Transcription
You and your friend Jeremy are fishing in a pond that contains ten trout and ten sunfish. Each time one of you catches a fish, you release it back into the water. Jeremy offers you the choice of two different bets.
Bet number one: If the next three fish he catches are all sunfish, you will pay him 100 dollars; otherwise, he will pay you 20 dollars.
Bet two: If you catch at least two sunfish of the next three fish that you catch, he will pay you 50 dollars; otherwise, you will pay him 25 dollars.
What is the expected value from bet one? Round your answer to the nearest cent. I encourage you to pause this video and try to think about it on your own.
Let's see. The expected value of bet one. Let's define a random variable here: let's say x is equal to what your profit is from bet one. It's a random variable. The expected value of x is going to be equal to: What's the probability? It's going to be negative 100 dollars times the probability that he catches three sunfish. The probability that Jeremy catches three sunfish—the next three fish he catches are going to be sunfish—times 100 dollars. Or, I should say, you're going to pay that. Since you're paying it, we'll put it as negative 100 because we're saying that this is your expected profit, so you're going to lose money there. That's going to be one minus this probability, the probability that Jeremy catches three sunfish. In that situation, he'll pay you 20 dollars. You get 20 dollars there.
The important thing is to figure out the probability that Jeremy catches three sunfish. Well, the sunfish are 10 out of the 20 fish, so any given time he's trying to catch fish, there's a 10 in 20 chance, or you could say a one-half probability that it's going to be a sunfish. The probability that you get three sunfish in a row is going to be one-half, times one-half, times one-half. They put the fish back in; that's why it stays 10 out of 20 fish. If he wasn't putting the fish back in, then the second sunfish you would have a nine out of 20 chance of the second one being a sunfish. In this case, they keep replacing the fish every time they catch it. There is a one-eighth chance that Jeremy catches three sunfish, so this right over here is one-eighth. And one minus one-eighth is seven-eighths. You have a one-eighth chance of paying 100 dollars and a seven-eighths chance of getting twenty dollars, so this gets us to... Your expected profit here: there's a one-eighth chance, one-eighth probability, that you lose 100 dollars here, so times negative 100. But then there is a seven-eighths chance that you get—I'll just put parentheses here to make it clear—seven-eighths, there's a seven-eighths chance that you get 20 dollars. Your expected payoff here is positive five dollars. Your expected payoff here is equal to five dollars. This is your expected value from bet one.
Now let's think about bet two. If you catch at least two sunfish of the next three fish you catch, he will pay you 50; otherwise, you will pay him 25. Let's think about the probability of catching at least two sunfish of the next three fish that you catch. There's a bunch of ways to think about this, but since there's only three times that you're trying to catch the fish and there's only one of two outcomes, you could actually write all the possible outcomes that are possible here. You could get sunfish, sunfish, sunfish. You could get—what's the other type of fish that you have?—oh, trout. You could have sunfish, sunfish, trout. You could have sunfish, trout, sunfish. You could have sunfish, trout, trout. You could have trout, sunfish, sunfish. You could have trout, sunfish, trout. You could have trout, trout, sunfish. Or you could have all trout. You see here that each of these, each time you go, there's two possibilities; each time you try to catch a fish, there's two possibilities, so if you're doing it three times, there's two times two times two possibilities. One, two, three, four, five, six, seven, eight possibilities here. Now, out of these eight equally likely possibilities, how many of them involve you catching at least two sunfish? You catch at least two sunfish in this one, in this one, in that one, in this one, and I think that is it. Yep, this is only one sunfish, one sunfish, one sunfish, and no sunfish. In four out of the eight equally likely outcomes, you catch at least two sunfish. Your probability of catching at least two sunfish is equal to four-eighths, or one-half.
Let's see, what's the expected value? Let's say Y is the expected profit from bet two. Let's let Y equals, another random variable, is equal to expected profit from bet two. The expected value of our random variable Y: you have a one-half chance that you win. You have a one-half chance of getting 50 dollars, and then you have the one-half chance—the rest of the probability. If there's a one-half chance you win, there's going to be a one minus one-half, or essentially a one-half chance that you lose. So you have a one-half chance of having to pay 25 dollars. Let's see what this is. This is one-half times 50 plus one-half times negative 25. This is going to be 25 minus 12.50, which is equal to 12.50. Your expected value from bet two is 12.50.
Your friend says he's willing to take both bets a combined total of 50 times. If you want to maximize your expected value, what should you do? Well, bet number two—actually, both of them are good bets; I guess your friend isn't that sophisticated—but bet number two has a higher expected payoff, so I would take bet two all of the time. I would take bet two all of the time.