The Kelly Criterion
Optimal stake sizing for positive-EV bets
The Kelly Criterion calculates the mathematically optimal fraction of your bankroll to wager on any bet where you have a genuine edge. This guide covers the formula, its derivation, why fractional Kelly is standard practice, and when the theory breaks down in real betting markets.
What is the Kelly Criterion?
The Kelly Criterion is a formula for calculating the optimal fraction of a bankroll to bet on a positive expected value wager. Applied consistently, it maximises the long-run growth rate of a bankroll. Staking more than Kelly recommends increases variance without increasing expected growth; staking less is safer but grows the bankroll more slowly.
Definition
The Kelly Criterion determines the fraction of your bankroll to stake on a bet, given your estimated edge and the offered odds. It was derived by physicist John L. Kelly Jr. at Bell Labs in 1956, originally as a formula for signal transmission. The mathematics were adapted for gambling by Ed Thorp, who applied them to blackjack and later to financial markets.
Kelly staking is grounded in information theory. Kelly’s insight was that a bettor with a genuine edge faces the same mathematical problem as a communications engineer: how to transmit the most information through a noisy channel. The answer in both cases is to bet in proportion to your advantage.
The formula produces a percentage of your current bankroll — not a fixed unit amount. This means stakes automatically shrink when the bankroll falls and grow when it rises. Over a losing run, Kelly reduces exposure; over a winning run, it compounds growth. This is the mathematical source of its advantage over flat staking.
The Kelly Criterion formula
The standard form of Kelly used in sports betting is:
Kelly Criterion formula
f* = (bp − q) / b- f* = optimal fraction of bankroll to bet
- b = net decimal odds (decimal odds minus 1)
- p = your estimated win probability
- q = probability of losing (1 − p)
Result is expressed as a decimal fraction. A result of 0.05 means bet 5% of your current bankroll. A negative result means the bet has negative expected value and should not be placed.
The formula can be rewritten equivalently as:
Equivalent form
f* = (p × (b + 1) − 1) / bThis form makes it explicit that Kelly is driven by expected return (p × decimal odds) minus the cost of losing (1), divided by the net odds received.
The two forms are mathematically identical. Some practitioners find the second form more intuitive because the numerator is the expected value per unit staked (which must be positive for Kelly to recommend any bet at all).
Key constraint: Kelly only makes a bet when expected value is positive. A negative f* is not “bet the other way” — it is a signal that the bet has negative EV and should be skipped entirely. There is no fraction of a negative-EV bet that produces positive expected growth.
Converting American and Fractional odds to b
The variable b in the formula is the net decimal odds — what you win per unit staked, not including your stake returned. Converting from other formats:
| Odds format | Example | Decimal odds | b (net decimal odds) |
|---|---|---|---|
| American (negative) | −110 | 100/110 + 1 = 1.909 | 1.909 − 1 = 0.909 |
| American (positive) | +150 | 150/100 + 1 = 2.50 | 2.50 − 1 = 1.500 |
| Fractional | 5/2 | 5/2 + 1 = 3.50 | 3.50 − 1 = 2.500 |
| Decimal | 1.91 | 1.91 | 1.91 − 1 = 0.910 |
Worked examples
Scenario: NFL game, odds −110 American. Your model estimates a 57% win probability.
−110 American → decimal:
100/110 + 1 = 1.909Net odds:
b = 1.909 − 1 = 0.909p = 0.57 | q = 1 − 0.57 = 0.43f* = (0.909 × 0.57 − 0.43) / 0.909f* = (0.518 − 0.43) / 0.909f* = 0.088 / 0.909 = 0.097Half Kelly = 4.85% — on a $1,000 bankroll: stake $48.50.
Quarter Kelly = 2.42% — stake $24.20.
Scenario: Soccer match, odds +200 American. Your model gives the team a 42% win probability.
+200 American → decimal:
200/100 + 1 = 3.00Net odds:
b = 3.00 − 1 = 2.00p = 0.42 | q = 0.58f* = (2.00 × 0.42 − 0.58) / 2.00f* = (0.84 − 0.58) / 2.00f* = 0.26 / 2.00 = 0.13Half Kelly = 6.5% — on a $1,000 bankroll: stake $65.
