1 00:00:02,340 --> 00:00:07,801 Hi, so far in this course we've discussed a wide variety of inductive arguments. 2 00:00:07,801 --> 00:00:12,434 First of all, we looked at statistical generalization from a sample. 3 00:00:12,434 --> 00:00:17,827 Then we looked at application of the generalization down to a particular case. 4 00:00:17,827 --> 00:00:23,480 Then we looked at inference to the best explanation, and argument from analogy. 5 00:00:23,480 --> 00:00:28,330 And last week we turned to causal reasoning, which involved the positive 6 00:00:28,330 --> 00:00:33,680 necessary and sufficient condition test. And, in addition, at the very end, we 7 00:00:33,680 --> 00:00:39,101 looked at concommitent or concommitent theoriation, which involved a kind of 8 00:00:39,101 --> 00:00:43,310 manipulation in drawing conclusions about what causes what. 9 00:00:43,310 --> 00:00:46,521 So we've seen a lot of kinds of induction, 10 00:00:46,521 --> 00:00:52,178 and that's gotta raise the question, what's in common to all the different 11 00:00:52,178 --> 00:00:55,619 types of induction that makes them induction? 12 00:00:55,619 --> 00:01:00,741 What we saw, the inductive arguments, unlike deductive arguments, are 13 00:01:00,741 --> 00:01:04,564 defeasible, and they don't even aim at being valid. 14 00:01:04,564 --> 00:01:09,857 Instead, they just try to be strong. But strength comes in different degrees, 15 00:01:09,857 --> 00:01:15,080 and we haven't really learned anything yet about, what exactly strength is? 16 00:01:15,080 --> 00:01:20,799 Well, it turns out that one way of understanding inductive strength is that 17 00:01:20,799 --> 00:01:27,521 an argument is strong in proportion to the probability that its conclusion is 18 00:01:27,521 --> 00:01:32,140 true, given its premises. So inductive strength can be understood 19 00:01:32,140 --> 00:01:36,093 in terms of probability. And so for this week, we're going to turn 20 00:01:36,093 --> 00:01:41,129 to the notion of probability, which will help us understand inductive arguments. 21 00:01:41,129 --> 00:01:44,636 But, probability is important in a lot of other ways too. 22 00:01:44,636 --> 00:01:47,887 I mean just think about going on a picnic tomorrow. 23 00:01:47,887 --> 00:01:53,115 You want to know what the probability of it raining is, because if the probability 24 00:01:53,115 --> 00:01:57,004 of rain is very high, you might not want to go on your picnic. 25 00:01:57,004 --> 00:02:01,548 Instead, stay inside. Okay, so probability affects our beliefs, 26 00:02:01,548 --> 00:02:06,605 and it affects our decisions, and that makes it important to understand 27 00:02:06,605 --> 00:02:11,282 probability, and try to figure it out. But that's a problem, 28 00:02:11,282 --> 00:02:15,843 because if you just rest on your intuitions about probability, you're 29 00:02:15,843 --> 00:02:19,022 going to get messed up, you're going to make mistakes. 30 00:02:19,022 --> 00:02:24,413 Almost everybody messes up probability until they start thinking about it a 31 00:02:24,413 --> 00:02:30,593 little bit more seriously. Let me give you a few examples where 32 00:02:30,593 --> 00:02:39,122 people get confused about probability. [SOUND] Seven, alright. 33 00:02:39,122 --> 00:02:43,387 [SOUND] Seven, alright. [SOUND] Another seven. 34 00:02:43,387 --> 00:02:45,900 Unbelievable. Cool. 35 00:02:45,900 --> 00:02:49,867 Three sevens in a row, and seven's a winning number. 36 00:02:49,867 --> 00:02:55,908 So, that's great but, you know, the odds of getting four sevens in a row are 37 00:02:55,908 --> 00:03:00,507 really slim, so I bet the next one's not going to be a 38 00:03:00,507 --> 00:03:01,589 seven. Right? 