WEBVTT 00:00:00.880 --> 00:00:03.053 You and your friend Jeremy are fishing in a 00:00:03.053 --> 00:00:06.164 pond that contains ten trout and ten sunfish. 00:00:06.568 --> 00:00:09.505 Each time one of you catches a fish 00:00:09.505 --> 00:00:11.568 you release it back into the water. 00:00:11.803 --> 00:00:14.928 Jeremy offers you the choice of two different bets. 00:00:15.193 --> 00:00:18.786 Bet number one. We don't encourage 00:00:18.786 --> 00:00:20.927 betting but I guess Jeremy wants to bet. 00:00:20.927 --> 00:00:25.477 If the next three fish he catches are all sunfish you will 00:00:25.477 --> 00:00:30.044 pay him 100 dollars, otherwise he will pay you 20 dollars. 00:00:30.334 --> 00:00:34.693 Bet two, if you catch at least two sunfish of the next 00:00:34.693 --> 00:00:39.959 three fish that you catch he will pay you 50 dollars, 00:00:40.177 --> 00:00:44.647 otherwise you will pay him 25 dollars. 00:00:45.021 --> 00:00:50.243 What is the expected value from bet one? 00:00:50.491 --> 00:00:52.915 Round your answer to the nearest cent. 00:00:52.915 --> 00:00:54.146 I encourage you to pause this video 00:00:54.146 --> 00:00:55.824 and try to think about it on your own. 00:00:56.380 --> 00:00:59.458 Let's see. The expected value of bet one. 00:00:59.741 --> 00:01:06.255 The expected value of bet one where we'll say bet one is -- 00:01:06.255 --> 00:01:08.772 Let's just define a random variable here 00:01:08.772 --> 00:01:10.646 just to be a little bit better about this. 00:01:10.646 --> 00:01:18.086 Let's say x is equal what you pay, or I guess 00:01:18.086 --> 00:01:19.601 you could say , because you might get something, 00:01:19.601 --> 00:01:28.023 what your profit is from bet one. 00:01:29.969 --> 00:01:31.391 It's a random variable. 00:01:31.625 --> 00:01:39.502 The expected value of x is going to be equal to, let's see. 00:01:39.502 --> 00:01:44.688 What's the probability, it's going to be negative 100 00:01:44.688 --> 00:01:48.750 dollars times the probability that he catches three fish. 00:01:49.187 --> 00:01:58.281 The probably that Jeremy catches three sunfish, 00:01:58.281 --> 00:02:02.718 the next three fish he catches are 00:02:02.718 --> 00:02:05.969 going to be sunfish, times 100 dollars. 00:02:07.522 --> 00:02:09.083 Or I should, well you're going to pay that. 00:02:09.083 --> 00:02:11.647 Since you're paying it we'll put it as negative 100 00:02:11.647 --> 00:02:13.832 because we're saying that this is your expected profit, 00:02:13.832 --> 00:02:15.319 so you're going to lose money there. 00:02:17.010 --> 00:02:20.501 That's going to be one minus this probability, 00:02:20.733 --> 00:02:27.541 the probability that Jeremy catches three sunfish. 00:02:29.002 --> 00:02:32.082 In that situation he'll pay you 20 dollars. 00:02:32.287 --> 00:02:34.002 You get 20 dollars there. 00:02:34.002 --> 00:02:35.941 The important thing is to figure out the 00:02:35.941 --> 00:02:38.613 probability that Jeremy catches three sunfish. 00:02:39.019 --> 00:02:43.768 Well the sunfish are 10 out of the 20 fish so any given 00:02:43.768 --> 00:02:49.221 time he's trying to catch fish there's a 10 in 20 chance, 00:02:49.221 --> 00:02:52.536 or you could say one half probability 00:02:52.536 --> 00:02:54.191 that it's going to be a sunfish. 00:02:54.460 --> 00:02:57.096 The probability that you get three sunfish in a row is 00:02:57.096 --> 00:03:01.165 going to be one half, times one half, times one half. 00:03:01.165 --> 00:03:04.421 They put the fish back in, that's why it stays 10 out 00:03:04.421 --> 00:03:07.046 of the 20 fish. If he wasn't putting the fish back in 00:03:07.046 --> 00:03:10.968 then the second sunfish you would have a nine out of 20 00:03:10.968 --> 00:03:13.860 chance of the second one being a sunfish. 00:03:13.860 --> 00:03:16.397 In this case they keep replacing the 00:03:16.397 --> 00:03:17.824 fish every time they catch it. 00:03:17.824 --> 00:03:21.132 There is a one eighth chance that Jeremy catches 00:03:21.132 --> 00:03:24.834 three sunfish, so this right over here is one eighth. 00:03:25.209 --> 00:03:28.772 And one minus one eighth, this is seven eighths. 00:03:31.010 --> 00:03:33.979 You have a one eighth chance of paying 100 dollars 00:03:33.979 --> 00:03:35.996 and a seven eights chance of getting 00:03:35.996 --> 00:03:38.875 twenty dollars so this gets us to ... 00:03:42.732 --> 00:03:45.562 Your expected profit here, 00:03:45.562 --> 00:03:50.044 there's a one eighths chance, one eighth probability, 00:03:50.044 --> 00:03:54.997 that you lose 100 dollars here, so times negative 100. 00:03:55.504 --> 00:04:00.508 But then there is a seven eighths chance that you get -- 00:04:01.261 --> 00:04:03.597 I'll just put parentheses here to make it clear. 