The Kelly Criterion in Sports Betting
The Kelly formula chooses a stake fraction to maximize expected logarithmic bankroll growth under a specified probability model. It does not estimate the probability of winning, establish an edge, or guarantee profit. This guide derives the simple win/loss formula and shows how estimation error changes its meaning.
All examples are hypothetical, with no fees, taxes, pushes, or voids. Stakes are fractions of the current bankroll. Use full precision in calculations and round only the displayed result; actual stake limits and rounding require separate treatment.
What the Formula Optimizes
Let p be an assumed win probability, q = 1 - p, and D the offered decimal price. Define b = D - 1: net winnings per dollar staked, not the decimal price including stake. At even money, D = 2 and b = 1.
For a fraction f of the bankroll staked, a win multiplies the bankroll by 1 + bf; a loss multiplies it by 1 - f. Expected log growth for this decision is:
g(f) = p × ln(1 + bf) + q × ln(1 - f).
For an interior maximum, set the derivative to zero:
g′(f) = pb / (1 + bf) - q / (1 - f) = 0.
pb(1 - f) = q(1 + bf), so f* = (bp - q) / b.
This is a single, divisible stake with 0 ≤ f < 1, no borrowing, and a win/loss model with 0 < p < 1. If the formula is zero or negative, the maximizing feasible stake is zero. The assumed probability must exceed the raw break-even probability 1 / D for a positive stake. Beating a normalized market estimate alone is not enough.
Maximizing expected log growth is different from maximizing expected dollar profit. Extending this simple calculation to repeated stakes requires appropriate conditional probabilities, reinvestment, and a model of dependence and changing prices. It is not a promise about realized growth.
A -110 Worked Example
Assume a 55% win probability for a selection offered at -110. The exact net payout ratio is b = 100 / 110 = 10 / 11; the decimal price is 21 / 11, approximately 1.9091. Do not replace b with the prematurely rounded 0.91.
f* = ((10 / 11) × 0.55 - 0.45) / (10 / 11) = 0.055 = 5.50%.
On a hypothetical $1,000 current bankroll, that full-Kelly calculation produces a $55.00 stake. It is a model output, not a recommended stake for every reader.
The sensitivity matters. If the win probability were actually 50%, the same formula would give -5.00%, making the feasible stake zero. A positive stake computed from an optimistic input cannot turn a negative-expectation price into a favorable one.
Fractional Kelly and Remaining-Balance Stakes
Assume a separate selection at exactly +110 and an illustrative 53% win probability. These are fixed inputs for this example, not a model's observed accuracy or an average American price across different bets.
b = 1.1; q = 0.47.
f* = (1.1 × 0.53 - 0.47) / 1.1 = 0.113 / 1.1 ≈ 10.27%.
On a $1,000 current bankroll, full Kelly is $102.73. Half Kelly uses half the unrounded fraction: approximately 5.14%, or $51.36. Rounding 5.14% before calculating the stake would incorrectly produce $51.40.
Fractional Kelly scales a model's output; it does not fix a wrong probability estimate. In the correctly specified single-bet model, using a smaller fraction than the optimum lowers expected log growth and reduces exposure to that bet's outcome. No universal drawdown or ruin probability follows from the fraction alone.
If the assumed inputs stay the same, recalculate the fraction from the remaining balance after each settlement. This differs from repeatedly staking a fixed dollar amount based on the initial balance. Simultaneous correlated bets also require a joint allocation model; applying the full single-bet fraction separately to each can overstate appropriate total exposure.
Apply the Assumptions Before the Formula
Record the offered price, source of the probability assumption, current available bankroll, settlement rules, other open positions, and any minimum stake. A practical risk-of-ruin calculation needs a stopping boundary and time horizon as well as a return model. Idealized fractional staking without minimum stakes can approach zero without reaching exactly zero; that does not rule out a severe drawdown or practical inability to continue.
Review Bankroll Management 101 for fixed versus recalculated stakes and Statistical Variance for the distinction between expectations and realized paths. The conversion guide separates the offered break-even threshold from probability estimates.
Common Kelly Questions
Q: Does b mean decimal odds in the Kelly formula?
A: No. In the formula, b is net winnings per dollar staked, equal to decimal odds minus one. At even money, decimal odds are 2 and b is 1.
Q: What does the simple Kelly formula optimize?
A: It maximizes expected logarithmic bankroll growth under the specified single-bet win/loss model. It does not maximize every possible measure of profit or guarantee realized growth.
Q: Is half Kelly safe when the probability estimate is wrong?
A: Half Kelly reduces the stake produced by the model but does not validate the probability estimate or create an edge. A practical loss or ruin forecast needs additional assumptions about dependence, the time horizon, and the stopping boundary.
Q: Can the formula use a parlay's probability?
A: Only if the payout and joint win probability correctly represent that ticket's settlement model. Multiplying marginal leg probabilities requires independence; correlated legs need conditional or joint probabilities. Other open positions also affect portfolio exposure.