39 00:03:01,589 --> 00:03:04,235 No. Just because the first three were a 40 00:03:04,235 --> 00:03:06,835 seven, doesn't tell you the next one's not going to be. 41 00:03:06,835 --> 00:03:10,812 Sure, it's unlikely to get four in a row, but once you got the first three, then 42 00:03:10,812 --> 00:03:14,788 the likelihood of getting the fourth seven is just the same as the likelihood 43 00:03:14,788 --> 00:03:16,930 of getting a seven on any roll whatsoever. 44 00:03:16,930 --> 00:03:21,278 Some people make that mistake. They think that just because they got 45 00:03:21,278 --> 00:03:25,687 three in a row, it's not going to come up that way again the fourth time. 46 00:03:25,687 --> 00:03:28,321 And other people make the opposite mistake. 47 00:03:28,321 --> 00:03:32,179 They say, hey the dice are hot tonight. I got three sevens in a row. 48 00:03:32,179 --> 00:03:34,751 The next one I'm going to get a seven too. 49 00:03:34,751 --> 00:03:37,896 I'm going to put all my money on seven. Don't do it. 50 00:03:37,896 --> 00:03:42,688 You'll lose your money, because even if the first three are sevens, dice don't 51 00:03:42,688 --> 00:03:45,841 get hot. The odds of getting a seven on the fourth 52 00:03:45,841 --> 00:03:50,508 one are exactly the same after you've gotten three sevens, and as if you didn't 53 00:03:50,508 --> 00:03:53,170 get three sevens. And both of these mistakes. 54 00:03:53,170 --> 00:03:55,879 Three sevens, so the next one's not going to be seven. 55 00:03:55,879 --> 00:03:58,353 Three sevens, so the next one's going to be seven. 56 00:03:58,353 --> 00:04:02,947 They're both mistakes, because the odds on the fourth roll are not affected by 57 00:04:02,947 --> 00:04:07,247 the odds on the first three rolls. And people who think otherwise, they're 58 00:04:07,247 --> 00:04:11,114 just going to get fooled by gamblers. It's not just in gambling, too. 59 00:04:11,114 --> 00:04:14,160 It's also in basketball. People think, oh he's hot. 60 00:04:14,160 --> 00:04:16,833 He's made a lot of shots in a row. He's hot. 61 00:04:16,833 --> 00:04:21,183 He's going to make the next one. And then they'll tell that player, you 62 00:04:21,183 --> 00:04:23,483 take the shot, because you're hot tonight. 63 00:04:23,483 --> 00:04:26,840 Well that's a little more controversial in basketball, 64 00:04:26,840 --> 00:04:31,315 but some statistics suggest that people don't really have hot hands in 65 00:04:31,315 --> 00:04:34,485 basketball. They just make the percentage that they 66 00:04:34,485 --> 00:04:37,344 make. And you're going to get strings, but that 67 00:04:37,344 --> 00:04:42,584 doesn't mean you're hot in any way. Like I said, that's controversial, but 68 00:04:42,584 --> 00:04:48,879 with dice, at least we know that dice do not get hot, and so don't waste your 69 00:04:48,879 --> 00:04:54,346 money making that mistake. That's called the gambler's fallacy, and 70 00:04:54,346 --> 00:04:59,035 don't do it. Okay, deck of cards, and here I am. 71 00:04:59,035 --> 00:05:03,666 I'm going to shuffle the cards, okay, there we go. 72 00:05:03,666 --> 00:05:08,399 There we go, I'm going to shuffle the cards again. 73 00:05:08,399 --> 00:05:15,808 We've got these cards all shuffled and now I'm going to deal out two hands, 74 00:05:15,808 --> 00:05:17,660 okay. Nine, a queen, 75 00:05:17,660 --> 00:05:20,952 an eight, a queen. A five, a queen. 76 00:05:20,952 --> 00:05:23,936 A jack, a queen. A four, an ace. 77 00:05:23,936 --> 00:05:27,950 Wow, look at those hands. That's amazing. 78 00:05:27,950 --> 00:05:33,272 You think they're about equally probable? What's the likelihood of that hand coming 79 00:05:33,272 --> 00:05:35,773 up? Most people would say, that's pretty 80 00:05:35,773 --> 00:05:38,531 slim. It's unlikely that you're going to get a 81 00:05:38,531 --> 00:05:41,993 hand with four queens and an Ace. What about this hand? 82 00:05:41,993 --> 00:05:45,905 That's the kind of hand I get all the time when I play poker. 