The Kelly criterion gives you the share of your bankroll to stake from an edge you assume is known: f* = (b·p − q) / b, where b is the decimal odds minus 1, p your estimated probability and q = 1 − p. With odds of 2.20, a 50% estimated probability and a £1,000 bankroll, you get f* = (1.20 × 0.50 − 0.50) / 1.20 = 8.33%. That's an £83.33 stake, or £41.67 at half-Kelly. The maths treats p as certain. It never is, and if you overrate it, the stake grows with it.
What Kelly is trying to optimise
The Kelly criterion doesn't work out an expected win. It gives you the share of your bankroll that grows your money fastest over a very long run of decisions. That's what people mean by the geometric growth rate.
Maximising the expected value of one bet is a different goal. So is chasing its most likely outcome. And the gap matters: even with a real edge, a fraction that's too big slows growth down instead of speeding it up. After a losing run, the bankroll climbs back more slowly.
So Kelly answers a narrow question. What share of the bankroll maximises compound growth? It doesn't answer "how much should I put on this match".
What "optimal" doesn't mean here. Kelly maximises a growth rate over an infinite number of theoretical repetitions. It doesn't maximise what you win on a bet, or even the expected value over a finite number of bets. Over a short record, a fraction smaller than f* often ends up with a better final result.
The formula on one line
You need two inputs, no more: the decimal odds and your estimated probability.
b = decimal odds − 1
p = estimated probability, q = 1 − p
f* = (b·p − q) / b
Take odds of 2.20, an estimated probability of 50% and a £1,000 bankroll. You get b = 1.20 and q = 0.50. So f* = (1.20 × 0.50 − 0.50) / 1.20 = 0.10 / 1.20 = 8.33%. That's an £83.33 stake.
Work out 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.
You can also read the same sum another way, as on the value bet page. The numerator is the expected value per unit staked. It's the quantity the expected value page walks through: the +10% edge at these odds, or their theoretical value. Divide it by b and you land back on f* = 0.10 / 1.20. A Kelly fraction is really just an edge rescaled to the odds.
A formula born a long way from the pitch
Kelly published this formula in 1956, in a paper on information theory. Nothing to do with sport.
Thorp then applied it to favourable games of chance. Blackjack first, then financial markets. He's the one who formalised the fractional version, as a practical answer to uncertainty about p. Sports betting picked it up as it was. The formula doesn't change. Only where the estimate comes from does.
Why compound growth changes the question
If your bankroll drops 50%, it needs a 100% gain to get back to where it started. Not 50%. Geometric growth accounts for that asymmetry. Arithmetic expected value ignores it completely.
Look only at the expected value of one bet and you can end up staking a huge chunk of your bankroll the moment an edge shows up. In that calculation, a win always cancels out a loss. Compound growth on a repeated bankroll adds a penalty, and that penalty is variance itself. Past a certain point, even with a positive edge, staking more drags growth down. Kelly finds the balance between the edge and that variance. That's all it does.
Why f* runs away with the smallest error in p
The formula has one source of uncertainty, the probability you type in. And it treats that number as certain. No real-world estimate can live up to that.
A few points too many on p is enough to turn a sensible fraction into an oversized stake. Stick with odds of 2.20. If you put the probability at 55% when it's really 50%, f* jumps from 8.33% to 17.50%. More than double, for a five-point miss.
That sensitivity, more than the formula itself, is what pulls you towards risk of ruin when you play full Kelly.
When f* ≤ 0: no edge, no stake
When your estimated probability doesn't beat the one the odds require, f* drops to zero or below. The formula then has nothing left to size. That's not a bug. It's the normal result whenever the theoretical edge isn't positive.
Same odds of 2.20, but now put the probability at 40% instead of 50%. The edge flips negative: b·p − q = 1.20 × 0.40 − 0.60 = −0.12, so f* = −0.12 / 1.20 = −10%. No stake comes out of it. Not a positive one, and not a reversed one either.
Try those 40% in the calculator above. You won't see "−10%" on screen. You'll get a banner saying there's no positive edge with these values. That's how I'd want this page to behave, because a negative number on screen always ends up being given a meaning it doesn't have.
A negative f* doesn't mean "bet against it" or "bet the other way round". Sports betting has no mirror position, unlike some other markets. An f* ≤ 0 just means: no stake, at this probability and these odds.
Nobody plays full Kelly on its own
Full Kelly maximises theoretical long-run growth, at the cost of huge variance. Even with a real edge, watching your bankroll halve or worse stays statistically possible.
| 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 | Much smaller drawdowns |
| Quarter-Kelly (f*/4) | Slower growth | Limited drawdowns, good tolerance to an error in p |
Fractional Kelly gives up some theoretical growth. In return, it shields you far better from an estimation error, the one variable the formula can't measure. That's the form people use in practice, for the simple reason that p is never as solid as the formula assumes.
