Question

A fair 20-sided die is rolled repeatedly, until a gambler decides to stop. The gambler pays $1 per roll, and receives the amount shown on the die when the gambler stops (e.g., if the die is rolled 7 times and the gambler decides to stop then, with an 18 as the value of the last roll, then the net payo↵ is $18 $7 = $11). Suppose the gambler uses the following strategy: keep rolling until a value of m or greater is obtained, and then stop (where m is a fixed integer between 1 and 20). (a) What is the expected net payoff? (b) Use R or other software to find the optimal value of m.

Answer 1

Answer Date:7/6/2019 (a) Define random variable X that marks the net payoff. Define the random variable N that marks the turn in which gambler stops. Define G as the number on die in the last throw. Observe that Geom(P(Gm) N equal to m since the Gambler stops when he gets the number greater simply have that or Observe that we 20- т + 1 P(G m) 20 Now, find the expectation of X conditioning on Е(X) E(X|N = k)P(N = k) k-1 Observe that given N = k, the expected amount of money that gambler gets is the sum all possibilities m, m + 1,..., 20 with equal probabilities mimus the money he puts on bid. Using the hint have that we 20 m Е(XIN 3D 3) 2 since he loses k dollars to bid. Now we have that

20m k P(N k) E(XIN k)P(N = k - _ 2 k-1 k-1 20m P(N k) kP(N = k) 2 k-1 k-1 20 20m 20m - E(N) 2 20 m1 2 (b) E(X) is Using R, code given below, we have that the plot of function m

C 20 15 10 5 Index LC 7. 0 f(1:20)

so the optimal value of m that maximizes the expectation is 15 function ) f 20/a(20-m+1) (20+m) /2 S return (s) } plot (f (1:20))