1 00:00:00,880 --> 00:00:03,053 You and your friend Jeremy are fishing in a 2 00:00:03,053 --> 00:00:06,164 pond that contains ten trout and ten sunfish. 3 00:00:06,568 --> 00:00:09,505 Each time one of you catches a fish 4 00:00:09,505 --> 00:00:11,568 you release it back into the water. 5 00:00:11,803 --> 00:00:14,928 Jeremy offers you the choice of two different bets. 6 00:00:15,193 --> 00:00:18,786 Bet number one. We don't encourage 7 00:00:18,786 --> 00:00:20,927 betting but I guess Jeremy wants to bet. 8 00:00:20,927 --> 00:00:25,477 If the next three fish he catches are all sunfish you will 9 00:00:25,477 --> 00:00:30,044 pay him 100 dollars, otherwise he will pay you 20 dollars. 10 00:00:30,334 --> 00:00:34,693 Bet two, if you catch at least two sunfish of the next 11 00:00:34,693 --> 00:00:39,959 three fish that you catch he will pay you 50 dollars, 12 00:00:40,177 --> 00:00:44,647 otherwise you will pay him 25 dollars. 13 00:00:45,021 --> 00:00:50,243 What is the expected value from bet one? 14 00:00:50,491 --> 00:00:52,915 Round your answer to the nearest cent. 15 00:00:52,915 --> 00:00:54,146 I encourage you to pause this video 16 00:00:54,146 --> 00:00:55,824 and try to think about it on your own. 17 00:00:56,380 --> 00:00:59,458 Let's see. The expected value of bet one. 18 00:00:59,741 --> 00:01:06,255 The expected value of bet one where we'll say bet one is -- 19 00:01:06,255 --> 00:01:08,772 Let's just define a random variable here 20 00:01:08,772 --> 00:01:10,646 just to be a little bit better about this. 21 00:01:10,646 --> 00:01:18,086 Let's say x is equal what you pay, or I guess 22 00:01:18,086 --> 00:01:19,601 you could say , because you might get something, 23 00:01:19,601 --> 00:01:28,023 what your profit is from bet one. 24 00:01:29,969 --> 00:01:31,391 It's a random variable. 25 00:01:31,625 --> 00:01:39,502 The expected value of x is going to be equal to, let's see. 26 00:01:39,502 --> 00:01:44,688 What's the probability, it's going to be negative 100 27 00:01:44,688 --> 00:01:48,750 dollars times the probability that he catches three fish. 28 00:01:49,187 --> 00:01:58,281 The probably that Jeremy catches three sunfish, 29 00:01:58,281 --> 00:02:02,718 the next three fish he catches are 30 00:02:02,718 --> 00:02:05,969 going to be sunfish, times 100 dollars. 31 00:02:07,522 --> 00:02:09,083 Or I should, well you're going to pay that. 32 00:02:09,083 --> 00:02:11,647 Since you're paying it we'll put it as negative 100 33 00:02:11,647 --> 00:02:13,832 because we're saying that this is your expected profit, 34 00:02:13,832 --> 00:02:15,319 so you're going to lose money there. 35 00:02:17,010 --> 00:02:20,501 That's going to be one minus this probability, 36 00:02:20,733 --> 00:02:27,541 the probability that Jeremy catches three sunfish. 37 00:02:29,002 --> 00:02:32,082 In that situation he'll pay you 20 dollars. 38 00:02:32,287 --> 00:02:34,002 You get 20 dollars there. 39 00:02:34,002 --> 00:02:35,941 The important thing is to figure out the 40 00:02:35,941 --> 00:02:38,613 probability that Jeremy catches three sunfish. 41 00:02:39,019 --> 00:02:43,768 Well the sunfish are 10 out of the 20 fish so any given 42 00:02:43,768 --> 00:02:49,221 time he's trying to catch fish there's a 10 in 20 chance, 43 00:02:49,221 --> 00:02:52,536 or you could say one half probability 44 00:02:52,536 --> 00:02:54,191 that it's going to be a sunfish. 