Odds, Probability & Bankroll

Kelly betting

Also called

Kelly criterion, Kelly strategy, proportional betting¹²

Binary-bet formula

f* = p – q / b, where p is win probability, q = 1 – p, and b is net odds per unit staked¹

50/50 example

For a 50/50 bet paying 2-to-1, the optimal fraction is 25% of bankroll; at 5-to-1 it is 40%³

Even-money example

With P(H) = 0.6 and even odds, the optimal wager is 20% of current wealth⁵

Key points

• The strategy maximises long-run growth rather than short-run profit, balancing risk and reward¹⁹.
• If the bet has positive edge (expected value), wager the Kelly fraction; if there is no edge, the optimal wager is zero¹⁰.
• For multiple outcomes, the criterion is to maximise the expected log of post-bet wealth, E[log Z]⁴.
• Kelly betting applies to repeated bets with known or estimated probabilities, including binary bets and horse-race markets¹⁶.
• Unlike flat betting or Martingale systems, Kelly adjusts stake size to current bankroll and edge, not to a fixed amount or a progression sequence.

How to calculate a Kelly bet

1. Estimate the true probability p of the outcome you want to back.
2. Note the net odds b offered per unit staked (e.g., decimal odds minus 1).
3. Plug into the formula: f* = p – (1 – p) / b.
4. If f* is positive, wager that fraction of your current bankroll; if zero or negative, do not bet.

Examples of Kelly bet sizes

How Kelly betting differs from other staking methods

Flat betting stakes the same amount each time, ignoring bankroll size and edge. Martingale systems double stakes after losses, risking large drawdowns. Kelly betting ties stake size directly to your estimated edge and current wealth, aiming for the fastest long-term growth without risking ruin. It is not a prediction system — it is a sizing rule used after you have identified a value bet. The method trades off growth against risk by maximising long-run growth rather than short-run profit¹⁹.

For general multi-outcome betting, the criterion is to maximise E[log Z], the expected log of post-bet wealth⁴⁷. This applies to contexts such as horse-race-style markets and other situations with multiple possible outcomes⁸.

Where this term is used

• Odds and probability

Related terms

Sources

1. Kelly criterion en.wikipedia.org Provides the standard binary-bet formula and the definition of the Kelly criterion.
2. Gambling and information theory en.wikipedia.org Supports the alternative name 'proportional betting'.
3. Proebsting's paradox - Wikipedia en.wikipedia.org Gives the 50/50 bet examples with 2-to-1 and 5-to-1 odds.
4. Kelly Betting as Bayesian Model Evaluation arxiv.org Defines the criterion as maximising expected log wealth and covers multi-outcome betting.
5. arXiv:1503.06535v2 [math.OC] 1 Aug 2017 arxiv.org Provides the even-money example with P(H)=0.6 yielding a 20% wager.
6. Kelly Betting with Quantum Payoff: a continuous variable ... arxiv.org Confirms that Kelly betting applies to binary bets and horse-race markets.
7. Gambling under unknown probabilities as a proxy for real ... arxiv.org Supports the multi-outcome formula for expected log wealth.
8. [PDF] Kelly betting on horse races with uncertainty in probability estimates arxiv.org Supports the application of Kelly betting to horse-race-style markets with multiple outcomes.
9. Full article: Analytical solution for Kelly's criterion for multiple outcomes tandfonline.com Supports the statement that Kelly maximises long-run growth.
10. Using the Kelly Criterion for Investing webhomes.maths.ed.ac.uk Explains that a positive edge leads to a Kelly wager and no edge means zero wager.

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.