# Problem 139 Solution

Denote the probability of winning any given get to be p=18/38.

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)!)) * pw * (1-p)n-w

The expected return of n bets is $\sigma$ (for w=0 to n) (n!/(w!*(n-w)!)) * pw * (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.

 Problem 139 Expected Gain Numberof Bets ExpectedGain 1 0.210526 2 0.171745 3 0.257618 4 0.219182 5 0.273977 6 0.235800 7 0.275100 8 0.237137 9 0.266779 10 0.229003 11 0.251903 12 0.214296 13 0.232154 14 0.194700 15 0.208607 16 0.171295 17 0.182001 18 0.144822 19 0.152868 20 0.115815 21 0.121605 22 0.084671 23 0.088519 24 0.051698 25 0.053852 26 0.017138 27 0.017797 28 -0.018813 29 -0.019482 30 -0.055994

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.

Michael Shackleford, ASA, August 19 1999