> 01:08:38,740 Oh, I do. 1302 01:08:38,740 --> 01:08:40,584 Yeah, I get all these. 1303 01:08:40,584 --> 01:08:41,380 That's good. 1304 01:08:41,380 --> 01:08:42,600 You're in trouble already. 1305 01:08:42,600 --> 01:08:43,957 What happens here, though? 1306 01:08:43,957 --> 01:08:44,540 Who wins that? 1307 01:08:48,279 --> 01:08:50,010 Your 6 beats 3, 6 beats 4. 1308 01:08:50,010 --> 01:08:51,240 You win those. 1309 01:08:51,240 --> 01:08:53,609 I got you here. 1310 01:08:53,609 --> 01:08:56,210 Then you win 7 beats 3. 1311 01:08:56,210 --> 01:08:57,350 A wins. 1312 01:08:57,350 --> 01:08:59,149 7 beats 4, A wins. 1313 01:08:59,149 --> 01:09:01,370 It's tied, but, oh, my 8 beat your 7. 1314 01:09:03,950 --> 01:09:04,609 All right. 1315 01:09:04,609 --> 01:09:07,939 So let's see here. 1316 01:09:07,939 --> 01:09:14,000 There's one, two, three, four, five for me, each at 1/9. 1317 01:09:14,000 --> 01:09:18,583 All right, so I'm going to win. 1318 01:09:18,583 --> 01:09:20,180 All right? 1319 01:09:20,180 --> 01:09:24,510 Now, in this case, the sample space is uniform. 1320 01:09:24,510 --> 01:09:27,140 Every probability is the same for the outcomes. 1321 01:09:27,140 --> 01:09:28,950 They're all 1/9. 1322 01:09:28,950 --> 01:09:29,700 1/3 times 1/3. 1323 01:09:29,700 --> 01:09:31,240 1/3 times 1/3. 1324 01:09:31,240 --> 01:09:31,740 All right? 1325 01:09:31,740 --> 01:09:36,090 So that's the case where we say the sample space is uniform. 1326 01:09:36,090 --> 01:09:36,960 Don't go anywhere. 1327 01:09:36,960 --> 01:09:39,160 We're going to do some more here. 1328 01:09:42,180 --> 01:09:53,689 The sample space is uniform if every sample point 1329 01:09:53,689 --> 01:09:54,730 has the same probability. 1330 01:10:01,880 --> 01:10:05,350 In which case, the probability of the sample points 1331 01:10:05,350 --> 01:10:08,760 would be 1 in the cardinality of the sample space. 1332 01:10:08,760 --> 01:10:10,320 In this case, there's nine points. 1333 01:10:10,320 --> 01:10:13,270 They each have probability 1/9. 1334 01:10:13,270 --> 01:10:21,190 And in this case, C beats A with probability of 5/9. 1335 01:10:23,970 --> 01:10:27,070 That means I win the dollar. 1336 01:10:27,070 --> 01:10:29,058 All right. 1337 01:10:29,058 --> 01:10:32,274 So do you have some more money in there? 1338 01:10:32,274 --> 01:10:33,300 In that wallet? 1339 01:10:33,300 --> 01:10:35,170 I'll make change for you. 1340 01:10:35,170 --> 01:10:38,220 So, what we're going to do is play again. 1341 01:10:38,220 --> 01:10:40,910 And you've learned something now, that C beats A. 1342 01:10:40,910 --> 01:10:42,750 So we play again, you're probably 1343 01:10:42,750 --> 01:10:46,070 not going to want to pick A. In fact, 1344 01:10:46,070 --> 01:10:49,050 C is surprisingly good there. 1345 01:10:49,050 --> 01:10:50,550 Now, I'll let you wager more, if you 1346 01:10:50,550 --> 01:10:51,800 want, to win your dollar back. 1347 01:10:51,800 --> 01:10:53,845 Do you want to do double or nothing? 1348 01:10:53,845 --> 01:10:55,745 [LAUGHTER] 1349 01:10:59,335 --> 01:11:02,184 AUDIENCE: So you always get to pick second, though? 