1 00:00:02,540 --> 00:00:06,519 We've talked about using probability to determine what to believe. 2 00:00:06,519 --> 00:00:10,981 Such as, whether to believe that a certain form of contraception will work 3 00:00:10,981 --> 00:00:15,322 or will probably work. Or whether to believe that you have cancer when a 4 00:00:15,322 --> 00:00:19,723 doctor has done a test and you got a positive result which can be pretty 5 00:00:19,723 --> 00:00:22,557 scary. But you still have to figure out whether 6 00:00:22,557 --> 00:00:25,090 to believe that you really do have cancer. 7 00:00:25,090 --> 00:00:29,672 Now, probability can be very useful in the formation of beliefs. 8 00:00:29,672 --> 00:00:33,381 But it's also extremely useful in making decisions. 9 00:00:33,381 --> 00:00:38,618 So, in this lecture, we want to talk about the use of probability to make 10 00:00:38,618 --> 00:00:42,618 decisions or choices. There are three kinds of decisions or 11 00:00:42,618 --> 00:00:47,710 choices that we need to discuss. First of all, decisions with certainty. 12 00:00:47,710 --> 00:00:53,661 Second, decisions with risks. And third, decisions with ignorance. 13 00:00:53,661 --> 00:00:59,330 Or, as those are sometimes called, decisions with uncertainty. 14 00:00:59,330 --> 00:01:04,253 And we'll talk about the first two and then I'll just say a few words about the 15 00:01:04,253 --> 00:01:06,714 third. First, let's look at decisions with 16 00:01:06,714 --> 00:01:09,780 certainty. A decision with certainty is one where 17 00:01:09,780 --> 00:01:14,556 you know exactly what's going to happen, you know what the effects are going to be 18 00:01:14,556 --> 00:01:18,264 and you know it definitely. Its not that its probably going to 19 00:01:18,264 --> 00:01:20,526 happen, you know what's going to happen. 20 00:01:20,526 --> 00:01:23,417 You know the good effects and the bad effects. 21 00:01:23,417 --> 00:01:26,433 So there's no uncertainty, everything is certain. 22 00:01:26,433 --> 00:01:30,958 For example, suppose you go to a restaurant, an Italian restaurant, with a 23 00:01:30,958 --> 00:01:34,477 friend of yours. And, you have to decide whether to order 24 00:01:34,477 --> 00:01:38,756 pasta or pizza. Some really nice eggplant Parmesan on top 25 00:01:38,756 --> 00:01:41,613 of pasta that you like at this restaurant. 26 00:01:41,613 --> 00:01:44,470 They also have really good mushroom pizza. 27 00:01:44,470 --> 00:01:49,061 So you have to think about that, but you know that if you order the eggplant 28 00:01:49,061 --> 00:01:52,082 Parmesan pasta, then they're going to bring it to you. 29 00:01:52,082 --> 00:01:56,613 And you know that if you order the mushroom pizza, they'll make one for you, 30 00:01:56,613 --> 00:02:00,782 and you've got enough money for both of them so there's no uncertainty. 31 00:02:00,782 --> 00:02:04,709 You know what you're going to get. Still might be a difficult choice 32 00:02:04,709 --> 00:02:09,241 because, well, you might not be sure whether you feel more like the eggplant 33 00:02:09,241 --> 00:02:11,960 Parmesan pasta tonight or the mushroom pizza. 34 00:02:11,960 --> 00:02:16,216 And, they're going to be also potentially a conflict of values. 35 00:02:16,216 --> 00:02:20,545 Maybe the eggplant Parmesan pasta costs more than the pizza. 36 00:02:20,545 --> 00:02:25,595 In which case you have to decide, well, even though I like it more, do I 37 00:02:25,595 --> 00:02:30,566 want to spend that money right now? You got a conflict between what you feel 38 00:02:30,566 --> 00:02:34,921 like and how much it's going to cost. Or, there might be a conflict with your 39 00:02:34,921 --> 00:02:37,845 friend. Your friend wants to buy the mushroom 40 00:02:37,845 --> 00:02:43,060 pizza and split it with you because he doesn't want an entire pizza himself. 