Odds, Probability & Bankroll

Kelly criterion

Formula (binary bet)

f* = (bp - q) / b, where b is net odds, p win probability, q loss probability⁴

Formula (continuous)

f* = μ / σ², where μ is expected return and σ² variance⁴

Goal

Maximise expected logarithmic growth of wealth, not single-bet profit¹³

Fractional Kelly

Using a fraction < 1 of the full Kelly stake to reduce volatility⁵

Also called

Kelly strategy, Kelly bet, proportional betting¹⁷

Key points

• The Kelly criterion chooses the stake that maximises the expected logarithm of wealth, equivalent to the long-run geometric growth rate.¹²
• For a simple binary bet, the optimal fraction is the advantage divided by the net odds paid on a 'to one' basis.³
• Kelly betting scales the stake with the current bankroll, so wins and losses compound over time.⁵⁶
• Fractional Kelly — using a fraction of the full Kelly stake — reduces volatility and drawdown risk while preserving some growth benefit.⁵
• The criterion applies to repeated betting and investment settings, including gambling, sports betting, and portfolio allocation.³

How the Kelly criterion works

The bettor estimates the edge (expected return) and stakes the Kelly fraction of the current bankroll rather than a fixed amount.³⁴ The logic is to maximise compounded growth over many repetitions, not to maximise the chance of winning any single bet.¹² Because the stake scales with bankroll, wins and losses both compound, which can lead to rapid growth but also to large drawdowns if the edge is overestimated.⁵⁶ In practice, bettors often use fractional Kelly to reduce risk, and the criterion has been extended to handle multiple simultaneous outcomes and continuous return distributions.⁸⁴ The growth rate G is defined as the limit of log wealth over time, and the formula's derivation assumes repeated independent or weakly dependent opportunities.²⁹ Sources note the criterion is sensitive to estimation error: overestimating edge leads to overbetting and potential ruin.⁶

Where this term is used

• Odds and probability

Not the same as

• Kelly criterion betting — the same concept applied specifically to sports and event betting
• Kelly criterion blackjack — the same staking rule used in blackjack when card counting gives a positive edge
• Kelly criterion poker — the same bankroll-growth rule applied to poker wager sizing under a positive edge

Related terms

Sources

1. Kelly criterion en.wikipedia.org Provides the definition and goal of maximising logarithmic growth.
2. Using the Kelly Criterion for Investing webhomes.maths.ed.ac.uk Supports the definition and the long-run growth objective.
3. The Kelly Criterion wizardofodds.com Explains the binary-bet formula and the concept of edge.
4. Leverage and Uncertainty arxiv.org Gives both the binary and continuous formulas for the optimal fraction.
5. March 25, 2025 arxiv.org Defines fractional Kelly and explains stake scaling with bankroll.
6. [2107.08827] Optimal sports betting strategies in practice - ar5iv ar5iv.labs.arxiv.org Supports the compounding effect of Kelly betting and estimation risk.
7. Gambling and information theory - Wikipedia en.wikipedia.org Provides alternate names for the criterion.
8. Analytical solution for Kelly's criterion for multiple outcomes tandfonline.com Supports extension to multiple simultaneous outcomes.
9. arXiv:1505.06216v2 [q-fin.GN] 18 Aug 2015 arxiv.org Supports the definition of growth rate G as the limit of log wealth.

Sources are drawn from regulators, universities and published research, and each one is labelled with what it actually is — a preprint is not called a paper. Bookmaker and affiliate pages are never cited here, because a page that sells betting is not a neutral authority on it.