1350 01:11:02,184 --> 01:11:02,850 PROFESSOR: Yeah. 1351 01:11:02,850 --> 01:11:04,092 I've got to be fair here. 1352 01:11:04,092 --> 01:11:05,425 I'm going to let you pick first. 1353 01:11:05,425 --> 01:11:07,470 [LAUGHTER] 1354 01:11:07,470 --> 01:11:09,866 All right, you got a couple bucks? 1355 01:11:09,866 --> 01:11:10,990 We'll do double or nothing. 1356 01:11:10,990 --> 01:11:11,656 What do you say? 1357 01:11:11,656 --> 01:11:13,270 I'll make change for you. 1358 01:11:13,270 --> 01:11:14,690 There you go. 1359 01:11:14,690 --> 01:11:15,565 Here's $5. 1360 01:11:15,565 --> 01:11:17,220 I'll change your $5. 1361 01:11:17,220 --> 01:11:18,270 And then we'll bet $2. 1362 01:11:18,270 --> 01:11:19,390 How's that? 1363 01:11:19,390 --> 01:11:23,166 I've actually got a $2 bill here. 1364 01:11:23,166 --> 01:11:23,780 There we go. 1365 01:11:23,780 --> 01:11:24,440 You've got $2. 1366 01:11:24,440 --> 01:11:25,640 There we go. 1367 01:11:25,640 --> 01:11:28,560 All right, double or nothing. 1368 01:11:28,560 --> 01:11:29,924 Will, what do you like? 1369 01:11:29,924 --> 01:11:31,340 You're obviously not going to pick 1370 01:11:31,340 --> 01:11:34,245 A. That just got beat by C. What do you like, B or C? 1371 01:11:42,724 --> 01:11:44,801 Well, one of them's got to be best. 1372 01:11:44,801 --> 01:11:46,217 One of them's got to be best here. 1373 01:11:46,217 --> 01:11:48,712 You've just got to figure out which one it is. 1374 01:11:48,712 --> 01:11:51,970 AUDIENCE: I feel like everyone's going to have [INAUDIBLE]. 1375 01:11:51,970 --> 01:11:54,500 If I choose C, let me see. 1376 01:11:54,500 --> 01:11:58,610 That would be [INAUDIBLE] so I guess. 1377 01:11:58,610 --> 01:12:00,604 PROFESSOR: I won't pick A, that's for sure. 1378 01:12:00,604 --> 01:12:02,354 AUDIENCE: If you pick B, I get one from 4, 1379 01:12:02,354 --> 01:12:03,854 I get two from the 8, which is four. 1380 01:12:03,854 --> 01:12:05,318 And you get the other five. 1381 01:12:05,318 --> 01:12:09,270 If I pick C and you pick A, that means I get one from the 3, 1382 01:12:09,270 --> 01:12:11,246 I get one from the 4. 1383 01:12:11,246 --> 01:12:14,704 If I get two from the 8, you still get 5. 1384 01:12:14,704 --> 01:12:16,186 PROFESSOR: That's good. 1385 01:12:16,186 --> 01:12:19,644 You've figured out you're screwed here. 1386 01:12:19,644 --> 01:12:20,650 That's very good. 1387 01:12:20,650 --> 01:12:21,980 All right. 1388 01:12:21,980 --> 01:12:24,120 So, yeah, so he's sort of-- he's doing great. 1389 01:12:24,120 --> 01:12:26,245 He's figured out, because he went through this sort 1390 01:12:26,245 --> 01:12:28,800 of like this method, what all the possible outcomes were. 1391 01:12:28,800 --> 01:12:31,680 And he's discovered that, no matter what he picks, 1392 01:12:31,680 --> 01:12:33,830 I'm going to beat him. 1393 01:12:33,830 --> 01:12:35,340 So let's see why that's the case. 1394 01:12:35,340 --> 01:12:36,440 So let's see. 1395 01:12:36,440 --> 01:12:39,130 So because you figured it out, I won't take your money 1396 01:12:39,130 --> 01:12:39,990 this time. 1397 01:12:39,990 --> 01:12:41,770 But we'll do some more in a minute. 