41 00:02:43,060 --> 00:02:45,869 But you really feel like the eggplant Parmesan. 42 00:02:45,869 --> 00:02:49,755 So now, you've got a conflict between your loyalty to your friend. 43 00:02:49,755 --> 00:02:53,880 You want to support your friend and help your friend get what he wants. 44 00:02:53,880 --> 00:02:57,048 And yet, you feel like the eggplant Parmesan yourself. 45 00:02:57,048 --> 00:03:01,950 So, these decisions can be very difficult in some cases but there's no uncertainty. 46 00:03:01,950 --> 00:03:06,254 You still know that if you order the eggplant Parmesan you're going to get 47 00:03:06,254 --> 00:03:09,662 eggplant Parmesan, or if you order the pizza you're going to 48 00:03:09,662 --> 00:03:11,327 get pizza. Right? 49 00:03:11,327 --> 00:03:16,544 There's not uncertainty at all. No, of course there's uncertainty. 50 00:03:16,544 --> 00:03:19,658 You might order the pizza and it comes burnt. 51 00:03:19,658 --> 00:03:21,527 Yuck. Oh, I hate burnt pizza. 52 00:03:21,527 --> 00:03:27,411 And you might order the eggplant Parmesan and it turns out that they really messed 53 00:03:27,411 --> 00:03:31,425 up the sauce, and it's got way too much salt in it tonight. 54 00:03:31,425 --> 00:03:36,824 Maybe they have a new chef in the kitchen who's going to mess up one but not the 55 00:03:36,824 --> 00:03:40,073 other. It's not really certain that when you 56 00:03:40,073 --> 00:03:44,360 order it, you're going to get what you wanted or that you're going to like it. 57 00:03:45,440 --> 00:03:50,341 Now, this is just a fact about life. There's nothing really certain in this 58 00:03:50,341 --> 00:03:53,190 life. You might think that you know exactly 59 00:03:53,190 --> 00:03:57,164 what's going to happen. But there's always at least some chance 60 00:03:57,164 --> 00:04:00,410 that it's not going to turn out the way you thought. 61 00:04:00,410 --> 00:04:04,716 And so, what do you do about that? Well, some people just stipulate. 62 00:04:04,716 --> 00:04:08,956 If I order the pizza I'll get the pizza. And it'll be good pizza. 63 00:04:08,956 --> 00:04:13,659 Well, you can of course stipulate that. But then, it just makes the whole 64 00:04:13,659 --> 00:04:17,236 situation unrealistic. Cause in the real life, you're never 65 00:04:17,236 --> 00:04:21,320 going to know for certain. So another possibility is to say, we're 66 00:04:21,320 --> 00:04:26,047 just going to ignore these slight chances that the pizza's going to come back 67 00:04:26,047 --> 00:04:29,068 burned. I've been at this restaurant a hundred 68 00:04:29,068 --> 00:04:32,350 times, before they've always done it just right. 69 00:04:32,350 --> 00:04:36,093 And so, what are the odds they're going to burn it this time? 70 00:04:36,093 --> 00:04:39,113 Pretty slim. 99% chance they're going to do it the 71 00:04:39,113 --> 00:04:41,748 right way. Well, you might want to say, I'm just 72 00:04:41,748 --> 00:04:46,285 going to ignore the 1% chance of getting burnt because it's not going to make any 73 00:04:46,285 --> 00:04:49,194 difference. I still want the pizza or I still want 74 00:04:49,194 --> 00:04:50,532 the eggplant, whichever. 75 00:04:50,532 --> 00:04:54,430 The other thing you might do is say, look, nothing's certain, but the 76 00:04:54,430 --> 00:04:57,281 probabilities are balanced. They go on both sides. 