What is the Kelly Criterion?
The Kelly Criterion is a mathematical formula for calculating the optimal fraction of your bankroll to stake on a bet with a positive expected value. Developed by John L. Kelly Jr. at Bell Labs in 1956 and later applied to gambling and financial investment, it maximizes the long-term growth rate of your bankroll while avoiding the risk of ruin.
**The Kelly formula:** `f* = (bp - q) / b`
Where: - `f*` = fraction of bankroll to stake (the optimal bet size) - `b` = decimal odds minus 1 (e.g. odds of 2.50 → b = 1.50) - `p` = your estimated probability of winning - `q` = probability of losing (1 - p)
**Worked example:** - Bookmaker odds: 2.50 (implied probability: 40%) - Your estimated true probability: 50% - b = 2.50 - 1 = 1.50 - p = 0.50, q = 0.50 - f* = (1.50 × 0.50 - 0.50) / 1.50 = (0.75 - 0.50) / 1.50 = **16.7% of bankroll**
If you estimate the bet has 50% probability but the market prices it at 40%, Kelly says stake 16.7% of your bankroll on this single bet.
**Why Kelly maximizes growth:** Kelly staking produces the highest geometric mean of returns over many bets. Over the long run, a Kelly bettor will grow their bankroll faster than any bettor using a different fixed-fraction strategy — assuming their probability estimates are accurate.
Fractional Kelly: the practical approach
Full Kelly staking (as calculated above) is mathematically optimal under ideal conditions, but practitioners consistently find it too aggressive in practice. Two reasons:
**Probability estimation error:** The Kelly formula is only as good as your probability estimates. If you think a team has 50% chance of winning but the true probability is 44%, full Kelly massively over-bets and increases risk of ruin. Small errors in probability assessment compound into large staking errors.
**Variance and drawdown:** Full Kelly produces extremely high bankroll variance. A string of losses at full Kelly stakes can halve a bankroll quickly, even with a genuine edge. This is psychologically difficult to sustain.
**Fractional Kelly solution:** Most professional bettors use half Kelly (f* / 2) or quarter Kelly (f* / 4). This: - Reduces variance substantially (by 75% at quarter Kelly) - Reduces peak expected growth rate by only 25% (at half Kelly) - Provides a large buffer for probability estimation errors - Is far easier to execute psychologically
**Half Kelly worked example (from above):** - Full Kelly stake: 16.7% of bankroll - Half Kelly stake: 8.3% of bankroll - Quarter Kelly stake: 4.2% of bankroll
**Recommended starting point:** Use quarter Kelly when first applying the Criterion. The reduction in variance is significant without substantially degrading long-term growth. Increase toward half Kelly once you've validated your probability estimation accuracy over 200+ bets.
Use our value bet calculator to calculate both full and fractional Kelly stakes from any combination of odds and estimated probability.
Estimating probabilities accurately
The Kelly Criterion is only as good as your probability estimates. This is the hardest part of the system — and where most bettors struggle.
**Methods for estimating true probabilities:**
**No-vig fair odds (market consensus):** Remove the bookmaker's margin from market odds to find the implied 'fair' probability. Example: Bet365 offers 1.85 on Team A and 2.05 on Team B. With vig removed (using Pinnacle's sharp market as reference), fair probability for Team A might be 54% rather than the 54.1% implied by 1.85.
**Sharp market reference:** Pinnacle Sports and Asian handicap markets (especially Asian bookmakers) are the most efficient pricing in the world. Their odds, once vig-adjusted, are the closest available approximation of true probability. Compare your target bookmaker's odds to Pinnacle's line — a discrepancy of 3%+ probability is a Kelly-worthy edge.
**Statistical models:** Building a statistical model (Elo ratings, Expected Goals for football, team form/travel/lineup factors) produces independent probability estimates. Comparing your model output to market odds identifies edges.
**Avoiding overconfidence:** Studies consistently show bettors over-estimate their edge. If your Kelly calculations regularly produce f* > 20%, your probability estimates are almost certainly inflated. A realistic edge in sports betting produces Kelly stakes of 2-6% per bet.
**Calibration:** Track your estimated probabilities versus actual outcomes over 500+ bets. If your 70% probability bets win 70% of the time, your calibration is good. Most bettors are overconfident — true win rates are lower than estimated probability. Recalibrate regularly.
Kelly Criterion for multiple simultaneous bets
The Kelly formula assumes one bet at a time. Real betting involves multiple simultaneous bets across different events. Adjustments are needed:
**Simultaneous bets reduction:** If placing N simultaneous bets, divide each Kelly stake by N as a starting approximation. This prevents over-betting when many positions are open at once. More sophisticated multi-bet Kelly formulas exist but are complex to implement.
**Portfolio Kelly:** For bettors with many simultaneous positions, treating the betting book as a portfolio (similar to financial portfolio theory) produces more accurate stake sizing. The correlation between bets matters — two bets on the same team in different markets are more correlated than bets on unrelated sports.
**Parlays and Kelly:** As discussed in our parlay betting guide, parlaying value bets can be Kelly-justified. The Kelly stake on a parlay should be calculated from the combined parlay odds and the product of your individual leg probabilities. This typically produces much smaller stakes than the sum of individual Kelly bets.
**Practical simplified approach for recreational bettors:** 1. Estimate probability for each identified value bet 2. Calculate quarter Kelly stake for each 3. If total simultaneous stakes exceed 20% of bankroll, reduce all by equal proportion 4. Never bet more than 5% of bankroll on any single bet regardless of Kelly output
**Kelly and matched betting:** Matched betting (covering all outcomes across bookmaker + exchange) effectively has p ≈ 1.0 for the combined position. Kelly for matched betting relates to the free bet or bonus value extraction rate rather than outcome probability — a different application of the same principle. See our matched betting guide for details.