John L. Kelly Jr.'s 1956 paper established the expected-log-growth result in an information-theory setting. Applying the idea to betting requires additional assumptions about the payoff, available capital, repeated opportunities, and probability estimates (Kelly, 1956).
The Kelly Criterion Formula
The formula is:
f = (bp - q) / b
Where:
- f = the fraction of the bankroll to stake
- b = the decimal odds minus 1 (the net profit per unit staked)
- p = your estimated probability of winning
- q = the probability of losing (1 - p)
Checked illustrative example
In this illustrative example, a model estimate is 60% and the available decimal price is 1.80.
- b = 1.80 - 1 = 0.80
- p = 0.60
- q = 0.40
f = (0.80 x 0.60 - 0.40) / 0.80 f = (0.48 - 0.40) / 0.80 f = 0.08 / 0.80 f = 0.10
Under those inputs, full Kelly is 10% of bankroll. On a 500-unit bankroll that is 50 units. This is the formula's conditional output, not evidence that the 60% estimate is correct or that 50 units is affordable.
When Kelly Says Do Not Bet
If the formula produces a negative number, the constrained Kelly allocation is zero: it does not recommend taking the other side unless that side has been modelled separately. At a 45% estimated win probability and odds of 1.80, (0.80 x 0.45 - 0.55) / 0.80 = -0.2375, so this binary calculation allocates nothing to the selection.
Full Kelly vs Fractional Kelly
Fractional Kelly multiplies the formula's result by a chosen fraction:
| Approach | Fraction | Stake (500 pound bankroll, f=0.10) |
|---|---|---|
| Full Kelly | 100% | 50 pounds |
| Three-quarter Kelly | 75% | 37.50 pounds |
| Half Kelly | 50% | 25 pounds |
| Quarter Kelly | 25% | 12.50 pounds |
Reducing the fraction reduces the amount exposed on this bet. The exact effect on long-run growth and drawdown depends on the true probabilities, sequence of results, available bets, and dependence between them. There is no universal claim that half Kelly gives one fixed percentage of full-Kelly growth.
Practical Limitations
The Kelly Criterion is mathematically elegant but has real-world challenges:
The probability is not known. The mathematical optimum assumes the input probability is correct. In practice it is estimated. Overstating the win probability can create a positive and large allocation where none is justified.
The binary formula is not a portfolio model. Applying it separately to simultaneous dependent positions omits their joint payoff structure. Independence and dependence change how joint probabilities are calculated (OpenStax independent events).
Drawdowns remain possible. Maximising expected logarithmic growth is not the same objective as minimising short-term losses, preserving money needed for living costs, or keeping drawdown below a chosen limit.
Bankroll must be defined honestly. The fraction applies to capital allocated to the strategy, not household money or funds required for essentials. Recalculating against an inflated bankroll defeats the risk constraint.
Prices and settlement matter. Commission, dead heats, refunds, and non-binary outcomes change the net payoff and therefore the expected-value calculation (OpenStax expected value).
Kelly Criterion in Practice
Use the calculation as a sensitivity test rather than a label of certainty:
- Calculate the output with the central probability estimate.
- Repeat with a materially lower win probability.
- Include all costs and settlement outcomes.
- Aggregate exposure across related positions.
- Apply a separate affordability cap, including a zero-stake option.
If a small change in probability turns the output from positive to zero, the decision is dominated by estimation uncertainty. Kelly does not create an edge; it only transforms assumed inputs into a stake fraction.
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Related resources
Continue with Bankroll: What It Means in Betting for the next part of this topic, or return to Betting Glossary: Every Betting Term Explained in Plain English to compare the other guides in this collection.
Continue learning
- Next guide: Lay Bet
- Related guide: Matched Betting
- Go deeper: Kelly Criterion for Betting
Assumptions and limitations
The binary Kelly formula assumes a known win probability, known net payoff, repeatable opportunities, and capital that can be allocated according to logarithmic utility. Real betting probabilities are estimates and simultaneous positions can be dependent. Kelly's original paper supports the theoretical result, not a recommended stake or promised growth rate.
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