1398 01:12:41,770 --> 01:12:46,900 So say Will picked-- say he picked C. 1399 01:12:46,900 --> 01:12:52,000 And say that then I will pick B. All right, 1400 01:12:52,000 --> 01:12:55,220 so on C, there's three choices. 1401 01:12:55,220 --> 01:12:55,830 3. 1402 01:12:55,830 --> 01:12:57,480 4, and 8. 1403 01:12:57,480 --> 01:13:01,775 And then, on my choice of B, there's 1, 5, and 9. 1404 01:13:06,160 --> 01:13:09,120 All right, so I get 1, 5, and 9. 1405 01:13:09,120 --> 01:13:12,070 And now we can see who wins here. 1406 01:13:12,070 --> 01:13:17,810 Well, C wins there, loses here, loses there. 1407 01:13:17,810 --> 01:13:22,370 C wins here, and B wins the next two. 1408 01:13:22,370 --> 01:13:24,000 And C wins these. 1409 01:13:24,000 --> 01:13:26,650 And B wins the tiebreaker. 1410 01:13:26,650 --> 01:13:30,280 So B beats C with probability 5/9. 1411 01:13:38,420 --> 01:13:38,920 All right. 1412 01:13:38,920 --> 01:13:43,650 So you don't want to pick A, you don't want to pick C, 1413 01:13:43,650 --> 01:13:45,094 you want to try a bet with B? 1414 01:13:48,363 --> 01:13:52,840 I mean, look, you've got B beats C. We just showed that. 1415 01:13:52,840 --> 01:13:57,847 C beats A. I mean, that's a sure bet if I ever saw one. 1416 01:13:57,847 --> 01:13:59,284 [LAUGHTER] 1417 01:13:59,284 --> 01:14:01,200 PROFESSOR: What do you think? 1418 01:14:01,200 --> 01:14:06,268 AUDIENCE: So if I pick B and you pick C, that means [INAUDIBLE] 1419 01:14:09,220 --> 01:14:12,172 [AUDIENCE CHATTER] 1420 01:14:12,172 --> 01:14:22,012 But if he picks A, then I get [INAUDIBLE] Yeah, 1421 01:14:22,012 --> 01:14:24,010 I'd still get 4, so-- 1422 01:14:24,010 --> 01:14:26,752 PROFESSOR: So if you pick B, what am I going to do? 1423 01:14:26,752 --> 01:14:28,499 AUDIENCE: Oh, you're going to pick A. 1424 01:14:28,499 --> 01:14:31,040 PROFESSOR: I'm going to pick A. Let's see how that works out. 1425 01:14:31,040 --> 01:14:36,850 All right, so B has 1, 5, and 9. 1426 01:14:36,850 --> 01:14:40,685 A is, yeah, 2, 6, and 7. 1427 01:14:46,780 --> 01:14:50,670 OK, so you're going to lose all three there. 1428 01:14:50,670 --> 01:14:53,320 A wins those. 1429 01:14:53,320 --> 01:14:56,950 A wins-- no, you win. 1430 01:14:56,950 --> 01:15:00,967 B wins there, but A wins the next couple. 1431 01:15:00,967 --> 01:15:02,341 And then, you win the last three. 1432 01:15:05,500 --> 01:15:13,090 So A wins, again, 5/9 of the time. 1433 01:15:13,090 --> 01:15:15,880 So how is this possible? 1434 01:15:15,880 --> 01:15:26,040 We've got A beats B, beats C, beats A. All 5/9. 1435 01:15:26,040 --> 01:15:27,885 How is that possible? 1436 01:15:27,885 --> 01:15:29,250 Did we make a mistake? 1437 01:15:32,440 --> 01:15:35,610 Well, why isn't it possible? 1438 01:15:35,610 --> 01:15:40,050 Whoever said probabilities were transitive? 1439 01:15:40,050 --> 01:15:40,912 Right? 1440 01:15:40,912 --> 01:15:42,370 I mean, you'd sort of think there's 1441 01:15:42,370 --> 01:15:44,260 a best of the three die. 1442 01:15:44,260 --> 01:15:46,220 But for any one that you pick, that he picks, 1443 01:15:46,220 --> 01:15:47,428 I can find one that's better. 