45 00:02:54,460 --> 00:02:57,096 The probability that you get three sunfish in a row is 46 00:02:57,096 --> 00:03:01,165 going to be one half, times one half, times one half. 47 00:03:01,165 --> 00:03:04,421 They put the fish back in, that's why it stays 10 out 48 00:03:04,421 --> 00:03:07,046 of the 20 fish. If he wasn't putting the fish back in 49 00:03:07,046 --> 00:03:10,968 then the second sunfish you would have a nine out of 20 50 00:03:10,968 --> 00:03:13,860 chance of the second one being a sunfish. 51 00:03:13,860 --> 00:03:16,397 In this case they keep replacing the 52 00:03:16,397 --> 00:03:17,824 fish every time they catch it. 53 00:03:17,824 --> 00:03:21,132 There is a one eighth chance that Jeremy catches 54 00:03:21,132 --> 00:03:24,834 three sunfish, so this right over here is one eighth. 55 00:03:25,209 --> 00:03:28,772 And one minus one eighth, this is seven eighths. 56 00:03:31,010 --> 00:03:33,979 You have a one eighth chance of paying 100 dollars 57 00:03:33,979 --> 00:03:35,996 and a seven eights chance of getting 58 00:03:35,996 --> 00:03:38,875 twenty dollars so this gets us to ... 59 00:03:42,732 --> 00:03:45,562 Your expected profit here, 60 00:03:45,562 --> 00:03:50,044 there's a one eighths chance, one eighth probability, 61 00:03:50,044 --> 00:03:54,997 that you lose 100 dollars here, so times negative 100. 62 00:03:55,504 --> 00:04:00,508 But then there is a seven eighths chance that you get -- 63 00:04:01,261 --> 00:04:03,597 I'll just put parentheses here to make it clear. 64 00:04:03,597 --> 00:04:05,597 I think the order of operations of the 65 00:04:05,597 --> 00:04:07,301 calculator would have taken care of it but 66 00:04:07,301 --> 00:04:09,301 I'll just do it so that it looks the same. 67 00:04:09,301 --> 00:04:11,472 Seven eighths, there's a seven eighths 68 00:04:11,472 --> 00:04:14,456 chance that you get 20 dollars. 69 00:04:15,597 --> 00:04:20,098 Your expected payoff here is positive five dollars. 70 00:04:20,960 --> 00:04:27,223 Your expected payoff here is equal to five dollars. 71 00:04:27,540 --> 00:04:30,239 This is your expected value from bet one. 72 00:04:30,629 --> 00:04:32,223 Now let's think about bet two. 73 00:04:35,323 --> 00:04:39,086 If you catch at least two sunfish of the next three fish 74 00:04:39,086 --> 00:04:44,211 you catch he will pay you 50, otherwise you will pay him 25. 75 00:04:44,725 --> 00:04:48,365 Let's think about the probability of catching at least 76 00:04:48,365 --> 00:04:53,131 two sunfish of the next three fish that you catch. 77 00:04:53,574 --> 00:04:56,257 There's a bunch of ways to think about this but since 78 00:04:56,257 --> 00:05:00,616 there's only three times that you're trying to catch the 79 00:05:00,616 --> 00:05:02,756 fish and there's only one of two outcomes you could actually 80 00:05:02,756 --> 00:05:06,304 write all the possible outcomes that are possible here. 81 00:05:06,648 --> 00:05:11,757 You could get sunfish, sunfish, sunfish. 82 00:05:12,506 --> 00:05:15,007 You could get, what's the other type of fish that you have? 83 00:05:15,007 --> 00:05:18,758 Oh, trout. You could have sunfish, sunfish, trout. 84 00:05:18,758 --> 00:05:22,413 You could have sunfish, trout, sunfish. 85 00:05:22,413 --> 00:05:25,789 You could have sunfish, trout, trout. 86 00:05:26,054 --> 00:05:29,476 You could have trout, sunfish, sunfish. 