The Kelly Criterion in Betting: Formula, Worked Example and Fractional Kelly

The Kelly criterion is a formula that answers the question "what share of my bank should I stake" when you know the probability of an outcome and the odds. It picks the stake size at which the bank grows fastest over the long run, and it says not to bet at all if the bet has no positive expected value. The Kelly criterion in betting is popular because it ties stake size to your edge: the bigger the edge, the bigger the stake.

One thing up front: betting carries the risk of losing money, and the formula knows nothing about the real probability — it trusts whatever number you give it. You can work out the share of the bank for your own numbers in the calculator below; it is also available on its own page — the Kelly calculator.

The Kelly formula

For a bet at decimal odds the formula is:

share of bank = (p × k − 1) / (k − 1)

Here p is your win probability and k is the odds. The numerator p × k − 1 is the expected value of the bet per unit staked: how much each unit returns on average on top of the unit itself. The denominator k − 1 is the net win per unit staked.

Example. You estimate the probability of an outcome at 55%, and the bookmaker offers odds of 2.00. Expected value: 0.55 × 2.00 − 1 = +0.10, that is +10% per bet. Share of bank: 0.10 / 1 = 10%. With a bank of 1,000, full Kelly suggests a stake of 100.

Second example. Probability 45%, odds 2.50. Expected value: 0.45 × 2.50 − 1 = +0.125. Share of bank: 0.125 / 1.5 ≈ 8.3%. The odds are higher, but so is the chance of losing, so the share is smaller than in the first example.

If the numerator is zero or negative, the formula gives zero: with that estimate the bet loses money on average, and Kelly says not to make it. This is the key condition — without positive expected value no stake size will help. What positive expected value is and how to look for it is covered in the article on value bets.

Where does p come from? It is your own estimate of the chance. It is worth comparing with the probability the bookmaker builds in (1 / k, adjusted for the margin) and with an independent estimate — for example, the estimated probabilities in match predictions. But each of these numbers is also an estimate, not knowledge.

Why Kelly grows the bank fastest

Kelly maximises not the average win of a single bet but the growth rate of the bank over many bets, each sized from the current bank. To see how it works, take the first example — a probability of 55% at odds of 2.00 — and work out the typical result after 100 bets for different shares of the bank. Typical means with the usual ratio of wins to losses for that probability.

Two things stand out. More than Kelly does not mean faster: at twice the calculated stake the bank stops growing, even though the edge is real. Less than Kelly is slower, but not by much: half Kelly keeps about three quarters of the growth rate.

The main weakness: the probability is never known exactly

The whole formula rests on p. In the examples above the probability was taken as exactly known, but in betting it never is. Models, statistics and intuition give an estimate with an error of a few percentage points, and that error is rarely in your favour: bets that look "obvious" are more often overrated than underrated.

Let us see what happens if you think the probability is 55% when it is really 50%. At odds of 2.00 there is no real edge, but you stake 10% of the bank by full Kelly. The typical bank after 100 bets is about ×0.6, that is minus 40%. Had you staked quarter Kelly, 2.5% of the bank, the typical result would be about ×0.97 — minus 3%. The same five-point error costs almost half the bank in the first case and a few percent in the second.

The calculator in this article shows this straight away: under the table it says what the right share would be with a probability 5 points below yours. For the example of 55% at odds of 2.00, the answer is zero.

Fractional Kelly: half and quarter

That is why full Kelly is rarely used in practice. Fractional Kelly is the same formula with the result divided by 2 (half Kelly) or by 4 (quarter Kelly).

• Half Kelly gives about 75% of the growth rate of full Kelly with half the swings of the bank. An error in the probability estimate costs noticeably less.
• Quarter Kelly grows the bank more slowly, but hardly suffers from a moderate overestimate of the probability and survives long losing streaks.

A good rule: the less you trust your probability estimate, the smaller the fraction of Kelly you use. If your edge is not confirmed by a log of hundreds of bets, it is wiser not to rely on the formula at all and to stake a small fixed share of the bank — a flat 1–2%, as described in the article on bankroll management.

The Kelly strategy in practice

If you do use the Kelly strategy, keep a few details in mind:

1. Work from the current bank. Kelly assumes stakes grow after a win and shrink after a loss. Staking a share of the starting bank is a different system.
2. Simultaneous bets share the bank. If you have five bets on one evening, the sum of their shares should not turn into half the bank. It is sensible to reduce each one.
3. Do not raise the share to win losses back. After a loss the bank is smaller, and the Kelly stake should be smaller too.
4. Record the probability you used. After a hundred bets the log will show whether your 55% really was 55%.
5. Zero is an answer. If the formula gives zero, there is no bet, even if the match looks interesting.

The short version

The Kelly criterion turns an edge into a stake size: the share of the bank is (p × k − 1) / (k − 1). It grows the bank faster than any other share, but only when the probability is known exactly. Since an estimate is always approximate, in practice people use half or quarter Kelly, and without a proven edge they bet a small flat stake.