77 00:04:57,281 --> 00:05:00,073 After all, there's a chance of burning the pizza, 78 00:05:00,073 --> 00:05:02,750 there's also a chance of burning the eggplant. 79 00:05:02,750 --> 00:05:08,496 So, in some cases it might make perfectly good sense to ignore these uncertainties 80 00:05:08,496 --> 00:05:13,050 because they're not going to effect, your decision making process. 81 00:05:13,050 --> 00:05:17,325 Still, it's important to realize that in life, throughout life, 82 00:05:17,325 --> 00:05:21,670 in all areas of life, there's always going to be some uncertainty. 83 00:05:21,670 --> 00:05:26,505 And that means, there's going to be a probability that if you look at it 84 00:05:26,505 --> 00:05:29,588 seriously, it's going to play into the decision 85 00:05:29,588 --> 00:05:30,850 making process. So, 86 00:05:30,850 --> 00:05:37,089 the kinds of cases that we want to focus on are the cases of risk where there is a 87 00:05:37,089 --> 00:05:43,178 probability of failure and a probability of success for the different plans that 88 00:05:43,178 --> 00:05:48,140 you have to choose among. All realistic examples then are actually 89 00:05:48,140 --> 00:05:52,500 decisions under a risk, especially the most important ones. 90 00:05:52,500 --> 00:05:57,688 So, what is a decision under risk? It's simply a case where you're not 91 00:05:57,688 --> 00:06:01,140 absolutely certain what's going to happen. 92 00:06:01,140 --> 00:06:05,231 It's not definitely this effect will occur, that effect will occur. 93 00:06:05,231 --> 00:06:07,649 You'll get your pizza, it'll taste good. 94 00:06:07,649 --> 00:06:10,066 You'll get your pasta, it'll taste good. 95 00:06:10,066 --> 00:06:12,918 Instead, there's a probability of each outcome. 96 00:06:12,918 --> 00:06:16,575 There's a probability that the pizza will get to the table. 97 00:06:16,575 --> 00:06:19,861 There's a probability that the pizza will taste good. 98 00:06:19,861 --> 00:06:24,634 There's a probability that the pizza will never make it to the table but get 99 00:06:24,634 --> 00:06:29,718 dropped by the waiter, or that the pizza will taste horrible because it was burnt. 100 00:06:29,718 --> 00:06:34,707 Then, you have a decision under risk. And how do we make these decisions under 101 00:06:34,707 --> 00:06:37,329 risk? Well, there are a lot of different 102 00:06:37,329 --> 00:06:41,970 theories,. But one, pretty common one, is to apply expected value theory. 103 00:06:41,970 --> 00:06:46,946 Now, we're going to see that there are lots of different values, and it's hard 104 00:06:46,946 --> 00:06:51,788 to see how to handle them all, but let's start with the really simple case. 105 00:06:51,788 --> 00:06:55,285 Let's assume for now that all that matters, is money. 106 00:06:55,285 --> 00:07:00,665 One common way, perhaps the most common way, of figuring this out is to calculate 107 00:07:00,665 --> 00:07:04,700 the expected monetary value or the expected financial value. 108 00:07:04,700 --> 00:07:11,861 And that calculation is pretty simple. It's the probability of winning times the 109 00:07:11,861 --> 00:07:17,270 net gain if you do win, minus the probability of losing times the 110 00:07:17,270 --> 00:07:22,295 net loss if you do lose. And here we're talking about gains and 111 00:07:22,295 --> 00:07:27,900 losses in terms of money. But why do we say net loss and net gain? 112 00:07:27,900 --> 00:07:34,900 Imagine, you buy a lottery ticket. And you win $20 million and they sent you 113 00:07:34,900 --> 00:07:39,442 a check for $20 million Well, how much did you win? 114 00:07:39,442 --> 00:07:43,706 Sounds like you won $20 million but not really. 