Kelly Criterion: Sizing Your Bets When You Have an Edge

Kelly Criterion is the only mathematically optimal bet-sizing formula. It was developed by John Kelly at Bell Labs in 1956, originally for signal transmission problems, and later applied to gambling and investment theory.

The key thing to understand about Kelly: it assumes you already have a positive expected value edge. If you don't have an edge, Kelly will (correctly) tell you not to bet.

The Formula

f = (bp - q) / b

Where:

• f = fraction of your bankroll to bet
• b = decimal odds minus 1 (your profit per unit staked)
• p = your estimated probability of winning
• q = probability of losing (1 - p)

Example: You believe a team has a 55% chance of winning. The bookmaker offers 2.10 decimal odds (paying $1.10 profit per $1 staked).

b = 2.10 - 1 = 1.10
p = 0.55
q = 0.45

f = (1.10 × 0.55 - 0.45) / 1.10
f = (0.605 - 0.45) / 1.10
f = 0.155 / 1.10
f = 0.141

Kelly says bet 14.1% of your bankroll.

Why Kelly Is Optimal

Kelly maximizes the logarithm of your bankroll, which is equivalent to maximizing the geometric growth rate. Over a long sequence of bets, no other strategy grows your bankroll faster than Kelly.

The intuition: bet too little and you leave money on the table. Bet too much and you risk ruin, because losing streaks will devastate your bankroll before your edge has time to express itself.

The Problem in Practice: You Don't Know p

The critical assumption of Kelly is that you know your true edge with accuracy. In reality, you don't. Your estimated probability is always wrong to some degree.

If you overestimate your edge, Kelly will tell you to bet more than you should, which increases your risk of ruin substantially. Doubling your Kelly fraction more than doubles your ruin risk.

This is why most serious bettors use Fractional Kelly — typically betting half Kelly (0.5f) or quarter Kelly (0.25f). This reduces growth rate but dramatically reduces the risk of bankroll destruction from edge estimation errors.

Kelly Assumes Independent Bets

Another practical limitation: Kelly assumes each bet is independent. In reality, if you're betting multiple events simultaneously, or your models share underlying assumptions, your bets may be correlated. Correlated losses can devastate a Kelly-sized bankroll in ways the formula doesn't account for.

When to Use Kelly

Kelly is most useful when:

• You have a clear, quantifiable edge
• You're making many similar bets over time
• You have an accurate model for estimating probabilities

It's less useful when:

• Your edge estimates are rough
• You're placing a small number of one-off bets
• You're betting for recreation and the precise optimal size doesn't matter much

The Bottom Line

Kelly Criterion is a powerful tool, but it solves the sizing problem, not the edge problem. You still need a genuine edge. If your probability estimates are no better than the bookmaker's implied probabilities, Kelly will correctly tell you: don't bet.

The formula is a tool for people who've already done the hard work of finding +EV situations. It tells you what to do next. Finding those situations is a different problem entirely.