1444 01:15:47,428 --> 01:15:49,630 And it goes in a cycle. 1445 01:15:49,630 --> 01:15:50,130 All right? 1446 01:15:50,130 --> 01:15:52,820 So you'd think picking first, you've got a shot 1447 01:15:52,820 --> 01:15:53,780 to pick the best one. 1448 01:15:53,780 --> 01:15:56,890 But picking second always wins this game, 1449 01:15:56,890 --> 01:15:59,411 if you pick the right one. 1450 01:15:59,411 --> 01:15:59,910 All right. 1451 01:15:59,910 --> 01:16:02,280 You know, since you did good and figured that out, 1452 01:16:02,280 --> 01:16:05,850 I'm going to give you a special opportunity here. 1453 01:16:05,850 --> 01:16:06,350 All right? 1454 01:16:06,350 --> 01:16:11,540 Now, I'm going to let you go second. 1455 01:16:11,540 --> 01:16:13,030 All right? 1456 01:16:13,030 --> 01:16:15,740 So I'm going to pick one, and I'm 1457 01:16:15,740 --> 01:16:17,360 going to let you pick after me. 1458 01:16:17,360 --> 01:16:19,480 And then we're going to bet. 1459 01:16:19,480 --> 01:16:22,812 There's going to be a little twist to it. 1460 01:16:22,812 --> 01:16:24,270 But you want to pick second, right? 1461 01:16:28,235 --> 01:16:30,485 All right, well, let me tell you what I'm going to do. 1462 01:16:30,485 --> 01:16:32,030 Just to make it-- I want to do a different analysis. 1463 01:16:32,030 --> 01:16:34,155 We already did these three, so what I'm going to do 1464 01:16:34,155 --> 01:16:37,289 is an analysis where we roll them twice. 1465 01:16:37,289 --> 01:16:38,830 Otherwise, it's the same basic thing, 1466 01:16:38,830 --> 01:16:41,510 but I want to do an analysis with more outcomes here. 1467 01:16:41,510 --> 01:16:43,310 We've sort of beat this one to death. 1468 01:16:43,310 --> 01:16:45,210 So I'm going to go first. 1469 01:16:45,210 --> 01:16:52,860 And let's see, what do I want to pick? 1470 01:16:52,860 --> 01:16:54,520 Hm. 1471 01:16:54,520 --> 01:16:56,720 Well, what should I pick, if I'm going to do this? 1472 01:16:56,720 --> 01:17:01,330 I guess I'll pick B, all right? 1473 01:17:01,330 --> 01:17:03,190 So I'm going to pick B here. 1474 01:17:06,940 --> 01:17:07,440 All right. 1475 01:17:07,440 --> 01:17:07,970 How much do you want to bet here? 1476 01:17:07,970 --> 01:17:09,678 I'm going to let you win your money back. 1477 01:17:09,678 --> 01:17:11,630 I'll let you bet up to $10. 1478 01:17:11,630 --> 01:17:12,245 Your choice. 1479 01:17:15,179 --> 01:17:16,970 What do you-- what do you want to bet here? 1480 01:17:16,970 --> 01:17:18,280 We've got to put some money on the table, 1481 01:17:18,280 --> 01:17:19,488 make it a little interesting. 1482 01:17:22,190 --> 01:17:23,314 Yeah, you pick-- 1483 01:17:23,314 --> 01:17:25,230 AUDIENCE: Do you roll a different die or what? 1484 01:17:25,230 --> 01:17:26,438 PROFESSOR: No, the same ones. 1485 01:17:26,438 --> 01:17:27,680 The die don't change. 1486 01:17:27,680 --> 01:17:31,220 I'm picking B. And I'm going to roll it a couple times. 1487 01:17:31,220 --> 01:17:32,700 And I'm going to add up my score. 