Kelly Criterion Calculator
The Kelly Criterion is a mathematical formula for optimal bet sizing that was developed by John L. Kelly Jr. at Bell Labs in 1956. Originally created to solve a problem in information theory, the Kelly formula has been adopted by professional bettors, investors, and mathematicians as the theoretically optimal approach to bankroll management—but only when the bettor has a positive edge.
This educational calculator demonstrates why the Kelly Criterion produces zero or negative results for casino games. Because all casino games have a built-in house edge, players have negative expected value, which means the mathematically optimal bet size according to Kelly is always zero. This tool helps illustrate why no bankroll management system can overcome the fundamental mathematics of gambling.
Calculate Kelly Criterion
Enter your probability of winning and the odds offered to calculate the optimal bet size according to Kelly.
Interpretation
Kelly Criterion for Casino Games
See why Kelly Criterion returns zero for casino games. Select a game to view its true probability and house edge, then calculate the Kelly result.
Mathematical Reality
The Kelly Criterion returns zero because you have no edge. When expected value is negative, the mathematically optimal bet size is always zero. This is why no betting system can overcome the house edge—the fundamental mathematics say you should not bet at all.
Edge Comparison: Positive vs. Negative
Compare how Kelly Criterion behaves with positive edge (theoretical winning scenarios) versus negative edge (all casino games). This illustrates why bankroll management only works when you have an advantage.
What Is the Kelly Criterion?
The Kelly Criterion was developed by John L. Kelly Jr. in his 1956 paper "A New Interpretation of Information Rate" published in the Bell System Technical Journal. Originally designed to optimize signal transmission, the formula was quickly recognized as applicable to gambling and investment scenarios where repeated bets with an edge are placed.
According to research published in the Journal of the American Statistical Association, the Kelly Criterion maximizes the geometric growth rate of wealth over time, making it theoretically optimal for long-term wealth accumulation—but only when the bettor has positive expected value.
f* = Optimal fraction of bankroll to bet
b = Net odds (decimal odds - 1)
p = Probability of winning
q = Probability of losing (1 - p)
Why Kelly Criterion Returns Zero for Casino Games
The critical insight about Kelly Criterion is that it requires positive expected value to work. When you examine the formula, the numerator (bp - q) represents your edge. For casino games:
- b = Net payout odds (e.g., 1.0 for even money)
- p = Your probability of winning (always less than fair odds due to house edge)
- q = Your probability of losing (1 - p)
When bp < q (which occurs in all casino games because the house edge ensures this), the numerator becomes negative. A negative Kelly fraction means the optimal bet is zero—or mathematically, you should be on the other side of the bet (which is why casinos are profitable).
Mathematical Example: European Roulette
For a red/black bet on European Roulette:
- b = 1.0 (even money payout)
- p = 18/37 = 0.4865 (probability of winning)
- q = 19/37 = 0.5135 (probability of losing)
Kelly = (1.0 × 0.4865 - 0.5135) / 1.0 = -0.027 or -2.7%
The negative result confirms what the house edge tells us: you should not bet. The 2.7% negative Kelly matches the 2.7% house edge exactly.
The Relationship Between Kelly and House Edge
The Kelly Criterion and house edge are mathematically connected. For even-money bets, the negative Kelly fraction exactly equals the house edge. This is not coincidence—both measure the same fundamental concept: the mathematical advantage one side has over the other.
Research from the University of Nevada, Las Vegas International Gaming Institute confirms that house edge is designed into games to ensure the casino always has positive expected value. Kelly Criterion simply quantifies this: when you have negative edge, optimal betting is zero.
This applies directly to gambling in South Korea. At Kangwon Land, the only legal casino for Korean citizens, all games operate with standard house edges. The Kelly Criterion would recommend Korean players bet zero at every game—a mathematical conclusion that aligns with the government's restrictive gambling laws.
When Kelly Criterion Works: Positive Edge Scenarios
The Kelly Criterion is designed for situations where the bettor has a genuine edge. These scenarios are rare but include:
- Card Counting in Blackjack: Skilled card counters can achieve 0.5-1.5% edge over the casino, making Kelly applicable. However, casinos actively ban card counters, as documented in enforcement literature.
- Sports Betting with Superior Analysis: If a bettor's win probability estimate significantly exceeds the implied probability of the odds, positive edge exists.
- Investment Decisions: Kelly is used in portfolio management when expected returns exceed risk-free rates.
- Poker: Skilled players can have edge over weaker opponents, making bankroll management meaningful.
