Kelly Criterion

Having a sound sports betting strategy is key to gambling success. Long-term risk assessment, bankroll management, and mathematically-backed probabilities analysis will help you to win more often and generate profits more consistently.

One such tool that can help you analyze both what to bet and how much to wager is the Kelly Criterion, which has practical applications for sports betting, casino gambling, stock market investing, and more.

What Is the Kelly Criterion in Sports Betting?

The Kelly Criterion is a mathematical formula used to determine the optimal size of a series of bets to maximize logarithmic wealth growth over time. The sports betting version of the formula relies on a binary outcome, so only bets with a win or loss option can be used in the equation.

In 1956, John Larry Kelly Jr. came up with the formula while working for Bell Labs. The Criterion focuses on the long-term geometric growth rate to help determine the optimum betting amount as a percentage of the existing bankroll as well as which sides of the wager to back (or whether to avoid a bet altogether).

Understanding the Kelly Criterion Formula

The Kelly Criterion formula is expressed as:

𝑓* = 𝑝 – (𝑞/𝑏) = 𝑝 – (1-𝑝)/𝑏

where

• 𝑓* is the fraction of the current bankroll to wager
• 𝑝 is the probability of a win
• 𝑞 = 1- 𝑝 is the probability of a loss
• 𝑏 is the proportion of the bet gained with a win. For example, if you were wagering $100 on a 3-to-1 odds bet, (returning $400 and winning you $300), so in this case, 𝑏 = $300/$100 = 3.0.

Successful Kelly betting relies on accurate calculation of win probabilities. You can use the available odds to determine an estimated win probability for each side, though this is still just an approximation to give you the numbers needed for the calculation.

How to Calculate the Kelly Criterion

Once you have done the research to calculate the win probability for each side of the game you’re betting on, you can apply the Kelly Criterion to determine the amount of money you should wager.

For example, let’s use an NFL game from the 2025 playoffs. The moneyline odds for the AFC Championship game saw the Buffalo Bills at +104 and the Kansas City Chiefs at -122; using those odds allows us to calculate the 𝑏 values for each team (Bills = 1.04, Chiefs = 0.82).

We can also use ESPN’s Matchup Predictor tool, which says the Bills have a 54.6% chance of winning (with a Chiefs win at 45.4%).

Using these figures, we can now put these numbers into the equation to determine the bankroll amount to wager on each side (for the sake of making math easy, let’s assume a $100 bankroll). A wager on Buffalo would produce this version of the formula:

𝑓* = .546 – (.454/1.04) = .1095, rounded to 0.11. Thus, the Kelly Criterion dictates your most profitable wager over 11% of your bankroll ($11).

Similarly, the formula for the Chiefs would be:

𝑓* = .454 – (.546/.82) = -0.212. In this case, because our final value was negative, the Kelly Criterion would indicate a wager on the Chiefs is unlikely to be the correct side to bet.

It’s important to acknowledge that the Kelly Criterion most effectively applies to a series of bets where the odds stay consistent over time. That doesn’t mean Kelly betting is impractical in this scenario, it just means you’ll need to keep these figures in mind as you research the matchups and prepare to make your wagers.

Fractional Kelly Betting

Many mathematicians have offered proofs of Kelly’s work, and as a result, criticisms arose around the practicality of the Criterion for gamblers. For example, a bettor may have financial constraints that would prevent them from being able to gamble at the optimal rate, encouraging them to opt for a Fractional Kelly instead.

When using fractional Kelly betting, gamblers wager a fraction of the amount recommended by Kelly’s model to reduce volatility. This is an important step in helping to manage gambling risk.

Additionally, the Kelly Criterion opens itself up to ergodicity breaking and Knightian uncertainty. Ergodicity refers to the idea that over time, the average point of a system can be calculated from the trajectory of a single point, while Knightian uncertainty addresses the lack of quantifiable knowledge of a subject rather than the presence of a quantifiable risk.

More simply, external factors that the sports bettor doesn’t know can cause the Kelly Criterion to be less effective, and single-game matchups don’t necessarily allow for sustained probabilities over time. A variety of factors that are beyond the bettor’s control or prior knowledge can also influence the outcome, so it is not always the most useful strategy.

Practical Applications of the Kelly Criterion

The Kelly Criterion has several non-sports betting applications, most notably stock market investments and portfolio management. In these applications, the formula changes to read as:

𝑓* = (𝑝/𝑙) – (𝑞/𝑔)

where

• 𝑓* is the fraction of assets to apply to the security
• 𝑝 is the probability that the investment will increase in value
• 𝑙 is the fraction that is lost in a negative outcome
• 𝑞 is the probability that the investment will decrease in value
• 𝑔 is the fraction that is gained in a positive outcome

While the formula is distinctly different, we can see how the overall principle is the same. With accurate dataand modeling to calculate the probabilities, it is possible to determine the amount of financial assets to spend on an investment to maximize earnings (or whether to avoid investment altogether). 

History and Origin of the Kelly Criterion

The Kelly Criterion was described by John Larry Kelly Jr. in 1956 when he was a researcher at Bell Labs in New York City. Though the Kelly Criterion is mathematically the same as Daniel Bernoulli’s 1738 article on choosing a betting or investment outcome with the highest geometric mean of outcomes, Kelly and Bernoulli were both framing their work to address different issues and solve different problems. 

In the 2000s, Kelly-style investment strategies became a mainstream form of analysis, most notably used by Warren Buffett and Bill Gross. 

Claude Shannon, a colleague of Kelly’s at Bell Labs, would later join Ed Thorpe, a mathematician at M.I.T., in testing Kelly’s work in Las Vegas while playing blackjack and roulette, making both of them large sums of money. They continued this success in the stock market, yielding even greater returns. 

Sadly, Kelly died of a stroke in 1965 at the age of 41. Kelly was a heavy smoker, at times as much as six packs of cigarettes in a day, which likely contributed to his premature death. 

Conclusion

Though there is not a “perfect” model for calculating how much money you should wager while sports betting, the Kelly Criterion is certainly one of the more reliable options. However, accurate data sets, especially in modeling the win probabilities, are essential to successful Kelly betting. 

With a proper understanding of how to apply the Kelly formula, you are now ready to practice Kelly betting as you make your wagers. If you’d prefer to try things out without the same financial involvement, we recommend using a sweepstake sportsbook as a practice area; for an idea of where to start, check out our social sportsbook reviews!). 

Good luck, have fun, and always gamble responsibly!