1488 01:17:32,700 --> 01:17:34,700 And then, you get to pick whatever die you want. 1489 01:17:34,700 --> 01:17:35,783 I'm going to let you pick. 1490 01:17:35,783 --> 01:17:39,770 I mean, I'm going to let you pick the one that beats it. 1491 01:17:39,770 --> 01:17:41,620 You roll yours a couple times. 1492 01:17:41,620 --> 01:17:43,310 And now we see which one has the higher 1493 01:17:43,310 --> 01:17:45,393 sum, who's more likely to win. 1494 01:17:45,393 --> 01:17:47,726 AUDIENCE: So it's the sum total, like you roll a 9 and 9 1495 01:17:47,726 --> 01:17:48,777 and I roll a 7 and 7. 1496 01:17:48,777 --> 01:17:51,110 PROFESSOR: Yeah, then I'm going to win, because I get 18 1497 01:17:51,110 --> 01:17:52,090 and you got 14. 1498 01:17:52,090 --> 01:17:54,270 If I roll a 9 and a 1, I get a 10. 1499 01:17:54,270 --> 01:17:56,884 You roll a 6 and a 7, you beat me at 13. 1500 01:17:56,884 --> 01:17:58,360 All right? 1501 01:17:58,360 --> 01:18:02,490 So which one do you want to pick? 1502 01:18:02,490 --> 01:18:05,740 You could pick C. But you've got to put the money on the table 1503 01:18:05,740 --> 01:18:08,739 here, whatever you want to bet. 1504 01:18:08,739 --> 01:18:11,280 You know, I want to give you a chance to win your money back. 1505 01:18:14,830 --> 01:18:18,032 What's the probability he's going to lose? 1506 01:18:18,032 --> 01:18:19,514 [LAUGHTER] 1507 01:18:19,514 --> 01:18:20,449 You got a T-Card? 1508 01:18:20,449 --> 01:18:21,490 What've you got in there? 1509 01:18:21,490 --> 01:18:23,512 Ah, I'll take the [INAUDIBLE] card. 1510 01:18:23,512 --> 01:18:24,985 What's that worth? 1511 01:18:24,985 --> 01:18:27,931 Ah, it's worth a few bucks here. 1512 01:18:27,931 --> 01:18:33,390 All right, so I'm picking B. Which are you picking? 1513 01:18:37,110 --> 01:18:40,304 I've really got his head messed up now. 1514 01:18:40,304 --> 01:18:41,720 AUDIENCE: There's $4 on the table. 1515 01:18:41,720 --> 01:18:43,690 Take that and run. 1516 01:18:43,690 --> 01:18:44,690 PROFESSOR: Yeah, really. 1517 01:18:44,690 --> 01:18:46,570 We already saw-- I'll give you a hint here. 1518 01:18:46,570 --> 01:18:50,860 A beats B. 5/9 of the time, A comes up higher than B. 1519 01:18:50,860 --> 01:18:54,030 So if we roll them twice, your odds 1520 01:18:54,030 --> 01:18:59,420 got even better that A beats B. All right, 1521 01:18:59,420 --> 01:19:03,850 so Will goes with A. Will, that was a big mistake. 1522 01:19:03,850 --> 01:19:05,020 All right. 1523 01:19:05,020 --> 01:19:06,020 How can it be a mistake? 1524 01:19:06,020 --> 01:19:09,220 A beats B 5/9 of the time. 1525 01:19:09,220 --> 01:19:12,220 Now what we're going to do is make the decision tree. 1526 01:19:12,220 --> 01:19:15,351 If we roll B twice, if we roll A twice. 1527 01:19:15,351 --> 01:19:15,850 All right. 1528 01:19:15,850 --> 01:19:17,530 Let's see what happens. 1529 01:19:17,530 --> 01:19:19,090 This is a nasty one to write down. 1530 01:19:19,090 --> 01:19:20,840 In fact, you'll start seeing this when you 1531 01:19:20,840 --> 01:19:22,100 do these tree method things. 