| f* | The fraction of your current bankroll to stake. The star marks it as the optimal value. |
| p | Probability you win the bet. |
| q = 1 − p | Probability you lose. |
| b | Net odds: profit per 1 staked if you win. Even money is b = 1. From decimal odds d, b = d − 1. |
| b·p − q | EV per 1 staked: you win b with probability p and lose 1 with probability q. This is your edge. |
| ÷ b | Scale down for long-shot payoffs, which swing more. So the result is edge divided by odds. |
Why not just maximise EV?
Maximising EV says stake everything on every +EV bet, and the first loss takes you to zero. Wealth compounds by multiplying, not adding: +50% followed by −50% leaves you at 1.5 × 0.5 = 0.75 of where you started. So Kelly maximises the average growth rate, which is the expected logarithm of the multiplier on each bet:
g(f) = p · ln(1 + f·b) + q · ln(1 − f)
Here ln(1 + f·b) is the log of your wealth multiplier on a win, and ln(1 − f) is the log of the multiplier on a loss. Set the slope of g to zero, p·b ÷ (1 + f·b) = q ÷ (1 − f), and solving for f gives f* = (b·p − q) ÷ b.
Example: p = 0.55, b = 1 → f* = 0.10. Growth per bet = 0.55 · ln 1.1 + 0.45 · ln 0.9 = 0.0524 − 0.0474 = 0.0050, about 0.5% per bet, compounding.