87 00:05:29,790 --> 00:05:31,975 You could have trout, sunfish, trout. 88 00:05:32,226 --> 00:05:35,133 You could have trout, trout, sunfish. 89 00:05:35,746 --> 00:05:38,024 Or you could have all trout. 90 00:05:38,292 --> 00:05:45,538 You see here that each of these, each time you go there's 91 00:05:45,771 --> 00:05:48,628 two possibilities, each time you try to catch a fish there's 92 00:05:48,628 --> 00:05:51,020 two possibilities, so if you're doing it three times 93 00:05:51,020 --> 00:05:53,113 there's two times two times two possibilities. 94 00:05:53,113 --> 00:05:55,991 One, two, three, four, five, six, 95 00:05:55,991 --> 00:05:57,785 seven, eight possibilities here. 96 00:05:57,785 --> 00:06:01,423 Now out of these eight equally likely possibilities how 97 00:06:01,423 --> 00:06:05,194 many of them involve you catching at least two sunfish? 98 00:06:05,799 --> 00:06:09,907 You catch at least two sunfish in this one, in this one, 99 00:06:09,907 --> 00:06:15,439 in that one, in this one and I think that is it. 100 00:06:15,819 --> 00:06:18,238 Yep, this is only one sunfish, one sunfish, 101 00:06:18,238 --> 00:06:21,174 one sunfish and no sunfish. 102 00:06:21,519 --> 00:06:24,876 In four out of the eight equally likely 103 00:06:24,876 --> 00:06:27,877 outcomes you catch at least two sunfish. 104 00:06:28,388 --> 00:06:36,764 Your probability of catching at least two sunfish 105 00:06:38,075 --> 00:06:42,857 is equal to four eighths or one half. 106 00:06:43,450 --> 00:06:45,699 Let's see, what's the expected value? 107 00:06:45,980 --> 00:06:49,219 Let's say Y is the expected profit from bet. 108 00:06:49,941 --> 00:06:53,692 Let's let Y equals, another random variable, 109 00:06:53,692 --> 00:07:02,568 is equal to expected profit from bet two. 110 00:07:02,973 --> 00:07:07,026 The expected value of our random variable Y, 111 00:07:07,363 --> 00:07:10,911 you have a one half chance that you win. 112 00:07:10,911 --> 00:07:16,411 You have a one half chance of getting 50 dollars and then 113 00:07:16,411 --> 00:07:20,024 you have the one half chance, the rest of the probability. 114 00:07:20,505 --> 00:07:21,770 If there's a one half chance you win 115 00:07:21,770 --> 00:07:23,761 there's going to be a one minus one half or 116 00:07:23,761 --> 00:07:26,099 essentially a one half chance that you lose. 117 00:07:26,556 --> 00:07:31,727 So you have a one half chance of having to pay 25 dollars. 118 00:07:32,679 --> 00:07:34,316 Let's see what this is. 119 00:07:34,316 --> 00:07:38,286 This is one half times 50 plus one half times negative 25. 120 00:07:38,598 --> 00:07:48,708 This is going to be 25 minus 12.50, which is equal to 12.50. 121 00:07:49,618 --> 00:07:52,926 Your expected value from bet two is 12.50. 122 00:07:53,692 --> 00:07:55,552 Your friend says he's willing to take 123 00:07:55,552 --> 00:07:59,616 both bets a combined total of 50 times. 124 00:08:00,038 --> 00:08:03,004 If you want to maximize your expected value what should you do? 125 00:08:04,025 --> 00:08:07,021 Well bet number two -- Actually both of them are good bets, 126 00:08:07,021 --> 00:08:09,801 I guess your friend isn't that sophisticated, 127 00:08:09,801 --> 00:08:14,740 but bet number two has a higher expected payoff, 128 00:08:14,740 --> 00:08:17,902 so I would take bet two all of the time. 129 00:08:18,396 --> 00:08:22,301 I would take bet two all of the time.