115 00:07:43,706 --> 00:07:49,268 You never got that dollar back that you paid for the ticket. 116 00:07:49,268 --> 00:07:56,874 So really, you only won $19,999,999. You know, that's pretty trivial if you 117 00:07:56,874 --> 00:07:59,445 just won the lottery. No big deal. 118 00:07:59,445 --> 00:08:02,640 But now, imagine a different circumstance. 119 00:08:02,640 --> 00:08:05,991 When you're playing poker and you bet $10. 120 00:08:05,991 --> 00:08:11,757 And the opponent raises it to twenty. And you're bluffing so you want to bet 121 00:08:11,757 --> 00:08:13,238 big. So you bet $200. 122 00:08:13,238 --> 00:08:18,614 Put that all in the pot, too. And your opponent backs down and you pull 123 00:08:18,614 --> 00:08:22,355 in the entire pot. How much has it got in it? $230. 124 00:08:22,355 --> 00:08:25,861 Well, did you win $230? No, because you put in 210. 125 00:08:25,861 --> 00:08:30,408 You only won twenty. The net winnings are how much you pull in 126 00:08:30,408 --> 00:08:34,748 minus how much you put in before you pulled in your winnings. 127 00:08:34,748 --> 00:08:39,016 So it's 230 minus the 210 you put in, you only gained twenty. 128 00:08:39,016 --> 00:08:44,424 A lot of people forget that in poker and they end up thinking that they win a lot 129 00:08:44,424 --> 00:08:49,760 by bluffing when actually they didn't win as much as they thought they did. 130 00:08:49,760 --> 00:08:55,371 In any case, we have to look at the net gain and the net loss when applying this 131 00:08:55,371 --> 00:09:00,702 formula for expected financial value. Let's do a few examples of calculating 132 00:09:00,702 --> 00:09:06,243 expected financial values, and let's do them with a deck of cards because we're 133 00:09:06,243 --> 00:09:10,312 familiar with that. There are thirteen different cards and 134 00:09:10,312 --> 00:09:14,100 four different suits, spades, hearts, diamonds and clubs. 135 00:09:14,100 --> 00:09:18,153 So we have to remember that to calculate the probabilities. 136 00:09:18,153 --> 00:09:24,138 Let's imagine that you can make this bet. If you bet a dollar, you'll win $5 net if 137 00:09:24,138 --> 00:09:28,264 you pick a spade. Now, notice that it's net. So that means 138 00:09:28,264 --> 00:09:34,086 either you put a dollar in, and then you take $6 out, and your net gain is five. 139 00:09:34,086 --> 00:09:39,907 Or you say, if it's not a spade I'll give you a dollar, and if it is a spade you 140 00:09:39,907 --> 00:09:43,665 give me $5. Either way, if you pick a spade, you're 141 00:09:43,665 --> 00:09:47,791 going to be $5 up. What's the expected financial value of 142 00:09:47,791 --> 00:09:51,402 this bet? Well, we know that the odds of picking a 143 00:09:51,402 --> 00:09:56,326 spade is one in four. That's the odds of winning because if you 144 00:09:56,326 --> 00:10:00,023 pick a spade you'll win. And the net gain is $5. 145 00:10:00,023 --> 00:10:05,650 So, to calculate the expected financial value, we need to multiply this 146 00:10:05,650 --> 00:10:11,920 one-quarter, the odds of picking a spade, times five, the number of dollars that 147 00:10:11,920 --> 00:10:17,455 you win, net if you win. And then, we have to subtract the odds of 148 00:10:17,455 --> 00:10:22,993 loosing three quarters, because if you pick a heart, a diamond, or a club, then 149 00:10:22,993 --> 00:10:26,581 you loose, and what do you loose? You loose $1. 