Scenario: Same −110 odds as Example 1, but your win probability estimate is only 50%.
f* = (0.909 × 0.50 − 0.50) / 0.909f* = (0.455 − 0.50) / 0.909f* = −0.045 / 0.909 = −0.049At −110 odds, you need a win probability above 52.38% before Kelly recommends any stake.
Run these calculations on any bet
Free browser-based calculator — American, Decimal, and Fractional oddsFractional Kelly: the practical standard
Full Kelly is theoretically optimal only under very specific conditions that almost never hold in practice: perfect probability estimates, no model uncertainty, and an infinite sample of bets. In real betting, all three assumptions fail to some degree. This is why fractional Kelly — betting some fixed proportion of the full Kelly stake — is the standard approach.
The four Kelly fractions
| Fraction | Stake | Expected growth | Variance | When to use |
|---|---|---|---|---|
| Quarter Kelly Conservative | 25% of full Kelly | ~56% of full Kelly | Very low | Uncertain models, high-volume bettors, early-stage tracking |
| Half Kelly Recommended | 50% of full Kelly | ~75% of full Kelly | Low–moderate | Most systematic bettors with validated probability estimates |
| Three-quarter Kelly | 75% of full Kelly | ~94% of full Kelly | Moderate | Experienced bettors with high model confidence |
| Full Kelly Aggressive | 100% of Kelly formula | Maximum | High | Only with near-perfectly calibrated models over long samples |
The relationship between fractional Kelly and growth is not linear. Half Kelly produces roughly 75% of full Kelly’s expected growth rate but only 50% of the variance. This asymmetry — giving up 25% of growth to reduce variance by 50% — is why half Kelly has the best risk-adjusted growth profile and why it is the default recommendation for most systematic bettors.
Rule of thumb: If your probability estimates might be off by 2–3 percentage points in either direction, use half Kelly. If you are still building and validating your model, start with quarter Kelly until you have 300+ bets to assess accuracy.
Growth rate vs variance: the core trade-off
The mathematical reason to use fractional Kelly comes from the behaviour of the Kelly growth function. Expected bankroll growth is maximised at full Kelly, but the variance in outcomes is also highest there. Overbet Kelly even slightly — staking 110% or 120% of the formula’s output — and long-run expected growth actually decreases despite higher individual stakes.
What happens at different fractions (5% edge, −110 odds, 1,000 bets)
Quarter Kelly
+74%
Approximate bankroll growth. Very low variance.
Half Kelly
+148%
Best risk-adjusted result. Standard recommendation.
Full Kelly
+197%
Higher growth, but significant drawdown risk.
1.5x Kelly
+119%
Overbetting: more variance, less growth than full Kelly.
These figures are illustrative approximations — real results depend on exact edge, odds distribution, and variance across the sample. The key observation is the 1.5x Kelly column: overbetting Kelly destroys growth. This is not a modest reduction; it can be severe. Ed Thorp observed that staking double Kelly produces the same expected growth as not betting at all.
Never overbet Kelly. If the formula returns 8% and you stake 12% “because you’re confident,” you are mathematically worse off than at 8%. Overconfidence in probability estimates is the most common cause of Kelly overbetting in practice.
Where Kelly breaks down in practice
Kelly is a precise mathematical result, but it rests on assumptions that real betting markets violate in ways that matter. Understanding the failure modes tells you when to deviate from the formula.
Imprecise probability estimates
Kelly assumes you know p exactly. In practice, you have an estimate with some error margin. If your true win probability is 55% but you estimate 58%, Kelly will recommend a larger stake than optimal. The result is systematic overbetting, which reduces long-run growth. This is the primary argument for fractional Kelly: it provides insurance against model overconfidence.
