Kelly Criterion Calculator

The Kelly Criterion is a mathematical formula used by professional gamblers and investors to determine the optimal size of a bet or investment. Developed by Bell Labs scientist John L. Kelly Jr. in 1956, this formula maximizes the expected geometric growth rate of wealth while minimizing the risk of ruin.

The Kelly Formula

f* = (bp - q) / b
f* = Fraction of bankroll to bet b = Decimal odds - 1 p = Probability of winning q = Probability of losing (1-p)

Calculate Your Optimal Bet Size

Enter your edge and odds to find the mathematically optimal wager

Example: 55% means you win 55 out of 100 bets on average

2.0 = even money (1:1), 3.0 = 2:1, 1.5 = 1:2. This is total return including stake.

1.0x
Âź Kelly
½ Kelly
Full Kelly
2x Kelly

Most professionals use ½ to Ÿ Kelly to reduce variance and account for edge uncertainty.

Example Scenarios

Click any scenario to load its values into the calculator:

📊 Slight Edge (Card Counting)

A small but real edge like a skilled blackjack card counter might achieve.

Win: 55% Odds: 2.0 (Even) Bankroll: $10,000

🎯 Sports Betting Edge

A realistic edge for a skilled sports bettor against the spread.

Win: 52% Odds: 2.0 (Even) Bankroll: $5,000

🎰 Strong Edge (Rare)

An unusually strong edge—rare in gambling but possible in some investments.

Win: 70% Odds: 3.0 (2:1) Bankroll: $10,000

⚠️ No Edge (Casino Games)

Typical casino game odds—Kelly says don't bet at all!

Win: 45% Odds: 2.5 Bankroll: $10,000

The History of Kelly Criterion

The Kelly Criterion was developed in 1956 by John L. Kelly Jr., a scientist at Bell Labs who was working on information theory. His original paper, "A New Interpretation of Information Rate," applied Claude Shannon's information theory to gambling.

Kelly's insight was elegant: if you have an edge, there's a mathematically optimal bet size that maximizes the growth rate of your bankroll over time. Bet too little and you're leaving money on the table. Bet too much and you risk catastrophic losses that take forever to recover from.

Did You Know? Edward Thorp, the mathematician who invented card counting and later became a legendary hedge fund manager, was one of the first to apply Kelly Criterion to both gambling and investing. His use of Kelly at the blackjack tables is documented in our story about the MIT Blackjack Team, who adopted many of his techniques.

Why Kelly Criterion Matters

The Kelly Criterion answers one of gambling's most important questions: "How much should I bet?" According to research published by the CFA Institute, proper bet sizing is often more important than finding edge in the first place.

Here's why optimal bet sizing is crucial:

  • Overbetting leads to ruin: Even with a positive edge, betting too aggressively will eventually wipe out your bankroll due to variance
  • Underbetting wastes edge: Betting too conservatively means you're not capitalizing on your advantage
  • Geometric growth: The Kelly formula specifically optimizes for compound growth, which is how wealth actually accumulates
  • Risk management: Kelly automatically balances risk and reward based on your actual edge

The Mathematics Behind Kelly

The Kelly Criterion derives from maximizing the expected logarithm of wealth. For a simple bet with win probability p and odds b (decimal odds minus 1), the optimal fraction f* is:

f* = p - (1-p)/b = (bp - q) / b

Where q = 1 - p is the probability of losing. According to research from Columbia University, this formula has several important properties:

  • f* = 0 when edge = 0: If you have no edge (bp = q), Kelly says don't bet at all
  • f* is negative when edge is negative: A negative Kelly means the bet is unfavorable—you should bet the other side if possible
  • f* approaches 1 as edge increases: With a massive edge, you could theoretically bet your entire bankroll
  • The formula is edge-dependent: More edge = larger optimal bet
Critical Caveat: The Kelly formula assumes you know your true edge with certainty. In practice, edge estimates are often wrong. This is why professional gamblers and investors typically use "fractional Kelly" (½ or Ÿ of the calculated amount) to build in a margin of safety. Overestimating your edge and overbetting is the fastest path to ruin.

Kelly Criterion in Practice

While the Kelly formula is mathematically elegant, applying it in the real world requires some adjustments:

Why Professionals Use Fractional Kelly

  • Edge uncertainty: If you think your edge is 5% but it's actually 2%, full Kelly will overbet significantly
  • Reduced variance: Half-Kelly has 75% of the growth rate but much smoother equity curves
  • Psychological factors: Large drawdowns are emotionally difficult even if mathematically optimal
  • Correlated bets: Multiple bets that might fail together require smaller individual sizing

As documented in our story about casino whales and high rollers, even professional gamblers who understand Kelly often fail when they ignore its principles. The math doesn't care about feelings—overbetting destroys even the most skilled players.

Kelly in Investing

The Kelly Criterion has crossed over from gambling to professional investing. Legendary investors like Warren Buffett (who calls it "the Charlie Munger approach") have referenced Kelly-style thinking when concentrating capital in high-conviction bets.

According to the financial literature reviewed by Investopedia, Kelly Criterion is used in:

  • Hedge fund position sizing
  • Options trading strategies
  • Sports betting syndicates
  • Poker bankroll management
  • Venture capital allocation

When Kelly Criterion Doesn't Apply

It's important to understand the limitations of Kelly Criterion:

  • No edge: For standard casino games, you have no edge. Kelly says don't bet. Period. Our Casino Odds Calculator shows why the house always has the advantage.
  • Unknown edge: If you can't reliably estimate your edge, Kelly can't help you
  • Single bets: Kelly optimizes for long-term growth across many bets. For one-time situations, other factors may dominate.
  • Utility considerations: Kelly maximizes geometric growth, but some situations (like betting rent money) have non-linear utility that Kelly ignores

This connects to the psychology explored in our article on near-miss psychology—humans are notoriously bad at estimating probabilities, which makes accurate Kelly calculations nearly impossible for most gamblers.

The Bottom Line: The Kelly Criterion is a powerful tool when you genuinely have an edge and can estimate it accurately. For casino games where the house has the edge, Kelly's answer is clear: the optimal bet size is zero. This calculator is educational—understanding why Kelly works helps explain why most gambling systems don't.

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Remember: This calculator is for educational purposes only. It demonstrates mathematical concepts about optimal bet sizing. In practice, most gambling situations offer no edge, making the Kelly-optimal bet zero. If you or someone you know is struggling with gambling, the National Problem Gambling Helpline is available 24/7 at 1-800-522-4700.