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CONTENTS
The Kelly criterion is a staking formula: f* = (b × p − q) ÷ b, where f* is the fraction of a bankroll to stake on one bet. It is built for bets with a positive expected value, and when the result is zero or negative, the formula’s answer is to stake nothing. Every standard casino bet has a negative expected value for the player, so for those bets Kelly never recommends a stake.
This guide explains the formula, defines each symbol, and works through examples with hypothetical numbers. It shows how the growth rate changes when the stake is too small or too large, why an uncertain probability makes full Kelly risky, and why the criterion cannot beat a negative edge. All probabilities here are assumed inputs, not predictions.
What the Kelly Criterion Does and Assumes
Kelly sizes a bet to maximize long-run growth. It answers one question: if the same favorable bet can be repeated many times and winnings are reinvested, what fraction of the current bankroll gives the fastest long-run growth? The answer is a percentage of the bankroll, so the stake shrinks after losses and grows after wins.
The criterion starts from an edge you already believe you have. It does not find edges. The input p is the probability of winning, and the formula is only as reliable as that number. For a coin that is known to be biased, p can be calculated. For a sports outcome, p is an estimate and can be wrong, which is the central practical problem.
Several assumptions sit behind the formula: the bets are independent and repeatable, the probability and the payout are known, the bankroll can be split into any fraction, and the goal is long-run growth of the logarithm of the bankroll rather than the biggest average profit. Those assumptions rarely hold exactly, so the output is a reference point, not an instruction.
The Kelly Criterion Formula Explained
The formula uses three inputs. The win probability p, the loss probability q = 1 − p, and the net odds b, which is the profit on a winning $1 stake. Net odds are decimal odds minus 1, a conversion covered when you convert American odds to decimal.
where:
f* = fraction of the current bankroll to stake
b = net odds, or decimal odds − 1 (profit per $1 staked)
p = estimated probability of winning
q = 1 − p, the probability of losing
The numerator is the expected value per $1 staked. From the expected value formula, EV per $1 = b × p − q. Dividing by b gives a second form of Kelly, which is easier to remember and check.
The growth rate shows why this stake is special. If a fraction f of the bankroll is staked on each bet, the average growth of the logarithm of the bankroll per bet is given below, using the natural logarithm (ln).
where:
g(f) = average log growth per bet, positive means the bankroll tends to grow
f = fraction of bankroll staked, between 0 and 1
The value of f that maximizes g(f) is f*. If f* is zero or negative, no positive stake makes g(f) positive, so the correct stake is zero. A negative f* does not mean taking the other side, because standard casino games do not offer one.
Worked Examples With Hypothetical Numbers
Example 1: even-money bet with an assumed 55% chance. Suppose a bet pays 1 to 1 (decimal odds 2.00) and p is assumed to be 0.55.
Step 1: b = 2.00 − 1 = 1; p = 0.55; q = 0.45.
Step 2: EV per $1 = 1 × 0.55 − 0.45 = 0.10.
Step 3: f* = 0.10 ÷ 1 = 0.10, so stake 10% of the bankroll.
Example 2: decimal odds of 2.10 and an assumed 50% chance. Suppose p is 0.50, so the bet pays a little more than fair.
Step 1: b = 2.10 − 1 = 1.10; q = 0.50.
Step 2: EV per $1 = 1.10 × 0.50 − 0.50 = 0.05.
Step 3: f* = 0.05 ÷ 1.10 ≈ 0.0455, so stake about 4.55% of the bankroll.
Example 3: a European roulette bet on red. The win probability is 18/37, the payout is 1 to 1 (b = 1) and q = 19/37.
Step 1: EV per $1 = 1 × 18/37 − 19/37 = −1/37 ≈ −0.0270.
Step 2: f* = (−1/37) ÷ 1 ≈ −0.0270, which is below zero.
Step 3: The Kelly stake is $0. No positive fraction raises the growth rate.
The result in the third example holds for every standard casino bet, because each has a negative expected value for the player. The formula is not a way around the roulette house edge; it simply confirms that no stake is justified.
What Happens When the Stake Is Too Big or Too Small
The table uses Example 1 (p = 0.55, b = 1, full Kelly = 10%) and computes g(f) for several stakes. The last column converts g into a typical growth factor after 100 bets, e^(100 × g). It describes a typical path, not a guarantee, and every figure is rounded.
| Stake (fraction of bankroll) | Name | g(f) per bet | Typical growth after 100 bets |
|---|---|---|---|
| 5% | Half Kelly | +0.375% | about ×1.46 |
| 10% | Full Kelly | +0.501% | about ×1.65 |
| 20% | Double Kelly | −0.014% | about ×0.99 |
| 25% | 2.5 × Kelly | −0.673% | about ×0.51 |
Staking twice the Kelly fraction removes the growth entirely, even though the bet is favorable. Staking less than Kelly grows more slowly but with smaller swings, which is why fractions such as half Kelly are popular. Higher stakes also raise the variance explained in variance in gambling.
Estimation error makes overbetting easy. Suppose the true probability is 0.52, but the stake of 10% was based on an estimate of 0.55. The true Kelly fraction is then 4%, and g(0.10) = −0.101% per bet, so the typical growth after 100 bets is about ×0.90. A real edge existed, yet the stake was too large to benefit from it.
Common Mistakes With the Kelly Criterion
Treating Kelly as a way to beat the casino is the first and most important mistake. The formula needs a positive edge, and standard casino games do not offer one. Using decimal odds as b is the second: b is the net profit per $1, so decimal odds of 2.00 mean b = 1, not 2.
Trusting the input probability is the third. In sports markets p is an estimate, and a small overestimate can turn a favorable stake into a losing one. Ignoring the swings of full Kelly is the fourth: at a 10% stake, three losses in a row leave 0.9 × 0.9 × 0.9 = 72.9% of the bankroll.
Believing a positive f* guarantees profit is the fifth. Kelly maximizes long-run growth under its assumptions, but any single sequence of bets can lose. Gambling should be treated as paid entertainment, and this guide describes arithmetic, not advice.
Quick Reference and Check Your Answer
For an even-money bet (b = 1), the formula simplifies to f* = 2p − 1. The table below shows a few assumed win probabilities.
| Assumed win probability (p) | EV per $1 | Kelly fraction f* |
|---|---|---|
| 0.50 | 0.00 | 0% |
| 0.52 | 0.04 | 4% |
| 0.55 | 0.10 | 10% |
| 0.60 | 0.20 | 20% |
| 18/37 ≈ 0.4865 (roulette red) | −0.027 | negative, so stake $0 |
To check an answer, compute the EV per $1 first, since f* has the same sign. Then divide by b and confirm that f* is below 1. If the EV is zero or negative, the stake must be zero.
Related Concepts and Next Steps
To see why a staking pattern cannot fix a negative edge, read about the Martingale system, which tests a different system with the same expected-value logic. The expected value formula guide explains the numerator of Kelly, and variance in gambling explains the swings that come with larger stakes. For converting odds into the b used here, see how to convert American odds to decimal.
Frequently Asked Questions