83 00:05:45,905 --> 00:05:50,522 So most people are going to think that kind of hand is a lot more likely, but 84 00:05:50,522 --> 00:05:52,938 actually, they're equally probable. 85 00:05:52,938 --> 00:05:58,380 Getting exactly this hand and exactly that hand have the same probability. 86 00:05:58,380 --> 00:06:04,213 The reason people think this one is less likely than this one, is that you get 87 00:06:04,213 --> 00:06:09,000 kind of hand a lot in poker, namely, a hand that is junk, it's useless. 88 00:06:09,000 --> 00:06:13,030 You get a winning hand like this, not very often in poker. 89 00:06:13,030 --> 00:06:18,469 And so people think that this one is less likely than this one because this hand is 90 00:06:18,469 --> 00:06:23,584 more a representative of the type of hand that I get when I'm playing, and that 91 00:06:23,584 --> 00:06:28,101 most people get when they're playing. But that's what's called the 92 00:06:28,101 --> 00:06:32,809 representativeness heuristic. You think it's more probable because it's 93 00:06:32,809 --> 00:06:37,980 more representative of the kind of thing that you encounter and experience. 94 00:06:37,980 --> 00:06:42,622 Another famous example of the representativeness heuristic comes from 95 00:06:42,622 --> 00:06:44,810 kind of Kahneman and. Tversky. 96 00:06:44,810 --> 00:06:49,584 It's about Linda the Bank Teller. They tested a large number of subjects. 97 00:06:49,584 --> 00:06:55,351 What they did is simply describe Linda. Linda is 31 years old, single, outspoken, 98 00:06:55,351 --> 00:06:59,838 and very bright. As a student, she majored in philosophy. 99 00:06:59,838 --> 00:07:05,876 She was deeply concerned with issues of discrimination and social justice, 100 00:07:05,876 --> 00:07:10,500 and sh also participated in anti nuclear demonstrations. 101 00:07:10,500 --> 00:07:15,671 And yes there's subjects to rank the following statements with respect to 102 00:07:15,671 --> 00:07:19,375 probability. You know, which one is more probable than 103 00:07:19,375 --> 00:07:23,289 which of the others. Linda is a teacher in an elementary 104 00:07:23,289 --> 00:07:26,434 school. Linda works in a book store, and takes 105 00:07:26,434 --> 00:07:30,208 yoga classes. Linda is active in the feminist movement. 106 00:07:30,208 --> 00:07:34,401 Linda is a psychiatric social worker. Linda is a bank teller. 107 00:07:34,401 --> 00:07:39,712 Linda is an insurance sales person, and Linda is a bank teller and is active 108 00:07:39,712 --> 00:07:45,600 in the feminist movement. And when people ranked all of these, 109 00:07:45,600 --> 00:07:52,906 they tended to put Linda as a bank teller and is active in the feminist movement as 110 00:07:52,906 --> 00:07:57,460 more likely than Linda is a banker teller. 111 00:07:57,460 --> 00:08:00,197 That can't be. I mean, just think about it. 112 00:08:00,197 --> 00:08:05,473 It's got to be less likely that she's a bank teller and active in the feminist 113 00:08:05,473 --> 00:08:10,615 movement, and simply that she's a bank teller, because in every possible state 114 00:08:10,615 --> 00:08:15,891 of affairs where she's a bank teller and active in the feminist movement, she's 115 00:08:15,891 --> 00:08:19,536 also a bank teller. So there have to be some possibilities 116 00:08:19,536 --> 00:08:23,558 where she's a bank teller, but not active in the feminist movement. 117 00:08:23,558 --> 00:08:28,494 So it has to be more likely that she's a bank teller than that she's both a bank 118 00:08:28,494 --> 00:08:31,420 teller and also active in the feminist movement. 