1532 01:19:22,100 --> 01:19:23,725 Sometimes the trees get a little hairy, 1533 01:19:23,725 --> 01:19:27,520 so we've got to sort of approximate it a little bit. 1534 01:19:27,520 --> 01:19:30,140 So here's B, right? 1535 01:19:30,140 --> 01:19:33,950 I got 1, 5, and 9 for my first roll. 1536 01:19:33,950 --> 01:19:36,430 And then I got 1, 5, and 9 for my second roll. 1537 01:19:42,100 --> 01:19:44,590 And I add these up. 1538 01:19:44,590 --> 01:19:47,990 See, if you read the notes ahead, sometimes that helps. 1539 01:19:47,990 --> 01:19:49,495 6-- here are my outcomes. 1540 01:19:49,495 --> 01:19:51,120 These are the numbers I'm going to get. 1541 01:19:54,300 --> 01:19:57,000 These are my nine values. 1542 01:19:57,000 --> 01:20:01,360 Your nine values, you got 2, 6, and 7. 1543 01:20:01,360 --> 01:20:05,170 Then each 2, 6, and 7 again. 1544 01:20:05,170 --> 01:20:15,165 So we add those up, you got 4, 8, 9, 8, 12, 13, 9, 13, and 14. 1545 01:20:15,165 --> 01:20:17,290 And now, we have each one of these 1546 01:20:17,290 --> 01:20:19,130 against each one of these. 1547 01:20:19,130 --> 01:20:20,830 Because there's 81 outcomes. 1548 01:20:20,830 --> 01:20:22,230 I got four levels to the tree. 1549 01:20:22,230 --> 01:20:25,370 And rather than draw four levels of this [INAUDIBLE] tree 1550 01:20:25,370 --> 01:20:27,250 to get 81 outcomes, you're going to say 1551 01:20:27,250 --> 01:20:30,280 that each one of these corresponds 1552 01:20:30,280 --> 01:20:34,330 to this tree, all right, that has nine leaves. 1553 01:20:34,330 --> 01:20:40,205 And now we can just count up how many times I win. 1554 01:20:40,205 --> 01:20:41,630 Well, 2 got nothing. 1555 01:20:41,630 --> 01:20:44,270 That's a 0. 1556 01:20:44,270 --> 01:20:46,120 Let's see how many times 6 wins. 1557 01:20:46,120 --> 01:20:50,064 One-- one time. 1558 01:20:50,064 --> 01:20:53,210 10-- one, two, three, four, five. 1559 01:20:55,890 --> 01:20:57,050 All right, we've got that. 1560 01:20:57,050 --> 01:21:01,260 6 wins, we already decided, once. 1561 01:21:01,260 --> 01:21:06,630 10, we decided already, wins five times. 1562 01:21:06,630 --> 01:21:09,612 14 wins one, two, three, four, five, six, seven, 1563 01:21:09,612 --> 01:21:12,290 eight, and ties once. 1564 01:21:12,290 --> 01:21:15,390 Tie-- one. 1565 01:21:15,390 --> 01:21:18,550 10, we already got was five. 1566 01:21:18,550 --> 01:21:23,260 14, we already got, was eight and a tie. 1567 01:21:23,260 --> 01:21:28,000 And finally, I get 18 wins every time. 1568 01:21:28,000 --> 01:21:28,780 Let's add this up. 1569 01:21:28,780 --> 01:21:36,210 I got 17, 22, 30, 35, 36, 41, 42 times 1570 01:21:36,210 --> 01:21:42,370 plus half the ties out of 81. 1571 01:21:42,370 --> 01:21:44,120 How much money's on this [INAUDIBLE] card? 1572 01:21:49,224 --> 01:21:50,440 Ah, smart guy. 1573 01:21:53,919 --> 01:21:55,410 How did this happen? 1574 01:21:55,410 --> 01:21:58,550 A is more likely to beat B, higher than B. 1575 01:21:58,550 --> 01:22:00,160 But you roll them twice, and it's 1576 01:22:00,160 --> 01:22:03,770 less likely to be as large on the sum? 1577 01:22:03,770 --> 01:22:05,020 Well, why not? 1578 01:22:05,020 --> 01:22:07,760 It's just our intuition's all screwed up. 1579 01:22:07,760 --> 01:22:10,160 But the math is straightforward. 