The Kelly Criterion, Explained Without the Calculus

Expected value tells you a bet is worth making. It does not tell you how much to put on it, and that turns out to be a separate problem with a real answer.

Bet too little and a genuine edge takes decades to compound into anything. Bet too much and an ordinary losing run — which, as the bankroll guide shows, is deeper than people expect — takes the bankroll to zero, from which no edge recovers.

In 1956, working at Bell Labs on information transmission, John Kelly derived the stake that maximises the long-run growth rate of a bankroll. It has a closed form, it is provably optimal by that measure, and it has been the reference point for staking ever since.

f = (b × p − q) / b

  b = decimal odds − 1   (profit per unit staked)
  p = your probability of winning
  q = 1 − p

Equivalently
  f = edge / odds  =  (p × decimal − 1) / (decimal − 1)

Worked at −110 with a 55% estimate:

decimal = 1.909  →  b = 0.909
p = 0.55, q = 0.45

f = (0.909 × 0.55 − 0.45) / 0.909
  = (0.500 − 0.450) / 0.909
  = 5.5% of bankroll

  on a $2,000 bankroll  →  $110

Note what that is: five and a half percent of everything, on a single bet, from a 2.62-point edge. Compare it with the 1% unit that flat betting recommends and the character of Kelly becomes obvious immediately. The Kelly calculator runs it for any price and estimate, with fractional Kelly built in.

Two things follow from this table.

First, Kelly is extremely sensitive to the edge. Moving the estimate from 53% to 55% — two percentage points, well inside anyone’s error bars — quadruples the stake. Whatever uncertainty exists in your probability estimate gets amplified, not absorbed.

Second, full Kelly is far more aggressive than it looks on paper. Staking a third of a bankroll on one game is mathematically optimal under the model’s assumptions and psychologically impossible for essentially everyone. A run of three losses at that size removes most of the bankroll.

Kelly is optimal given one assumption that never holds: that your probability estimate is exactly correct. It is not an estimate to the formula — it is a fact. In reality it is a guess with error around it, and the consequences of that error are asymmetric.

Underestimate your edge and you stake too little, growing more slowly than you could have. Overestimate it and you stake too much — and past a certain point the growth rate does not merely fall, it goes negative. Betting twice the Kelly fraction produces zero expected growth even with a real edge. Beyond that you lose money while being right.

Quarter to half Kelly is the common range in practice. A quarter is reasonable when the probability estimate comes from judgement; a half when it comes from a model validated out of sample or from a sharper book’s no-vig price.

• Cap it. Whatever the formula says, set a ceiling — 2% or 3% of bankroll — and never exceed it. A single bad estimate then costs a bad day rather than the account.
• Recalculate against the current bankroll. Kelly is a fraction, not an amount. Applying it to a bankroll that has halved is what makes it self-correcting; applying yesterday’s dollar figure is what makes it dangerous.
• Simultaneous bets are one bet. The formula assumes bets resolve one after another. Five Kelly-sized bets on the same afternoon are a much larger exposure than the formula priced, and correlated ones are worse still.
• Be honest about where p came from. Kelly applied to a made-up probability produces a confidently oversized bet. The formula cannot tell the difference and will not warn you.
• If in doubt, flat bet. Kelly with a bad estimate is worse than flat betting with the same bad estimate, because it stakes most where the error is largest.

The honest summary: Kelly is the correct answer to a question that assumes you know something you do not. Fractional Kelly is how that gets handled, and the risk of ruin calculator is where to check what a given fraction actually exposes you to.