Problem 176 Solution

From problem 116 we know that with the probability of success of reaching $x before going broke is ((10/9)25-1)/((10/9)x-1). The following table shows this probability for various values of x, as well as the expected return.

Problem 176
Goal Probability
of Success		 Expected
Return
$260 0.893093 $8.93
$270 0.798269 $15.97
$280 0.714034 $21.42
$290 0.639101 $25.56
$300 0.572362 $28.62
$310 0.512855 $30.77
$320 0.459746 $32.18
$330 0.412306 $32.98
$340 0.369895 $33.29
$350 0.331956 $33.20
$360 0.297996 $32.78
$370 0.267579 $32.11
$380 0.240324 $31.24
$390 0.215890 $30.22
$400 0.193977 $29.10
$410 0.174318 $27.89
$420 0.156675 $26.63
$430 0.140837 $25.35
$440 0.126615 $24.06
$450 0.113842 $22.77

The expected value is maximized at a goal of $340.

Michael Shackleford, A.S.A.