Chapter 7, Problem 7.1P

To determine

To find: Minimum probability to win.

Explanation of Solution

Given the even money placed on bulls = $100,000

The current wealth = $1,000,000

$\begin{array}{l}\text{Let the probability of winning = p}\\ \text{Thus, expected utility, EU(bet) = p}\times \text{U(1100000) + (1 - p)}\times \text{U}\left(900000\right)\\ \text{Expected utility in the case of no bet}\text{.}\\ \text{EU(no bet) = U(1000000)}\end{array}$

Since the utility function is logarithmic, expected utility from betting and not betting should be equal.

$\begin{array}{l}\text{EU(bet) = EU(no bet)}\\ \text{p}\times \text{U(1100000) + (1 - p) }\times \text{ U(900000) = U(1000000)}\\ \text{p}\times \text{ln(1100000) + (1 - p) }\times \text{ ln(900000) = ln(1000000)}\\ \text{p}\times \text{ln(11/9) = ln(10/9)}\\ \text{p = 0}\text{.525}\end{array}$

The minimum probability to win is 0.525.