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$2$ -to-one payout

Expected valueEasy

A bet wins $2$ dollars for each dollar staked with probability $\frac{1}{2}$ , and otherwise loses the stake. What fraction of your bankroll maximises the expected logarithm of wealth?

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Hint

Use the formula with net odds b, win chance p and loss chance q.

Solution

Here $b = 2$ , $p = \frac{1}{2}$ and $q = \frac{1}{2}$ .
The optimal fraction is $\frac{b p - q}{b} = \frac{1 - \frac{1}{2}}{2}$ .
That equals $\frac{1}{4}$ .

Answer: $\frac{1}{4}$

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