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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 bettin number one bet number one we don't encourage betting but I guess Jeremy wants to bet if the next three fish he catches are all sunfish you will pay him $100 otherwise he will pay you $20 bet two if you catch at least two sunfish of the next three fish that you catch he will pay you $50 otherwise you will pay him $25 what is your expected value from problem from what is the expected value from bet one round your answer to the nearest cent and encourage you to pause this video and try to think about it on your own so let's see the expected value of bet one so the expected value of let's just say bet one or it will say bet one is the let's just define a random variable here just to be a little bit better about this so let's say X is equal to is equal to what you pay what you or I guess you could say because you might get something well what what your profit is your profit is from bet one from vet one and it's a random variable and so the expected value the expected value of X is going to be equal to well let's see what's the probability it's going to be it's going to be one hundred it's going to be negative one hundred dollars times the probability that he catches three fish so the probability that Jeremy Jeremy catches three sunfish the next three three the next three the next three fishy catches are going to be sunfish times 100 dollars times or I should say well you're going to pay that so since you're paying it we'll put it as negative 100 because saying that this is your expected profit you're going to lose money there and otherwise so that's going to be one minus this probability the probability and I'll just that Jeremy catches catches three sunfish in that situation he'll pay you $20 you get $20 there so the important thing is is to figure out the probability that Jeremy catches three sunfish well the sunfish are our 10 out of the 20 fish so at any given time he's trying to catch fish it's going there's a 10 and 20 chance or you could say 1/2 chant 1/2 probability that's going to be a sunfish so the probability that you get 3 sunfish in a row is going to be 1/2 times 1/2 times 1/2 and they put the fish back in so that's why it stays 10 out of the 20 fish if you wasn't putting the fish back in then the the second sunfish you would have a 9 out of 20 chance of the second one being a sunfish but in this case they keep replacing the fish every time they catch it so there is a 1/8 chance that Jeremy catches 3 sunfish so this right over here is 1/8 and 1 minus 1/8 this is 7/8 7/8 so you have a 1/8 chance of paying $100 and a 7 a chance of getting $20 and so this gets us to this gets us to so your expected I guess you could say profit here there's a 1/8 chance 1/8 probability that you lose $100 here so times negative 100 but then there is a 7/8 chance 7/8 chance that you get that you get I'll just put parentheses make it a little clearer I think the order of operations on the calculator would have taken care of it but I'll just do it just so that it looks the same 7/8 there's a 7h chance that you get $20 and so your expected payoff here is positive $5 so your expected payoff here is equal to is equal to $5 so this is your expected value from bet 1 now let's think about bet to bet to 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 so let's let's think about the probability of catching at least at least two sunfish of the next three fish that you catch now there's a bunch of ways to think about this but since since there's only since there's only kind of 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 you that that that are possible here you could get you could get sunfish sunfish sunfish you could get what's the other type of fish that you have or the 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 and you see here that each of these each time you go there's there's there there are there's there's two possibilities and so when your each time you try to catch a fish there's two possibilities so if you're doing it three times there's 2 times 2 times 2 possibilities 1 1 2 3 4 5 6 7 8 possibilities here now out of these eight equally likely possibilities how many of them involve you catching at least two sunfish well you catch at least two sunfish in this one in this one in that one in this one and I think that is it that is yep this is only one sunfish one sunfish one sunfish and no sunfish so when four out of the eight equally likely outcomes you catch at least two sunfish so your probability of catching at least two sunfish probability of at least two sunfish sunfish is equal to four eighths or or one half so let's see what's the expected value let's say why is the expected profit from bet so let's let y equal this is another random variable is equal to expected profit from vet 2 so the expected value of our random variable Y you have a 1/2 chance that you win so you have a 1/2 chance of getting $50 and then you have the 1/2 chance the rest of the probability and there's a 1/2 chance you win there's going to be a 1 minus 1/2 or essentially a 1/2 chance that you lose and so this is going in so you have a 1/2 chance of having to pay $25.00 so let's see what this is this is 1/2 times 50 plus 1/2 times negative 25 this is going to be 25 minus 1250 minus 1250 which is equal to 1250 so your expected value from bet 2 is 1250 your friend says he's willing to take both bets 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 2 I actually both of them are good bets I guess your friend isn't that sophisticated but bet number 2 has a higher expected payoff so I would take bet 2 all of the time so bad I would take vet two all of the time