A bet pays 3 to 1 and you believe it wins 32% of the time. What fraction of bankroll does the Kelly criterion stake, and what is the expected log growth per bet?

A bet pays 3 to 1 and you believe it wins 32% of the time. What fraction of bankroll does the Kelly criterion stake, and what is the expected log growth per bet?

Approach: Compute the edge as the expected profit per unit staked, divide by the odds offered, then evaluate the probability weighted log wealth change at that stake.

7/75. The Kelly fraction is f = (b*p - q)/b with b = 3, p = 0.32 and q = 0.68, so f = (0.96 - 0.68)/3 = 0.28/3, which is 7/75 or 9.33% of bankroll. Read the numerator as the edge: the bet returns 0.28 units of expected profit per unit staked, and dividing edge over odds converts that edge into a size. The expected log growth is 0.32 * ln(1.28) + 0.68 * ln(0.90667) = 0.0790 - 0.0666, which is 1.24% of bankroll per bet, so full Kelly compounds slowly even on a 28% edge. Half Kelly at 4.67% keeps three quarters of the growth, 0.94%, for half the volatility.

Follow-up: Your probability estimate is itself uncertain with a standard deviation of 4 percentage points. What does that do to the Kelly stake?

Key concepts: Kelly fraction, edge over odds, log growth.