1580 01:22:10,160 --> 01:22:12,410 In fact, an amazing thing happens. 1581 01:22:12,410 --> 01:22:17,710 If you roll them twice, they reverse completely. 1582 01:22:17,710 --> 01:22:22,590 A is worse than B, is worse than C, is worse than A 1583 01:22:22,590 --> 01:22:24,150 if you roll them twice. 1584 01:22:24,150 --> 01:22:27,570 If you roll them three times, something else happens. 1585 01:22:27,570 --> 01:22:29,540 In fact, it was known for a long time, 1586 01:22:29,540 --> 01:22:31,665 this phenomenon happens with one roll. 1587 01:22:31,665 --> 01:22:34,070 And when I was making lecture notes a few years ago, 1588 01:22:34,070 --> 01:22:35,869 I noticed they reverse with two rolls. 1589 01:22:35,869 --> 01:22:37,660 And then we did a class project to discover 1590 01:22:37,660 --> 01:22:39,830 what happens with k rolls. 1591 01:22:39,830 --> 01:22:44,034 And it turns out that you can get every possible combination. 1592 01:22:44,034 --> 01:22:45,450 You know, on some number of rolls, 1593 01:22:45,450 --> 01:22:50,230 A beats both B and C and B beats C. Other number of rolls, 1594 01:22:50,230 --> 01:22:53,010 A beats them both and C beats B. 1595 01:22:53,010 --> 01:22:55,600 And then, motivated by that, some researchers 1596 01:22:55,600 --> 01:22:59,330 out in San Diego showed that there's arbitrarily 1597 01:22:59,330 --> 01:23:03,600 strange dice, now, not with just three sides, 1598 01:23:03,600 --> 01:23:05,830 but with more sides. 1599 01:23:05,830 --> 01:23:09,025 Such that, for any tournaments-- remember a tournament 1600 01:23:09,025 --> 01:23:11,410 from graph theory, where everybody beats, 1601 01:23:11,410 --> 01:23:14,580 you know, between a pair of players, one player wins. 1602 01:23:14,580 --> 01:23:18,130 You can make the relation for any tournament 1603 01:23:18,130 --> 01:23:20,110 show up for a particular set of dice 1604 01:23:20,110 --> 01:23:22,489 by rolling a certain number of times. 1605 01:23:22,489 --> 01:23:24,280 All right, which is just really fascinating 1606 01:23:24,280 --> 01:23:25,280 that you can prove that. 1607 01:23:25,280 --> 01:23:27,660 So these dice can get arbitrarily weird 1608 01:23:27,660 --> 01:23:30,900 in terms of which die is likely to come up and be 1609 01:23:30,900 --> 01:23:32,989 better than another die. 1610 01:23:32,989 --> 01:23:34,280 All right, that's it for today. 1611 01:23:34,280 --> 01:23:35,430 Let me give you your money back here. 1612 01:23:35,430 --> 01:23:37,055 And I've got a gift certificate you can 1613 01:23:37,055 --> 01:23:40,170 have for being a good sport. 1614 01:23:40,170 --> 01:23:40,670 Here you go. 1615 01:23:40,670 --> 01:23:42,160 How much money did I take here? 1616 01:23:42,160 --> 01:23:43,697 AUDIENCE: I think only like $2. 1617 01:23:43,697 --> 01:23:44,280 PROFESSOR: $2? 1618 01:23:44,280 --> 01:23:45,620 All right, there you go. 1619 01:23:45,620 --> 01:23:46,270 Very good. 1620 01:23:46,270 --> 01:23:46,770 Thanks. 1621 01:23:49,480 --> 01:23:53,140 So we'll do a lot more examples over the next week or two.