The Kelly criterion calculates the share of bankroll to commit from an assumed known edge: f* = (b·p − q) / b, where b is the decimal odds minus 1, p is your estimated probability and q = 1 − p. With odds of 2.20, an estimated probability of 50% and a €1,000 bankroll: f* = (1.20 × 0.50 − 0.50) / 1.20 = 8.33%, a €83.33 stake (€41.67 at half-Kelly). The calculation assumes p is known with certainty — in practice it's always estimated, and overestimating it oversizes the stake.
What the formula optimises
The Kelly criterion doesn't calculate an expected gain: it sizes the share of your bankroll that maximises the geometric growth rate of a capital repeated over many decisions. That's not the same as maximising the expected value of a single bet, or its most likely outcome.
The distinction matters because too large a fraction, even with a real edge, slows long-run growth instead of speeding it up: the capital bounces back less well after a losing run. Kelly answers a specific question — what share of the bankroll maximises compound growth — not the broader question of "how much should I bet".
What "optimal" doesn't mean here. The Kelly criterion maximises a geometric growth rate over an infinite number of theoretical repetitions, not the gain from a bet, nor even the expected value over a finite number of bets. Over a short record, a fraction smaller than f* often produces a better final result.
The formula
Three variables are enough: decimal odds, estimated probability, nothing else.
b = decimal odds − 1
p = estimated probability, q = 1 − p
f* = (b·p − q) / b
With odds of 2.20, an estimated probability of 50% and a bankroll of €1,000: b = 1.20, q = 0.50, f* = (1.20 × 0.50 − 0.50) / 1.20 = 0.10 / 1.20 = 8.33%, a €83.33 stake.
Calculate the Kelly fraction and the stake it implies
Enter the odds on offer, your estimated probability and your bankroll.
Comma or dot accepted: 2,20 as well as 2.20.
Your own estimate, which must come from a source other than the odds themselves.
Only used to illustrate the stake. Nothing is stored.
Result
- Kelly fraction
- 8.33%
- Stake from the formula
- €83.33
- Half-Kelly stake
- €41.67
Warning:
The fraction depends entirely on your estimated probability: overestimating p oversizes the stake. This result is never a recommended stake.
Another way to read the same calculation, consistent with the value bet page: f* = expected value per unit staked ÷ b. This numerator is the same quantity covered on the expected value page — the +10% edge on this odds figure, or its theoretical value — so f* = 0.10 / 1.20, the same result. The Kelly fraction is never anything more than an edge rescaled by the odds.
Where the formula comes from
Kelly published this formula in 1956, in a paper about information theory, not sports betting. Thorp applied it to favourable games of chance — blackjack first, then financial markets — and formalised its fractional version as a practical answer to uncertainty about p. Sports betting inherits it directly: the formula stays the same, only the source of the estimate changes.
Why geometric growth changes the question
A capital that loses 50% needs a 100% gain to get back to where it started — not 50%. That's the asymmetry geometric growth accounts for, and one an arithmetic expected value ignores completely.
Maximising the arithmetic expected value of a single bet can lead to committing huge shares of a bankroll the moment an edge exists, because in that calculation a single gain always offsets a single loss. Maximising the geometric growth rate of a repeated bankroll, by contrast, introduces a penalty for variance itself: too large a fraction, even with a positive edge, ends up reducing compound growth instead of increasing it. That trade-off between edge and variance is exactly what the Kelly formula solves — and only it.
Why f* is hypersensitive to errors in p
The formula recognises only one source of uncertainty: the probability entered. Yet it treats that figure as if it were known with certainty, which no real-world estimate can guarantee.
Overestimating p by just a few points is enough to turn a reasonable fraction into oversizing. At the same odds of 2.20, an estimated probability of 55% instead of a real 50% pushes f* from 8.33% to 16.67% — double, for a five-point estimation gap. It's this sensitivity, not the formula itself, that brings you closer to risk of ruin when applied at full Kelly.
The f* ≤ 0 case: no edge, no stake
When the estimated probability doesn't exceed what the odds require, f* is negative or zero — and the formula has nothing to size. That's not a calculation error: it's the expected result the moment the theoretical edge isn't positive.
At the same odds of 2.20, an estimated probability of 40% instead of 50% pushes the edge negative: b·p − q = 1.20 × 0.40 − 0.60 = −0.12, so f* = −0.12 / 1.20 = −10%. The formula then produces no stake to commit — not a positive one, not a reversed position either.
A negative f* doesn't mean "bet against" or "bet the other way". There's no symmetrical position in sports betting, unlike some other markets. An f* ≤ 0 simply means: no stake, given this probability and these odds.
