What the idea is
The Kelly criterion comes from information theory and probability. It answers a narrow, theoretical question: if you face a repeated bet whose probabilities and payout you know exactly, and the bet is genuinely in your favour, what fraction of your capital maximises its long-run growth rate?
It is a piece of mathematics about growth, not a system for finding bets or a promise of profit. Real situations rarely satisfy its assumptions.
The formula for a simple bet
For a bet that returns b units of profit for each unit staked on a win, and loses the stake otherwise, with win probability p and loss probability q = 1 - p, the fraction of capital is:
f = (b x p - q) / b
The top of the fraction, b x p - q, is the expected profit per unit staked. In that sense the formula scales the edge by the payout. When the expected profit is positive, f is positive. When it is zero, f is zero. When it is negative, the formula returns a negative number, which is read as: do not bet.
Worked example with a positive edge
This is a hypothetical, with probabilities assumed known. Suppose an even-money bet (b = 1) wins with probability 0.55.
f = (1 x 0.55 - 0.45) / 1 = 0.10
The formula suggests 10% of capital per bet. Why that figure? Kelly maximises the average growth of the logarithm of capital per bet:
- At 10%: 0.55 x ln(1.1) + 0.45 x ln(0.9) is about 0.55 x 0.0953 + 0.45 x (-0.1054) = 0.0524 - 0.0474, roughly +0.005 per bet.
- At 20%: 0.55 x ln(1.2) + 0.45 x ln(0.8) is about 0.55 x 0.1823 + 0.45 x (-0.2231) = 0.1003 - 0.1004, roughly 0, slightly below.
So twice the Kelly fraction produces about zero long-run growth, even though the edge is still positive. Staking more than the theory suggests can destroy growth. This is the sense in which over-sizing a stake is costly. It also shows why swings are large: at 10% per bet, a few losses in a row remove a noticeable part of the capital.
What happens in casino games
Now apply the formula to a standard casino bet. An even-money bet on a single-zero roulette wheel wins with probability 18/37 and loses with probability 19/37.
f = (18/37 - 19/37) / 1 = -1/37, about -0.027
The result is negative. The Kelly answer to a negative edge is a stake of zero. No stake size gives positive growth, and betting any positive fraction reduces expected log-growth. This holds for every casino game with a house edge. The criterion does not rescue a negative expected value; it simply confirms the sign of the edge. It is also why a stake-sizing rule cannot create an advantage, and why betting systems fail to change the underlying expectation.
Assumptions that rarely hold
Even where a real positive edge could exist, using the formula properly demands a lot.
- The true probability must be known, not guessed. A small error in p moves f a lot, and overestimating an edge leads to overbetting, which as shown above can erase growth.
- Bets must be repeatable and independent, with capital freely divisible.
- The goal must be maximum long-run growth, which accepts big swings. Many people would be uncomfortable with them, which is a matter of risk tolerance.
- There is no stopping rule, ruin by drawdown, or limit on bet size.
Because of these limits, people who study the idea often discuss using a smaller fraction of the Kelly amount to reduce swings. That is a theoretical discussion about volatility versus growth, not advice.
How it connects to the rest of bankroll maths
Kelly is one view of the broader relationship between stake size and swings. The same pattern appears in gamblers ruin, where a finite bankroll meets a negative edge and runs out on average. In a game with a house edge, the stake-size question is moot: the lesson of the criterion is that the first requirement is a positive edge, and in casino games there is none.
It is also a reminder to separate a good theoretical tool from a practical promise. Outcomes are uncertain even with a genuine edge, and results in the short run can differ widely from the average.
If gambling is causing harm, or you are tempted to borrow or chase losses, a local support service can help.