Problem 139 Solution

Denote the number of bets made to be n.

Denote the number of winning bets to be w.

The probability of a w wins out of n bets is (n!/(w!*(n-w)!)) * p^w * (1-p)^n-w

The expected return of n bets is $\sigma$ (for w=0 to n) (n!/(w!*(n-w)!)) * p^w * (1-p)^n-w * if(w>n/2,2w-n,w-n/2)

The table below shows the expected gain given the number of wagers made.

From the table it can be seen that the maximum profit occurs when the number of bets is 7.

Thanks to Extra Stuff: Gambling Rambling by Peter Griffin for this problem. See chapter 7.