Ben is in a generous mood. He chooses a number X, uniformly drawn at random from

Question

Ben is in a generous mood. He chooses a number X, uniformly drawn at random from the set {1, 2, ..., 100}. He puts $X into one box and $2X into another identical box. He asks Lem to choose a box; Lem's chosen box is opened to reveal the amount it of money (call it Y) that it contains. Ben then gives Lem the option of either taking the box he chose, or trading boxes. Lem's goal is to maximize the amount in the box he finally decides on. Suppose Lem sees Y=$7 in the box he first chose. Should he with this box or trade? Why? Suppose Lem sees Y=$8 in the box he first chose. Given this, what is the probability that the other box contains $16? Prove your answer. Lem sees Y = $8 in the box he first chose. What is the expected value of his winnings if he sticks with his first choice? If Lem sees Y=$8 in the first box, what is the expected value of his winnings if he trades his box for the other one? Should he stick with his first choice, or trade? Are there any values of Y for which trading boxes decreases Lem's expected winnings?

Explanation / Answer

a) we have three options, single, double, half each probability is equal to P= 1/3 =.3333 he can stay with this amount of money cause if he change, he can selec the half of the value he got

b) its the same option P= 1/3 =.3333

c) E(x) = 8(.33)+16(.33)+4(.33)= 9.33

d)same prob as c

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