The Kelly Criterion
Finding an edge and sizing it are different problems, and the second has an exact answer when the bet repeats.
Why expected value is the wrong objective
Suppose a bet doubles your stake with probability 0.55 and loses it otherwise. Maximising expected wealth says stake everything, every time, because each bet has a positive expected value and a bigger stake scales it.
Follow that advice and you go bankrupt with probability 1. Any single loss ends you, and a loss arrives eventually.
The problem is that wealth compounds, so what accumulates is the product of the outcomes, not the sum. Products are governed by the average logarithm, and expected wealth is dominated by a vanishingly small set of paths that never lose, which nobody experiences.
The growth rate
Stake a fraction of the bankroll each time, winning per unit staked with probability . The long-run growth rate per bet is
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