Expected value I Nick right? Dan's payoff probabi Eric right? Y Dan right? $1 114 112 114 1/2 1/2 N 1/2 $2 114 112 114 From this revised tree diagram, we can work out Dan's actual expected payoff Ex( payoff=0:0+1·元+1·元+4.0+(-2)·0+(-2)+(-2)·元+00 1 1 So he loses an average of a half-dollar per game! Similar opportunities for subtle cheating come up in many betting games. For exam le, a group of friends once organized a football pool where each participant would guess the outcome of every game each week relative to the spread. This may mean nothing to you, but the upshot is that everyone was effectively betting on the outcomes of 12 or 13 coin tosses each week. The person who correctly predicts the most coin tosses won a lot of thinking in terms of the first tree diagram, swore up and down that there was no way to get an unfair"edge". But actually the number of participants was small enough that just two players betting oppositely could gain a substantial advantage Anothe aple involves a former MIT professor of statistics, Herman Chernoff State lotteries are the worst gambling games around because the state pays out only a fraction of the money it takes in. But Chernoff figured out a way to win! Here are rules for a typical lottery All players pay $1 to play and select 4 numbers from 1 to 36 The state draws 4 numbers from 1 to 36 uniformly at randomExpected Value I 3 probability Dan right? Eric right? Nick right? Dan’s payoff Y N Y Y Y Y Y Y N N N N N N $0 −$2 −$2 −$2 $4 $1 $1 $0 1/2 1/2 1/2 1/2 1/2 1/2 0 1 0 1/4 0 0 1/4 1/4 0 1/4 0 1 0 1 1 0 From this revised tree diagram, we can work out Dan’s actual expected payoff: 1 1 1 1 Ex (payoff) = 0 · 0 + 1 · + 1 · + 4 · 0 + (−2) · 0 + (−2) · + (−2) · + 0 · 0 4 4 4 4 1 = −2 So he loses an average of a half­dollar per game! Similar opportunities for subtle cheating come up in many betting games. For exam￾ple, a group of friends once organized a football pool where each participant would guess the outcome of every game each week relative to the spread. This may mean nothing to you, but the upshot is that everyone was effectively betting on the outcomes of 12 or 13 coin tosses each week. The person who correctly predicts the most coin tosses won a lot of money. The organizer, thinking in terms of the first tree diagram, swore up and down that there was no way to get an unfair “edge”. But actually the number of participants was small enough that just two players betting oppositely could gain a substantial advantage! Another example involves a former MIT professor of statistics, Herman Chernoff. State lotteries are the worst gambling games around because the state pays out only a fraction of the money it takes in. But Chernoff figured out a way to win! Here are rules for a typical lottery: • All players pay $1 to play and select 4 numbers from 1 to 36. • The state draws 4 numbers from 1 to 36 uniformly at random