00:04:03.597 --> 00:04:05.597 I think the order of operations of the 00:04:05.597 --> 00:04:07.301 calculator would have taken care of it but 00:04:07.301 --> 00:04:09.301 I'll just do it so that it looks the same. 00:04:09.301 --> 00:04:11.472 Seven eighths, there's a seven eighths 00:04:11.472 --> 00:04:14.456 chance that you get 20 dollars. 00:04:15.597 --> 00:04:20.098 Your expected payoff here is positive five dollars. 00:04:20.960 --> 00:04:27.223 Your expected payoff here is equal to five dollars. 00:04:27.540 --> 00:04:30.239 This is your expected value from bet one. 00:04:30.629 --> 00:04:32.223 Now let's think about bet two. 00:04:35.323 --> 00:04:39.086 If you catch at least two sunfish of the next three fish 00:04:39.086 --> 00:04:44.211 you catch he will pay you 50, otherwise you will pay him 25. 00:04:44.725 --> 00:04:48.365 Let's think about the probability of catching at least 00:04:48.365 --> 00:04:53.131 two sunfish of the next three fish that you catch. 00:04:53.574 --> 00:04:56.257 There's a bunch of ways to think about this but since 00:04:56.257 --> 00:05:00.616 there's only three times that you're trying to catch the 00:05:00.616 --> 00:05:02.756 fish and there's only one of two outcomes you could actually 00:05:02.756 --> 00:05:06.304 write all the possible outcomes that are possible here. 00:05:06.648 --> 00:05:11.757 You could get sunfish, sunfish, sunfish. 00:05:12.506 --> 00:05:15.007 You could get, what's the other type of fish that you have? 00:05:15.007 --> 00:05:18.758 Oh, trout. You could have sunfish, sunfish, trout. 00:05:18.758 --> 00:05:22.413 You could have sunfish, trout, sunfish. 00:05:22.413 --> 00:05:25.789 You could have sunfish, trout, trout. 00:05:26.054 --> 00:05:29.476 You could have trout, sunfish, sunfish. 00:05:29.790 --> 00:05:31.975 You could have trout, sunfish, trout. 00:05:32.226 --> 00:05:35.133 You could have trout, trout, sunfish. 00:05:35.746 --> 00:05:38.024 Or you could have all trout. 00:05:38.292 --> 00:05:45.538 You see here that each of these, each time you go there's 00:05:45.771 --> 00:05:48.628 two possibilities, each time you try to catch a fish there's 00:05:48.628 --> 00:05:51.020 two possibilities, so if you're doing it three times 00:05:51.020 --> 00:05:53.113 there's two times two times two possibilities. 00:05:53.113 --> 00:05:55.991 One, two, three, four, five, six, 00:05:55.991 --> 00:05:57.785 seven, eight possibilities here. 00:05:57.785 --> 00:06:01.423 Now out of these eight equally likely possibilities how 00:06:01.423 --> 00:06:05.194 many of them involve you catching at least two sunfish? 00:06:05.799 --> 00:06:09.907 You catch at least two sunfish in this one, in this one, 00:06:09.907 --> 00:06:15.439 in that one, in this one and I think that is it. 00:06:15.819 --> 00:06:18.238 Yep, this is only one sunfish, one sunfish, 00:06:18.238 --> 00:06:21.174 one sunfish and no sunfish. 00:06:21.519 --> 00:06:24.876 In four out of the eight equally likely 00:06:24.876 --> 00:06:27.877 outcomes you catch at least two sunfish. 00:06:28.388 --> 00:06:36.764 Your probability of catching at least two sunfish 00:06:38.075 --> 00:06:42.857 is equal to four eighths or one half. 00:06:43.450 --> 00:06:45.699 Let's see, what's the expected value? 00:06:45.980 --> 00:06:49.219 Let's say Y is the expected profit from bet. 00:06:49.941 --> 00:06:53.692 Let's let Y equals, another random variable, 00:06:53.692 --> 00:07:02.568 is equal to expected profit from bet two. 00:07:02.973 --> 00:07:07.026 The expected value of our random variable Y, 00:07:07.363 --> 00:07:10.911 you have a one half chance that you win. 00:07:10.911 --> 00:07:16.411 You have a one half chance of getting 50 dollars and then 00:07:16.411 --> 00:07:20.024 you have the one half chance, the rest of the probability. 00:07:20.505 --> 00:07:21.770 If there's a one half chance you win 00:07:21.770 --> 00:07:23.761 there's going to be a one minus one half or 00:07:23.761 --> 00:07:26.099 essentially a one half chance that you lose. 00:07:26.556 --> 00:07:31.727 So you have a one half chance of having to pay 25 dollars. 00:07:32.679 --> 00:07:34.316 Let's see what this is. 00:07:34.316 --> 00:07:38.286 This is one half times 50 plus one half times negative 25. 00:07:38.598 --> 00:07:48.708 This is going to be 25 minus 12.50, which is equal to 12.50. 00:07:49.618 --> 00:07:52.926 Your expected value from bet two is 12.50. 00:07:53.692 --> 00:07:55.552 Your friend says he's willing to take 00:07:55.552 --> 00:07:59.616 both bets a combined total of 50 times. 00:08:00.038 --> 00:08:03.004 If you want to maximize your expected value what should you do? 00:08:04.025 --> 00:08:07.021 Well bet number two -- Actually both of them are good bets, 00:08:07.021 --> 00:08:09.801 I guess your friend isn't that sophisticated, 00:08:09.801 --> 00:08:14.740 but bet number two has a higher expected payoff, 00:08:14.740 --> 00:08:17.902 so I would take bet two all of the time. 00:08:18.396 --> 00:08:22.301 I would take bet two all of the time.