The simultaneous bets problem
Classic Kelly assumes one bet at a time. If you are tracking multiple markets simultaneously — three NFL games on Sunday, five soccer matches on Saturday — the individual Kelly recommendations do not account for the combined risk across all open positions. Summing the individual Kelly fractions can produce a total exposure of 20–30% of bankroll or more, which is significantly higher than Kelly would recommend for a single combined position.
Common fix: When running multiple simultaneous bets, divide each individual Kelly recommendation by the number of concurrent open positions, or apply a global bankroll cap (e.g. maximum 20% of bankroll across all open bets at any time).
Sportsbook account limits
Kelly may recommend staking 6% of bankroll on a bet. At that percentage on a meaningful bankroll, the dollar amount may trigger sportsbook limits or flag the account for restriction. In practice, maximum bet sizes imposed by the book constrain the Kelly recommendation independent of what the formula produces. Sharp bettors operating at scale often cap individual bets at a fixed dollar amount that keeps them under sportsbook radar, regardless of what Kelly says.
Correlated bets
Kelly assumes each bet is independent. Betting multiple outcomes in the same game (a spread and a total, for example) or multiple games in the same sport on the same day introduces correlation. A systematic shock that hits one bet (a key injury, weather, referee decisions) can hit correlated bets simultaneously. Kelly does not account for this, and the formula will understate true risk.
Short-run experience
Kelly’s mathematical optimality holds over a theoretically infinite number of bets. Over 50 or 100 bets — a realistic short-term sample — the variance at full Kelly is high enough that the experience can be psychologically destructive even with a genuine edge. Bettors who adopt full Kelly often abandon the approach after a bad run before the mathematics have a chance to work. Half or quarter Kelly produces a more sustainable variance profile for the vast majority of practitioners.
Practical application: how to use Kelly
The following process describes how to integrate Kelly into a real betting workflow.
- Establish an independent win probability. This must come from your model or analysis, not from the bookmaker’s odds. If you use the implied probability of the offered odds as your p, the Kelly formula always returns zero (by definition, the expected value is zero). Your probability estimate must be independently derived.
- Convert odds to decimal format and calculate b. b = decimal odds − 1. Use the Kelly Calculator to automate this step.
- Check that EV is positive before sizing. If f* is negative, do not bet. A negative Kelly result is not a sizing problem — it is a bet selection problem. Find a better line or a better probability estimate.
- Apply your chosen fraction. Default to half Kelly unless you have strong reasons to deviate. If you are new to a market or model, start at quarter Kelly.
- Cap the stake at a reasonable maximum. Most practitioners cap individual bets at 5–10% of bankroll regardless of Kelly output. A Kelly recommendation above 15% usually reflects either an unusually large edge or an overestimated probability — both warrant extra scrutiny.
- Recalculate on current bankroll, not original bankroll. Kelly is a percentage of your current bankroll. After a significant win or loss, the dollar amounts change even if the fraction does not.
- Track results and calibrate. After 200–300 bets, compare your probability estimates to actual outcomes. If you are systematically winning 54% when you estimated 57%, reduce your model’s inputs accordingly to correct for overconfidence.
Do not use Kelly to rationalise larger bets on high-confidence picks. The formula exists to discipline bet sizing according to measured edge, not to provide mathematical cover for gut conviction. If the formula returns 3% and you believe in the pick “more than 3% worth,” the problem is with your probability estimate, not the formula.
Kelly Criterion vs flat staking
Flat staking means betting a fixed unit (e.g. 1% of starting bankroll, or $50) on every bet regardless of the estimated edge or odds. It is the dominant approach among recreational bettors and is simple to execute. Kelly is more complex but mathematically superior over large samples when probability estimates are accurate.
| Dimension | Flat staking | Kelly staking |
|---|---|---|
| Complexity | Simple — same unit every bet | Requires probability estimate and calculation per bet |
| Stake variation | Constant | Variable — larger stakes on higher-edge bets |
| Drawdown during losing runs | Fixed unit loss per bet | Shrinking units reduce drawdown automatically |
| Long-run growth | Linear growth | Geometric growth — superior over large samples |
| Requires accurate probability estimates | No | Yes — inaccurate estimates degrade performance |
| Works well for | Recreational bettors; bettors without calibrated models | Systematic bettors with validated probability models |
The honest summary: flat staking with a genuine edge is fine. Kelly with a genuine edge is better over time. Kelly with a poor probability estimate is worse than flat staking. The quality of your win probability estimate determines which approach serves you better.