Full Kelly is never used alone
Full Kelly maximises theoretical long-run growth, at the cost of considerable variance — bankroll drawdowns of 50% or more remain statistically possible even with a real edge.
| Fraction committed | Effect on growth | Effect on drawdowns |
|---|---|---|
| Full Kelly (f*) | Maximum theoretical growth | Very sharp bankroll drawdowns, even with a real edge |
| Half-Kelly (f*/2) | Roughly three-quarters of theoretical growth | Substantially reduced drawdowns |
| Quarter-Kelly (f*/4) | Slower growth | Limited drawdowns, high tolerance to an error in p |
Fractional Kelly sacrifices some theoretical growth to sharply cut exposure to an estimation error — the one variable the formula can't measure on its own. That's why only the fractional form is actually applied in practice, precisely because p is never known with the certainty the formula assumes.
The formula has another counterintuitive property: past f*, every extra fraction reduces compound growth rather than increasing it, down to zero at 2f*. Betting double the calculated fraction doesn't double the risk for proportional gain — it brings theoretical growth to zero, while carrying all the variance of a stake twice as large.
"Kelly maximises the gain."
No: Kelly maximises a theoretical geometric growth rate, calculated over an infinite number of repetitions and a probability assumed exact. Over a real, finite record built on an approximate estimate, a fraction smaller than f* very often produces a better result.
What the calculator's result isn't
The result shown by the calculator above is the fraction the formula implies for the probability entered — never a stake recommended by OddScore. The platform produces no sports probability independent of the market: it can't validate or correct the value of p the reader enters.
The quality of the fraction shown depends entirely on the quality of that estimate, exactly as the edge of a value bet depends on the probability feeding it. No one — not the formula, not the platform — can guarantee that quality on the reader's behalf.
Kelly among staking plans
The Kelly criterion is only one way to size a stake from a bankroll. Other methods exist — flat stakes, a fixed percentage, progressions — each with a different effect on risk of ruin and the size of drawdowns.
Compare staking plans, judged on risk rather than a promised return →
Responsible gambling. A stake can be lost in full, whatever fraction the formula produces. Never commit money to betting that you need for daily life, housing, bills or your emergency savings, and use the deposit limits or self-exclusion tools offered by licensed operators. Learn more about responsible gambling.
Dig into the market
Odds movements are only part of the story. Here are the next topics to read.
Profitability and risk: the complete guide
The full picture: profit, ROI, expected value, variance, sample size, drawdown, risk.
Read the guideValue bet: what the gap actually measures
The edge the Kelly formula consumes, without calculating it itself.
Understand value betsExpected value: the theoretical edge of a bet
What the edge means in expected-value terms, before any stake sizing.
Understand EVVariance: why results swing
What fractional Kelly tries to dampen, without ever removing it.
Understand varianceRisk of ruin: exhausting a bankroll
What an overestimated Kelly fraction can trigger.
Understand risk of ruinStaking plans: comparing the methods
Where Kelly sits among the other ways to size a stake.
Compare staking plansFrequently asked questions
What is the Kelly criterion?
It's a formula that calculates the share of your bankroll to commit to a bet, from odds and an estimated probability. It maximises the geometric growth rate of a bankroll repeated over many decisions, not the gain from a single bet.
How do you calculate the Kelly fraction?
f* = (b·p − q) / b, where b is the decimal odds minus 1, p your estimated probability and q = 1 − p. At odds of 2.20 with a 50% estimated probability, f* works out to (0.60 − 0.50) / 1.20, or 8.33% of the bankroll.
Does the Kelly criterion maximise the gain from a bet?
No. It maximises the geometric growth rate of a bankroll over a long series of repeated decisions, not the expected value of a single bet or its most likely outcome. A single bet can perfectly well lose.
What happens when f* is negative or zero?
No stake is committed. An f* ≤ 0 means the estimated probability doesn't exceed what the odds on offer would require: the formula has nothing to size.
Why use fractional Kelly instead of full Kelly?
Because the formula is hypersensitive to errors in p, which is almost never known precisely in practice. Betting a fraction of the calculated f* — half or a quarter — sharply cuts variance in exchange for slower theoretical growth.
Does the Kelly criterion protect against risk of ruin?
No, not if it's built on an overestimated probability. An overly optimistic estimate produces too large a fraction, which brings you closer to the very risk of ruin the formula is meant to avoid when p is accurate.
Does OddScore recommend a stake from the Kelly criterion?
No. The calculator shows the fraction implied by the probability the reader enters, never a stake recommended by the platform. The quality of the result depends entirely on the quality of the estimate supplied.
Sources & methodology
This page draws on Kelly's founding paper on optimal sizing of a repeated stake and on Thorp's work applying it to favourable games and markets, together with the odds-analysis methodology developed by OddScore.
- Present the formula as a sizing calculation conditional on a probability the reader supplies, never as a stake recommended by the platform.
- Treat the f* ≤ 0 case explicitly rather than hiding it behind a silent zero stake.
- Route the sensitivity to estimation error in p to the dedicated risk-of-ruin page rather than duplicating it here.
- Focus on fractional Kelly, the only form actually used, rather than the theoretical full-Kelly fraction.