KEY TAKEAWAYS
CONTENTS
The Kelly Criterion is a formula that turns two inputs, the odds a sportsbook offers and a bettor’s own estimate of the true win probability, into a recommended stake expressed as a percentage of bankroll. It is a bet-sizing concept, not a method for picking winners: the formula assumes the bettor already believes a wager has an edge, and it only answers how much of the bankroll to risk on it. When the estimated edge is zero or negative, the formula recommends no stake at all. Its appeal is that stakes scale with the size of the perceived edge, but that is also its weakness, because the output is only as reliable as the probability estimate behind it, and an estimate is an assumption, not a fact. This article explains the formula, works through hypothetical examples (including one where the estimate is wrong), presents full and fractional Kelly as a risk trade-off, and covers the limitations that keep Kelly from being a profit system.
What the Kelly Criterion Is and What It Is Not
The Kelly Criterion is a bet-sizing formula that maximizes the theoretical long-run growth rate of a bankroll, given known odds and a known probability of winning. John L. Kelly Jr. described it in a 1956 paper in the Bell System Technical Journal, originally in the context of information theory, and gamblers and investors later adapted it to wagering. The key phrase is “known probability”: the math is exact only when the win probability is known, which is almost never true in sports betting.
Kelly sits between two ideas already covered on this blog, and it builds directly on both. Expected value answers whether a wager has an edge at all, by comparing an estimated probability with the price. Flat units and percentage staking, covered in bankroll management and unit sizing, apply a fixed rule to every bet regardless of how big the edge looks. Kelly takes the edge that expected value identifies and converts it into a stake that grows or shrinks with that edge, which makes it a third, more mathematical approach to the same question of how much to risk.
It helps to be clear about what Kelly does not do. It does not find value, it does not improve the chance that any single bet wins, and it does not remove variance. If the inputs are wrong, the output is wrong with the same mathematical confidence, so Kelly is best understood as a way of sizing an opinion rather than a strategy that produces edges by itself.
The Kelly Formula, Step by Step
The standard form of the formula is f* = (bp − q) ÷ b. Here f* is the fraction of the current bankroll to stake, b is the net odds received on a win (decimal odds minus 1), p is the bettor’s estimated probability of winning, and q is the probability of losing, which equals 1 − p. A result of 0.05 means staking 5% of the bankroll, and a result of zero or below means no bet.
Applying it takes four steps. First, convert the offered price to decimal odds and subtract 1 to get b. Second, write down an estimated win probability p, and recognize that this number is the single most consequential input. Third, compute q as 1 − p and evaluate (bp − q) ÷ b. Fourth, multiply f* by the current bankroll to get a dollar stake. The stake is recalculated for each new bet because the bankroll changes after every result.
The numerator, bp − q, is the bettor’s expected profit per $1 staked, so Kelly recommends a positive stake only when expected value is positive. Dividing by b scales the stake to the price: for the same expected profit per $1, a longer price (larger b) produces a smaller recommended stake than a shorter one, because the wager is more volatile. The break-even probability for any price is 1 ÷ decimal odds. If the estimated p is at or below that number, the numerator is zero or negative and Kelly says to pass. For example, at -110 (decimal 1.9091, so b = 0.9091) and an estimated p of 55%, f* works out to 5.50% of the bankroll, while an estimated p of 52.38%, the break-even probability, gives 0%.
A Worked Example With Hypothetical +110 Odds
Suppose a sportsbook lists hypothetical odds of +110, which is 2.10 in decimal odds, so b = 1.10. The break-even probability is 1 ÷ 2.10 = 47.62%. Assume a bettor, after their own analysis, estimates the true win probability at 50%, so q = 0.50. These are illustrative numbers, not a current market price, and the 50% is an assumption.
The formula gives f* = (1.10 × 0.50 − 0.50) ÷ 1.10 = 0.05 ÷ 1.10 = 4.545% of bankroll. With a hypothetical $1,000 bankroll, full Kelly is a $45.45 stake. If the bet wins, the profit is $45.45 × 1.10 = $50.00 and the total payout is $95.45, which includes the returned stake. If it loses, the bettor is down $45.45, or 4.55% of the bankroll, leaving $954.55.
Expected value per $1 staked, on the bettor’s own estimate, is 2.10 × 0.50 − 1 = +$0.05, or 5% of each dollar risked. That is a theoretical average over many bets that assume p is correct. Any single bet still loses half the time under this assumption, and a run of losses is entirely normal, which is why the rest of this article focuses on what happens when the assumptions fail.
