The Kelly formula is a way of choosing your stake size from two inputs: your estimate of the true probability of an outcome, and the decimal odds the bookmaker offers. Its purpose is to grow a bankroll as fast as possible if you have a real edge, while scaling stakes down when your edge is small. It is not a selection tool and it is not a way to make a losing market profitable.
The formula
The formula is: f* = (bp – q) / b
- f* – the fraction of your bankroll the formula says to stake.
- b – the net decimal odds, i.e. the profit per £1 staked (decimal odds minus 1).
- p – your estimate of the probability of winning.
- q – the probability of losing (1 minus p).
If your estimated probability matches the price the bookmaker offers, the edge is zero and the formula correctly says to stake nothing. That is the most important behaviour in the whole system: Kelly refuses to bet you money on a fair or negative price.
A worked example
Bankroll: £1,000. You estimate a 55% chance of an outcome (p = 0.55, q = 0.45). The bookmaker offers decimal odds of 2.00, so b = 2.00 – 1 = 1.00.
Plug in: f* = (1.00 × 0.55 – 0.45) / 1.00 = 0.10. The formula says to stake 10% of the bankroll: £100. If your estimate is right, that is the growth-optimal size. If your estimate is wrong, that is exactly the size that hurts the most.
Why full Kelly is dangerous
The formula assumes you know the true probability. You do not. Your estimate is a guess with error in it. If you overestimate your edge, Kelly tells you to stake more than is justified – and because the stake is a percentage of the bankroll, the resulting drawdowns are large and slow to recover. If your edge is actually negative, full Kelly bleeds the bankroll faster than almost any other sizing rule, because it stakes aggressively and keeps re-sizing upward as prices look tempting.
That is why most people who use Kelly deliberately bet a fraction of it: half-Kelly stakes 5% in the example above, quarter-Kelly stakes 2.5%. A fraction sacrifices some theoretical growth in exchange for far smaller swings and far more tolerance for estimation error. The trade is almost always worth it, because estimation error is not a hypothetical – it is the normal state of affairs.
What Kelly does and does not do
- It scales stakes with your bankroll, so a shrinking bankroll automatically gets smaller bets. That limits – but does not eliminate – the chance of a ruinous run.
- It is maximised only when the probability estimate is correct. Garbage in, overconfident stakes out.
- It cannot create an edge. If the prices you find are all worse than the true probabilities, every Kelly stake is negative expected value, only sized differently.
Common mistakes
The classic failure is emotional: after a losing run the bankroll has shrunk, Kelly says the stakes should shrink with it, and the bettor raises them instead to “win it back”. That is the exact opposite of what the formula prescribes, and it is the fastest route from a controlled strategy to a blown bankroll. The second failure is using full Kelly on soft estimates – if you are not confident in the probability, the fraction should be small, and sometimes zero.
Verdict
Kelly is the right framework for stake sizing when you believe you have found value. Used fractionally, with honest probabilities and a pre-set loss limit, it keeps your exposure proportional to your edge. Used “fully” on confident-sounding guesses, it turns small errors into large losses. It is a discipline tool for finding and managing value, not a generator of it. If you do not yet have a reliable way to estimate probabilities better than the bookmaker does, the correct Kelly stake is zero.
Gambling can be addictive. Free and confidential help is available from GambleAware and GamCare. Our safer gambling page lists the limit tools and self-exclusion options. If you are in immediate distress, call 999.