119 00:08:31,420 --> 00:08:34,195 And why do people make this mistake so often? 120 00:08:34,195 --> 00:08:39,005 It's largely because, when they trust their intuitions about probability, those 121 00:08:39,005 --> 00:08:42,706 intuitions are based on what they take to be representative. 122 00:08:42,706 --> 00:08:47,393 Someone with a background like Linda, who is deeply concerned with issues of 123 00:08:47,393 --> 00:08:51,957 discrimination and social justice, are likely to be active in the feminist 124 00:08:51,957 --> 00:08:54,917 movement, whereas bank tellers are not typically 125 00:08:54,917 --> 00:08:57,877 known for being active in the feminist movement. 126 00:08:57,877 --> 00:09:02,688 So it would be more typical for her to be both than for her to just be a bank 127 00:09:02,688 --> 00:09:06,647 teller. And so they based their probability 128 00:09:06,647 --> 00:09:09,272 judgment on the representativeness heuristic, 129 00:09:09,272 --> 00:09:12,090 and that's what leads to the common mistake. 130 00:09:12,090 --> 00:09:17,062 One final common mistake about probability is illustrated by what has 131 00:09:17,062 --> 00:09:22,105 come to be known as The Monty Hall problem or sometimes the three door 132 00:09:22,105 --> 00:09:25,585 problem. Here's the setup, which comes from an old 133 00:09:25,585 --> 00:09:30,700 TV show called Let's Make A Deal. There are three doors on the stage, lets 134 00:09:30,700 --> 00:09:35,814 call them door A, door B and door C. And behind one of the doors is a car, 135 00:09:35,814 --> 00:09:39,650 and if you get the right door, you get to keep the car. 136 00:09:39,650 --> 00:09:45,323 But behind the other two doors is a goat, and if you get those doors, you go home 137 00:09:45,323 --> 00:09:48,621 with a goat. Now, we're assuming that you'd rather 138 00:09:48,621 --> 00:09:52,861 have a car than a goat. You might be one of those weird peoples, 139 00:09:52,861 --> 00:09:56,025 but let's assume you want a car and not a goat. 140 00:09:56,025 --> 00:10:01,208 Imagine that you picked door A, and then the host, Monty Hall opens door C, and 141 00:10:01,208 --> 00:10:06,868 reveals behind door C a goat. Now you know that there's door A and door 142 00:10:06,868 --> 00:10:09,791 B left. One of them has a car, one of them has a 143 00:10:09,791 --> 00:10:12,584 goat. And Monty Hall turns to you and he offers 144 00:10:12,584 --> 00:10:16,222 you the chance to switch from door A, the one you picked, 145 00:10:16,222 --> 00:10:20,240 to door B. And the question is, should you switch? 146 00:10:20,240 --> 00:10:23,562 And this has actually caused a lot of controversy, 147 00:10:23,562 --> 00:10:28,081 because some people think you should switch, some people think you shouldn't 148 00:10:28,081 --> 00:10:31,004 switch. I'm not going to tell you, because that's 149 00:10:31,004 --> 00:10:35,722 going to be one of our exercises. And the point of this lecture is just to 150 00:10:35,722 --> 00:10:38,779 show you that you can't trust your intuitions. 151 00:10:38,779 --> 00:10:43,497 Because so many people will get it wrong in the Monty Hall case, in the 152 00:10:43,497 --> 00:10:48,880 representativeness heuristic, and so many people commit the gambler's fallacy that 153 00:10:48,880 --> 00:10:52,826 you need to take probability seriously, and in the next lecture we'll start 154 00:10:52,826 --> 00:10:56,825 looking at some definitions of what it is, and the lectures after that we'll 155 00:10:56,825 --> 00:11:00,982 look at rules for probability, because you need that kind of thing if you can't 156 00:11:00,982 --> 00:11:02,140 trust your intuitions.