The Stanford Encyclopedia of Philosophy's entry on Decision Theory discusses how optimal betting strategies like Kelly assume rational actors with accurate probability estimates—conditions rarely met in recreational gambling.
Fractional Kelly and Risk Management
Even with positive edge, many practitioners use "fractional Kelly"—betting only a portion (typically 25-50%) of the full Kelly recommendation. This reduces volatility at the cost of lower expected growth.
| Kelly Fraction | Growth Rate | Volatility | Use Case |
|---|---|---|---|
| Full Kelly (100%) | Maximum | Very High | Theoretical optimum |
| Half Kelly (50%) | 75% of max | Moderate | Common practice |
| Quarter Kelly (25%) | 56% of max | Low | Conservative approach |
| Zero Kelly (Casino) | Negative | N/A | No betting recommended |
Connection to Other Gambling Mathematics
The Kelly Criterion is one of several mathematical frameworks that demonstrate the fundamental disadvantage facing casino gamblers:
- House Edge: Directly related to Kelly—house edge is essentially the casino's Kelly advantage.
- Risk of Ruin: With negative edge, risk of ruin approaches 100% over time regardless of bet sizing.
- Required Win Rates: Shows the impossible win rates needed to overcome house edge.
- Betting Systems: No system can create positive edge where none exists—Kelly confirms this mathematically.
Our Probability Calculator and Variance Calculator help illustrate the mathematical foundations underlying these concepts. Together, these tools demonstrate why gambling is designed to extract money from players systematically.
The Myth of Bankroll Management "Systems"
Some gambling literature promotes "bankroll management" as if it could improve outcomes. The Kelly Criterion exposes this myth: without positive edge, no money management system changes the fundamental mathematics. As research in the Journal of Gambling Studies documents, belief in betting systems is a cognitive distortion associated with problem gambling.
Our Fallacy Analyzer explores these cognitive biases in detail. The illusion that proper bet sizing can overcome negative expected value is one of the most dangerous misconceptions in gambling psychology.
South Korean Context: Why Understanding Kelly Matters
In South Korea, where online gambling is heavily restricted and penalties are severe, understanding the Kelly Criterion provides mathematical justification for the government's protective stance. The formula proves that:
- Casino games cannot be beaten through any betting strategy
- The mathematically optimal behavior is not to gamble
- Players are guaranteed to lose money over time
- No amount of bankroll management changes the outcome
This mathematical reality underlies the social costs of gambling that drive South Korea's restrictive gambling laws. When citizens gamble despite understanding these mathematics, it may indicate problematic gambling behavior—our Problem Gambling Self-Assessment provides a confidential screening tool.
Important Notice
This calculator is for educational purposes only. It demonstrates that the Kelly Criterion returns zero for all casino games because players have negative expected value. This tool should not be used to develop betting strategies, as no strategy can overcome the mathematical house edge. If you or someone you know has a gambling problem, visit our responsible gambling resources page for support information.
Frequently Asked Questions
What is the Kelly Criterion?
The Kelly Criterion is a mathematical formula developed by John L. Kelly Jr. at Bell Labs in 1956 for determining the optimal size of a series of bets to maximize long-term wealth growth. The formula calculates what fraction of your bankroll to wager based on your edge (probability advantage) and the odds being offered.
Can I use Kelly Criterion for casino gambling?
No. The Kelly Criterion requires a positive edge (expected value) to calculate a betting fraction. In casino games, the house edge means players have negative expected value, so the Kelly formula produces zero or negative results—mathematically confirming you should not bet at all.
Why does Kelly Criterion give zero for negative edge bets?
When your expected value is negative (as in all casino games), the Kelly formula's numerator becomes negative or zero. This mathematically indicates that the optimal bet size is zero—you should not bet because any bet has negative long-term expectation.
What is the Kelly Criterion formula?
The Kelly formula is: f* = (bp - q) / b, where f* is the fraction of bankroll to bet, b is the decimal odds minus 1 (net payout), p is the probability of winning, and q is the probability of losing (1-p). When edge is negative, this formula returns zero or negative values.
Additional Resources
For more information about gambling mathematics and responsible gambling:
- House Edge Calculator - Understand casino advantages
- Probability Calculator - Calculate true odds and expected value
- Session Simulator - Visualize gambling outcomes
- Budget Calculator - Plan entertainment spending responsibly
- Knowledge Quiz - Test your understanding of gambling mathematics
- Responsible Gambling Resources - Help and support information