There's another property that feels backwards. Past f*, every extra slice of bankroll cuts compound growth. At 2f*, growth hits zero. So staking double the calculated fraction doesn't double your upside. It drags theoretical growth down to zero, with all the variance of a stake twice the size.
"Kelly maximises your winnings" gets said a lot. Trouble is, Kelly maximises a theoretical growth rate, worked out over infinite repetitions with a probability assumed to be exact. On a real, finite record built on a rough estimate, a fraction smaller than f* very often does better.
What the calculator won't tell you
The calculator's result is the fraction your probability implies. It isn't a stake recommended by OddScore.
OddScore doesn't produce a sports probability of its own, independent of the market. So the platform can't check or correct the p you enter. Your fraction is worth exactly what your estimate is worth, just as the edge of a value bet depends on the probability behind it. Neither the formula nor the platform can vouch for that estimate on your behalf.
Kelly next to other staking plans
Kelly is just one way to size a stake from a bankroll. There's also flat staking, a fixed percentage, progressions. Each one weighs differently on risk of ruin and on how deep the drawdowns go.
Compare staking plans, judged on risk rather than a promised return →
Responsible gambling: no fraction, however carefully worked out, protects your stake. Sports betting carries a risk of financial loss. Set yourself a budget, don't chase your losses, and use the deposit limits and self-exclusion tools available. Learn more about responsible gambling.
Put it into practice
Apply this guide to real odds.
See a market reading
A real match with its odds, movements and 0-100 signal.
Open the demoMargin calculator
Measure a bookmaker's margin on any market.
Calculate the marginDig 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 whole picture: profit, ROI, expected value, variance, sample size, drawdown and risk.
Read the guideValue bet: what the gap actually measures
The edge Kelly uses without ever working it out itself.
Understand value betsExpected value: the theoretical edge of a bet
What an edge is worth in expected-value terms, before any talk of stake size.
Understand EVVariance: why results swing about
What fractional Kelly smooths out without ever getting rid of.
Understand varianceRisk of ruin: when a bankroll runs dry
What a Kelly fraction that's too big can set off.
Understand risk of ruinStaking plans: the methods side by side
Where Kelly sits among the other ways to size a stake.
Compare staking plansFrequently asked questions
What is the Kelly criterion?
A formula that works out what share of your bankroll to stake on a bet, from the odds and your estimated probability. It aims for the fastest growth of a bankroll over a long run of decisions. The result of any single bet isn't what it's after.
How do you work out the Kelly fraction?
f* = (b·p − q) / b. Here b is the decimal odds minus 1, p your estimated probability and q = 1 − p. At odds of 2.20 with a 50% probability, f* comes to (0.60 − 0.50) / 1.20, so 8.33% of the bankroll.
Does Kelly maximise what you win on a bet?
No. It maximises the geometric growth rate of a bankroll over a long series of repeated bets. Not the expected value of one bet, and not its most likely outcome. A bet sized with Kelly can lose just like any other.
What if f* comes out negative or zero?
Then you stake nothing. An f* ≤ 0 means your estimated probability doesn't beat the one the odds require. There's nothing for the formula to size.
Why does everyone talk about fractional Kelly?
Because the formula reacts hard to the smallest error in p, and p is almost never known well. Staking a fraction of the calculated f* (a half, a quarter) cuts variance a lot. The trade-off is slower theoretical growth.
Does Kelly protect you from risk of ruin?
Not if your probability is too optimistic. An inflated estimate gives you a fraction that's too big. And that's exactly what pushes you towards the ruin the formula avoids when p is right.
Does OddScore recommend a stake using Kelly?
No. The calculator shows the fraction that follows from the probability you type in, nothing more. It isn't a recommended stake, and the result is only as good as your estimate.
Sources and method
This page builds on Kelly's founding paper on sizing a repeated stake, on Thorp's work applying it to favourable games, and on OddScore's own odds-analysis method. The UK regulatory and player-support links come from the Gambling Commission and BeGambleAware.
- Present the formula as a calculation that depends on a probability you supply, never as a stake recommended by the platform.
- Give the f* ≤ 0 case its own section instead of hiding it behind a zero stake.
- Leave the risk-of-ruin page to go deep on sensitivity to errors in p, rather than repeating it here.
- Focus on fractional Kelly, the only form used in practice, rather than theoretical full Kelly.
- Kelly, J. L. (1956). A new interpretation of information rate. Bell System Technical Journal, 35(4), 917-926.
- Thorp, E. O. (1969). Optimal gambling systems for favorable games. Review of the International Statistical Institute, 37(3), 273-293.
- Gambling Commission, Public and players
- BeGambleAware, free and confidential support