Need accurate win probabilities to plug into Kelly?
Kelly staking only works when the win probability input is independently estimated — not derived from the odds you are evaluating. ZCode System provides AI-generated probability scores across NFL, NBA, MLB, NHL, and major soccer markets, calibrated against historical data across 80+ parameters. These probability estimates give you a quantitative starting point for Kelly calculations, though whether they generate genuine edge in your specific markets is something you should verify with a tracked sample before committing significant bankroll.
Explore ZCode SystemAffiliate disclosure: we earn a commission if you sign up via this link at no extra cost to you. We recommend ZCode because it provides probability-based analysis, not because of commission rate.
FAQ
Frequently asked questions
What does the Kelly Criterion calculate?
The Kelly Criterion calculates the optimal fraction of a bankroll to stake on a bet where you have a positive expected value. The result is a percentage of your current bankroll, not a fixed dollar amount. A result of 0.05 means stake 5% of your bankroll. A negative result means the bet has negative expected value and no stake should be placed.
Why do most bettors use half Kelly instead of full Kelly?
Full Kelly is mathematically optimal only with perfect probability estimates. In practice, all probability estimates contain some error. Overbetting relative to your true edge reduces long-run growth. Half Kelly provides approximately 75% of full Kelly’s expected growth rate while reducing variance by roughly 50%, giving a better risk-adjusted outcome when probability estimates are imperfect — which they always are.
What does a negative Kelly result mean?
A negative Kelly result means the bet has negative expected value: your estimated win probability is below the break-even probability implied by the odds. Do not bet. A negative result is not an instruction to bet the other side — it is confirmation that the bet, as structured, is not worth placing. Either the odds are too short, or your probability estimate is too low. There is no fraction of a negative-EV bet that produces positive expected growth.
What happens if you overbet Kelly?
Overbetting Kelly reduces long-run expected growth despite the higher individual stakes. At 1.5x Kelly, expected growth is lower than at full Kelly. At 2x Kelly, the expected long-run growth rate is mathematically zero — equivalent to not betting. Beyond 2x Kelly, you are in negative expected growth territory. This is not intuitive, but it is a mathematical consequence of the compounding structure Kelly operates within.
Can Kelly Criterion be used for multiple simultaneous bets?
Classic Kelly assumes one bet at a time. When placing multiple concurrent bets, summing individual Kelly recommendations can result in combined exposure of 20–30% or more of bankroll, which is higher than optimal. Common solutions include dividing each individual Kelly recommendation by the number of concurrent open positions, or applying a global cap such as a maximum of 20% of total bankroll across all open bets simultaneously.
What is the difference between the Kelly Criterion and flat staking?
Flat staking means betting the same unit on every bet regardless of edge or odds. Kelly scales each stake to the size of the estimated edge. Kelly produces superior long-run growth when probability estimates are accurate, because it bets more on higher-edge opportunities and less on marginal ones. Flat staking is simpler and tolerates imprecise probability estimates better. The right choice depends on the quality of your probability estimation process.
How does Kelly Criterion relate to expected value?
Expected value (EV) determines whether to place a bet at all. The Kelly Criterion determines how much to stake once you have confirmed positive EV. They are sequential steps in a complete betting decision: first calculate EV to filter bets, then apply Kelly to size the ones worth placing. A positive EV bet with no Kelly sizing system will eventually reach the correct long-run result, but more slowly and with greater variance than Kelly staking on the same bets. You can use the EV Calculator and the Kelly Calculator together for this workflow.
Responsible gambling notice. The Kelly Criterion is a mathematical staking framework, not a guarantee of profit. Sports betting involves financial risk and variance. Even correctly sized positive-EV bets lose. Never bet more than you can afford to lose. NCPG | BeGambleAware | Gambling Therapy