What Is the Kelly Criterion? Optimal Bet Sizing Explained

The Kelly Criterion is a formula that calculates the mathematically optimal fraction of your bankroll to wager on a bet with positive expected value. Developed by John L. Kelly Jr. at Bell Labs in 1956, it maximizes the long-term growth rate of capital. Sizing too large risks ruin; sizing too small leaves profit on the table. Kelly finds the precise balance.

Key Takeaways

• The Kelly formula is f* = (bp - q) / b, where b = net odds, p = win probability, q = loss probability
• Full Kelly maximizes geometric growth but produces large drawdowns in practice
• Fractional Kelly (half or quarter) is standard among professionals to account for estimation error
• The formula requires a positive expected value bet — Kelly on a -EV bet outputs zero or negative (meaning do not bet)
• Probability estimation error is the primary risk — small errors in p produce large errors in bet size

How Does the Kelly Criterion Formula Work?

The formula takes three inputs:

• b = net odds received on the wager (profit per $1 risked)
• p = probability of winning
• q = probability of losing (1 - p)

f = (bp - q) / b*

The output f* is the fraction of your total bankroll to wager. If f* is negative, the bet is -EV and Kelly says to pass entirely.

How Do You Convert Odds to the “b” Variable?

For American odds:

• Positive odds: b = odds / 100 (e.g., +200 means b = 2.0)
• Negative odds: b = 100 / |odds| (e.g., -150 means b = 0.667)

For decimal odds: b = decimal odds - 1 (e.g., 2.50 means b = 1.5)

For prediction market contracts: b = (payout - cost) / cost (e.g., buy at $0.40, payout $1.00, so b = 0.60 / 0.40 = 1.5)

What Does a Kelly Calculation Look Like?

Scenario: You estimate a 55% chance on an even-money bet (+100, b = 1.0).

• f* = (1.0 x 0.55 - 0.45) / 1.0
• f* = (0.55 - 0.45) / 1.0
• f = 0.10 = 10% of bankroll*

On a $10,000 bankroll, Kelly says to bet $1,000. This maximizes long-term growth given your edge.

Scenario: You estimate a 45% chance on a +150 line (b = 1.5).

• f* = (1.5 x 0.45 - 0.55) / 1.5
• f* = (0.675 - 0.55) / 1.5
• f = 0.083 = 8.3% of bankroll*

Despite a sub-50% win rate, the payout odds are favorable enough that Kelly recommends a meaningful bet.

How Does Kelly Sizing Change Across Different Edges?

The table below shows Kelly bet sizes for common edge and odds combinations. All values represent the percentage of bankroll to wager at full Kelly.

Notice how Kelly outputs 0% whenever the true probability does not exceed the implied probability from the odds. Kelly never recommends betting on -EV propositions.

What Is Fractional Kelly and Why Do Professionals Use It?

Full Kelly maximizes geometric growth rate, but the ride is brutal. Simulations show that full Kelly produces drawdowns of 50% or more with uncomfortable frequency. A string of losses — which will happen — can halve your bankroll before the edge reasserts itself.

Fractional Kelly reduces the wager to a fixed percentage of the full Kelly recommendation:

• Half Kelly (0.5x): Wager 50% of the Kelly amount. Sacrifices roughly 25% of the theoretical growth rate but cuts variance nearly in half.
• Quarter Kelly (0.25x): Wager 25% of the Kelly amount. Much smoother equity curve, slower growth.

Most professional sports bettors and prediction market traders use half Kelly or less. The reason is pragmatic: your probability estimates are never perfectly calibrated. If your true edge is 5% but you think it is 10%, full Kelly will size far too aggressively, accelerating losses rather than preventing them.

The Odds Reference Kelly calculator calculates optimal position size using live prediction market probability data. It supports both full and fractional Kelly outputs.

When Does the Kelly Criterion Break Down?

Kelly has well-known failure modes:

Probability estimation error. Kelly assumes you know p with certainty. You do not. A 2-point overestimate on a 55% true probability (betting as if 57%) increases Kelly sizing by 40%. This is the strongest argument for fractional Kelly.

Correlated bets. Kelly is designed for independent sequential bets. If you have three NFL bets in the same afternoon that share correlated outcomes (e.g., team totals in a parlay-like structure), applying Kelly to each one independently overstates the safe bet size.

Simultaneous wagers. Placing multiple Kelly-sized bets at once is not the same as placing them sequentially. Simultaneous bets compound the risk of drawdown beyond what the single-bet formula accounts for.

Illiquid bankroll. Kelly assumes your entire bankroll is available and fungible. If capital is locked in pending bets, the effective bankroll is smaller than the nominal figure.

For a broader discussion of managing bankroll constraints, see bankroll management.

How Does Kelly Apply to Prediction Markets?

The mechanics are identical. A prediction market contract at $0.40 with a $1.00 payout gives b = 0.60 / 0.40 = 1.5. If you estimate the true probability at 55%:

• f* = (1.5 x 0.55 - 0.45) / 1.5
• f* = (0.825 - 0.45) / 1.5
• f = 0.25 = 25% of bankroll (full Kelly)*

At half Kelly, that becomes 12.5%. At quarter Kelly, 6.25%.

Because prediction markets price contracts as direct probabilities with minimal vig, the edge calculation is cleaner than in sports betting. Your only uncertainty is the true probability — there is no vig distortion to remove first.

For additional context on prediction market mechanics, see the prediction markets glossary.

How Should You Use Kelly in Practice?

1. Confirm positive EV — Never apply Kelly to a -EV bet. The formula itself will output zero or negative, telling you to pass.
2. Estimate probability honestly — Kelly is only as good as your p estimate. Overconfidence is the most common error.
3. Apply fractional Kelly — Half Kelly is a sensible default. Quarter Kelly if your estimates are uncertain.
4. Track results — Over 500+ bets, compare your actual win rate to your estimated probabilities. If they diverge significantly, recalibrate before continuing to use Kelly.
5. Never exceed Kelly — Betting more than the Kelly fraction reduces long-term growth rate while increasing variance. There is no mathematical upside to overbetting.

Kelly is not a magic formula. It is a disciplined framework for translating edge into action without risking ruin. Used with honest probability estimates and fractional scaling, it remains the standard for optimal bet sizing across sports betting, prediction markets, and financial trading.