Why the Probability Estimate Is the Weak Point
Kelly’s recommended stake reacts sharply to small changes in p. The table below holds the same hypothetical +110 price (b = 1.10) and $1,000 bankroll constant and changes only the bettor’s estimated win probability. Notice that a move from 50% to 55% more than triples the recommended stake, and that every estimate at or below 47.62% produces no stake.
| Estimated win probability (p) | Full Kelly (f*) | Stake on $1,000 |
|---|---|---|
| 46.00% | 0% (negative edge) | $0.00 |
| 47.62% (break-even) | 0.00% | $0.00 |
| 48.00% | 0.73% | $7.27 |
| 50.00% | 4.55% | $45.45 |
| 52.00% | 8.36% | $83.64 |
| 55.00% | 14.09% | $140.91 |
The table shows that small errors in p become large errors in stake size. Now suppose the bettor estimates 55% but the true probability is only 50%. Full Kelly on the estimate stakes $140.91, while the correct Kelly stake for a true 50% is $45.45, so the bettor risks about 3.1 times the intended amount. The bet still has positive expected value (+$0.05 per $1), but the oversized stake makes the bankroll far more exposed to a losing run.
The worse case is an estimate that is wrong enough to erase the edge. If the bettor estimates 50% but the true probability is 46%, the expected value per $1 is 2.10 × 0.46 − 1 = −$0.034, and a $45.45 stake carries an expected loss of about $1.55. Kelly had no way to know, because it trusts whatever probability it is given.
Full Kelly, Half Kelly, and Volatility
The formula above is called full Kelly. Its theoretical appeal is the highest long-run growth rate when the probability is known exactly, but the path is bumpy. Using the same hypothetical +110 example, five consecutive losses at full Kelly (4.545% each time) would leave a $1,000 bankroll at about $792, a drawdown of roughly 21%. If the bettor had staked 14.09% each time because of an inflated estimate, five losses would leave about $468, a decline of more than half. Streaks like this are a normal feature of variance in sports betting, not evidence that something has broken.
A fractional Kelly strategy stakes only a set share of the full-Kelly amount. Half Kelly would be $22.73 in the original example, and quarter Kelly $11.36. The same five-loss streak at half Kelly leaves about $891, a drawdown of roughly 11%. Practitioners commonly use fractions for two reasons: to reduce volatility, and to leave a cushion against an overestimated p.
This is a risk-reduction trade-off, not a way to win. A smaller fraction lowers the theoretical growth rate and lengthens the time needed to reach any given bankroll target, in exchange for smaller swings and less sensitivity to estimation error. Staking more than full Kelly increases the risk of ruin without a matching benefit. No fraction makes a bet with no real edge profitable, and none removes the possibility of losing money.
Common Mistakes and Misconceptions
The most common mistake is treating an estimated probability as a known fact. A gut-feel 55% or a model output is still a guess, and the formula converts it directly into a stake. Bettors who feed it a confident number without testing that number against results are effectively sizing their own overconfidence.
A second misconception is that Kelly guarantees bankroll growth. The growth claim applies only to an accurate probability, an unlimited series of repeated bets, and stakes that can be sized freely. Real bettors have none of those conditions, and a positive expected value can still coexist with a losing year.
A third error is using Kelly to justify bigger stakes after a loss. Kelly is recalculated from the current bankroll, so a loss actually shrinks the next recommended stake; scaling up to win money back is chasing losses, and it is the opposite of what the formula does. A fourth is applying the formula to a bet with no real edge because the stake “looks small.” If p is at or below break-even, the answer is zero, however small the fraction would be. Finally, some bettors read the output as a target rather than a ceiling, forgetting that the output is already an optimistic number.
Limitations: Where Kelly Fits in Practice
Kelly’s clean math rests on assumptions that real betting breaks. Probabilities are uncertain, so the stake inherits that uncertainty. Bets are often placed at the same time rather than one after another, and the simple formula treats each bet in isolation, which ignores overlap when outcomes are connected, as with correlated parlay legs. Sportsbooks also impose limits and can restrict accounts, so a recommended stake may not be placeable, and prices can move before a bet is entered.
In practice, Kelly appears as a reference point for the size of a stake in the decision process that follows probability estimation. A bettor first estimates p and checks whether the price offers an edge, then compares the Kelly output with their own unit rules and tolerance for drawdowns, and commonly caps the stake well below the raw number. Stakes should also come only from money set aside for betting, never from funds needed for essentials. The formula informs a decision; it does not make it, and it never substitutes for judging whether the estimate itself deserves trust.
Related Concepts and Next Learning Steps
Kelly connects several ideas in this library. Expected value defines the edge, bankroll management provides the simpler staking frameworks, and variance explains why results swing even when the math is sound. The natural next step is learning how much data is needed to judge an estimate, which is the subject of why sample size matters in sports betting results. A bettor who cannot show that their probability estimates are reliable has little basis for trusting any formula built on top of them, and the Kelly Criterion is best treated as a tool for understanding risk rather than a promise about results.
