The Kelly Criterion: How to Size Your Bets Mathematically

Most gamblers and investors focus on whether a bet is good — whether the expected value is positive. Fewer think carefully about how much to bet. The Kelly Criterion is the mathematical answer to that second question.

The formula

Kelly was developed by physicist John L. Kelly Jr. at Bell Labs in 1956. The formula tells you what fraction of your bankroll to wager on a bet to maximise the long-run growth rate of your wealth.

f* = (bp - q) / b

Where:
  f* = fraction of bankroll to bet
  b  = net odds received (e.g. 2 for a 2-to-1 bet)
  p  = probability of winning
  q  = probability of losing (1 - p)

A concrete example

Suppose you find a bet where you win $2 for every $1 wagered, and you believe your probability of winning is 50%. The Kelly fraction is:

f* = (2 × 0.50 - 0.50) / 2
f* = (1.00 - 0.50) / 2
f* = 0.25

Bet 25% of your bankroll.

Why not bet more?

Betting more than Kelly — overbetting — reduces long-run growth even when you have a genuine edge. This seems counterintuitive. If the expected value is positive, why not bet everything? Because variance matters. A sequence of losses can wipe out a large bankroll before the edge has time to assert itself.

Half Kelly in practice

Many professional gamblers use half Kelly — betting half the Kelly-optimal fraction. This reduces variance significantly while sacrificing only a modest amount of long-run growth. The reasoning: your edge estimates are rarely perfect. If you think your win probability is 55% but it is actually 52%, full Kelly will overbet. Half Kelly provides a margin of safety.

At Kelly:      growth rate = maximum possible
At 2× Kelly:   growth rate = 0 (same as not betting)
At >2× Kelly:  growth rate = negative (guaranteed ruin)

Kelly and casino games

In standard casino games the Kelly fraction is negative — because the expected value is negative. Kelly tells you not to bet at all, which is the correct mathematical answer. The criterion only recommends positive bet sizes when you have a genuine positive edge: card counting in blackjack, sports betting with better information than the bookmaker, or poker against weaker opponents.

Kelly and investing

The Kelly Criterion is used by investors including Warren Buffett and Charlie Munger, who have described position sizing in Kelly-like terms. In investing, estimating the "probability of winning" is far harder than in gambling — which is why half Kelly or quarter Kelly is standard among practitioners who apply it to markets.

The criterion does not tell you which bets to take. It tells you, given a bet worth taking, exactly how much of your capital to put at risk.