150 00:10:26,581 --> 00:10:32,041 So, one quarter times five is five quarters, three quarter times one is 151 00:10:32,041 --> 00:10:38,202 three quarters, subtract three quarters from five quarters, you get two quarters 152 00:10:38,202 --> 00:10:43,584 and that of course is a half. So the expected financial value of this 153 00:10:43,584 --> 00:10:48,220 bet is one-half. Now, compare that bet to another bet. 154 00:10:48,220 --> 00:10:52,419 In this bet, if you bet a dollar, you're going to win ten. 155 00:10:52,419 --> 00:10:57,252 That sounds pretty good. Instead of winning five, you're going to 156 00:10:57,252 --> 00:11:01,056 win ten. But, you only win $10 net if you pick an 157 00:11:01,056 --> 00:11:04,542 ace. And there are thirteen cards, so we have 158 00:11:04,542 --> 00:11:09,772 to redo the calculations. And the probability of picking an ace is 159 00:11:09,772 --> 00:11:13,144 one in thirteen. The amount you win is ten. 160 00:11:13,144 --> 00:11:19,134 That's your net winnings, remember. You have to subtract the probability of 161 00:11:19,134 --> 00:11:25,853 losing, twelve in thirteen because you're going to lose if it's any card other than 162 00:11:25,853 --> 00:11:31,580 an ace. And the amount you'll lose is $1 net. 163 00:11:31,580 --> 00:11:39,867 10 * 13 = 13/10. Subtract 12/13 * 1, which is 12/13, and 164 00:11:39,867 --> 00:11:44,164 you get -2/13. Notice that's -2/13. 165 00:11:45,391 --> 00:11:50,302 That means the expected financial value of that bet is negative. 166 00:11:50,302 --> 00:11:56,518 You ought to expect to lose money over the long run if you play that kind of bet 167 00:11:56,518 --> 00:11:58,052 a lot. Third example. 168 00:11:58,052 --> 00:12:08,409 This time you can bet a dollar and win $51 net if you pick an ace of spades. 169 00:12:08,409 --> 00:12:12,547 But you can't pick any old ace, or any old spades. 170 00:12:12,547 --> 00:12:18,459 It's got to be the ace of spades. So now, what's the financial value of 171 00:12:18,459 --> 00:12:21,922 that bet? The expected financial value is 172 00:12:21,922 --> 00:12:28,341 probability of winning one over 52, times the amount you win net, which is 51, 173 00:12:28,341 --> 00:12:35,097 minus the probability of losing 51 over 52, times the amount you lose net, if you 174 00:12:35,097 --> 00:12:38,785 lose. 51 * 1/52 = 51/52. 175 00:12:40,343 --> 00:12:47,520 51/52 * 1 is 51/52. Subtract them and you get zero. 176 00:12:47,520 --> 00:12:54,000 So in this bet, the expected financial value of the bet is zero. 177 00:12:55,220 --> 00:12:58,978 Okay? Now, we can compare these two bets by 178 00:12:58,978 --> 00:13:04,531 asking what bet should you make. Well, the first bet was favorable. 179 00:13:04,531 --> 00:13:09,656 A bet's favorable if its expected value is greater than zero. 180 00:13:09,656 --> 00:13:16,661 And that means that it's in your interest to bet, at least if your only interest is 181 00:13:16,661 --> 00:13:20,762 money. A bet's unfavorable if its expected value 182 00:13:20,762 --> 00:13:25,375 is less than zero. And a bet is neutral or fair if its 183 00:13:25,375 --> 00:13:30,493 expected value is zero. And if you have a bet that's favorable, a 184 00:13:30,493 --> 00:13:35,748 bet that's unfavorable, and a bet that's neutral, it shouldn't be surprising that 185 00:13:35,748 --> 00:13:40,807 the one you should pick is the one that's favorable where the expected value is 186 00:13:40,807 --> 00:13:43,763 greater than zero, if you're going to bet at all. 187 00:13:43,763 --> 00:13:47,245 And that's the next question that we have to turn to.