The Kelly Criterion
Updated May 18, 2026 by Sam Smith
The Kelly criterion is a bet-sizing formula that calculates the optimal fraction of a bankroll to wager on a known +EV bet. Used correctly, it maximizes the long-run growth rate of the bankroll while keeping the risk of ruin at zero — assuming the bettor's probability estimates are accurate.
The expanded definition
The Kelly formula, developed by John Kelly Jr. at Bell Labs in 1956, answers a specific question: given a positive-expectation bet, what fraction of the bankroll should be wagered to maximize long-run growth? The standard form is:
f = (b × p − q) ÷ b
where b is the decimal odds minus 1, p is the bettor's estimated win probability, and q is 1 − p. The output f is the fraction of the bankroll to stake.
Worked example: a bet at +120 with a 55% estimated win probability and a $10,000 bankroll. b = 1.2, p = 0.55, q = 0.45:
f = (1.2 × 0.55 − 0.45) ÷ 1.2 = 0.21 ÷ 1.2 = 0.175. Full-Kelly stake: 17.5% × $10,000 = $1,750.
That is the bet size that maximizes the long-run growth rate of the bankroll if the 55% probability estimate is exactly right. Most professionals stake at half-Kelly ($875 in this example) or quarter-Kelly to absorb the fact that probability estimates are never perfectly calibrated.
Why the Kelly criterion matters
Flat staking — betting the same amount on every play — ignores the size of the edge. Two bets at +EV are not equivalent: a bettor with a 10% edge should risk far more than a bettor with a 1% edge. Kelly is the mathematical answer to that question. It also explains why winning bettors who size bets by gut feel eventually go broke: over-sizing a real edge produces ruinous drawdowns; under-sizing leaves money on the table. Disciplined +EV bettors track their bankroll, estimate the edge per bet, and size accordingly — and almost always at a fraction of full Kelly.
Related terms
- Positive expected value (+EV) — the precondition for Kelly sizing; Kelly only applies to bets with a positive expectation.
- No-vig odds — the fair-price benchmark used to compute the edge that feeds into the Kelly formula.
- Closing line value (CLV) — the long-run evidence that the probability estimates driving Kelly sizing are sound.
- Hedging — taking the opposite side of an open position; Kelly framing shows why reflexive hedging on +EV positions shrinks long-run bankroll growth.
Frequently Asked Questions
What is the Kelly criterion in sports betting?
The Kelly criterion is a bet-sizing formula that calculates the optimal fraction of a bankroll to wager on a known +EV bet. It was originally developed by John Kelly Jr. at Bell Labs in 1956 and was later popularized in gambling and investing. Used correctly, it maximizes the long-run growth rate of the bankroll while keeping the risk of ruin at zero.
How is the Kelly criterion calculated?
The Kelly criterion is calculated as f = (bp − q) ÷ b, where b is the decimal odds minus 1, p is the bettor's estimated win probability, and q is 1 − p. For +120 odds (b = 1.2) and a 55% win probability: f = (1.2 × 0.55 − 0.45) ÷ 1.2 = 17.5%. That is the full-Kelly stake — the share of bankroll Kelly says is optimal for that single bet.
What is fractional Kelly and why is it used?
Fractional Kelly is sizing each bet at a fixed fraction of the full Kelly stake — most commonly half-Kelly or quarter-Kelly. It is used because full-Kelly assumes the bettor's probability estimate is perfectly calibrated; even small estimation errors at full-Kelly cause large bankroll swings. Half-Kelly gives up about 25% of the long-run growth rate in exchange for roughly half the variance, which most bettors prefer.
What are the risks of using full Kelly stakes?
The risks of using full Kelly stakes are aggressive variance and severe drawdowns when probability estimates are slightly miscalibrated. Full-Kelly assumes the bettor knows the true win probability; in practice nobody does. Over-estimating an edge by even a few percentage points at full-Kelly can produce 50%+ bankroll swings. Most professionals use half-Kelly or smaller as a margin of safety.
Does OddsShopper recommend a Kelly bet size?
Yes, OddsShopper's Portfolio EV tool computes a modified Kelly stake for every +EV bet it surfaces, using the gap between its no-vig fair price and the offered price as the edge input. The default sizing is fractional — closer to quarter-Kelly than full-Kelly — to absorb model uncertainty. Bettors can override the sizing or use the suggested stake directly when placing bets.
Sam Smith
Sam Smith is a writer and editor with Stokastic and OddsShopper. He has been immersed in the world of professional sports data since 2015 while also writing extensively on the NFL for a multitude of blogs and websites. With OddsShopper, Sam looks to blend his sports and editorial expertise with OddsShopper's data to bring you the best betting information possible.
