The Kelly Criterion for Gamblers
The Kelly criterion is the most famous bet-sizing formula in gambling — and the most misunderstood. It tells you exactly what fraction of your bankroll to stake when you have an edge, and it delivers one brutally honest verdict about casino games that most articles skip. Here is the formula, the derivation in plain language, and what it actually means for how you should play.
Play with a plan →The formula in plain language
Developed by John Kelly at Bell Labs in 1956, the criterion answers one question: what fraction of my bankroll should I risk on a favorable bet to maximize long-run growth? The formula is f = (bp minus q) divided by b, where b is the net odds received (profit per unit staked), p is your probability of winning, and q = 1 minus p. A cleaner way to remember it: f equals edge divided by odds. The numerator, bp minus q, is your expected profit per unit bet — your edge. The intuition behind the derivation: bet too little and you leave growth on the table; bet too much and losing streaks crater your bankroll faster than winning streaks rebuild it, because losses compound geometrically (lose 50% and you need +100% to recover). Kelly finds the exact fraction that maximizes the expected logarithm of wealth — the compounding growth rate — rather than the expected value of a single bet. It is the mathematical boundary between aggressive and reckless.
A worked example with a real edge
Suppose you find a genuinely favorable even-money proposition: you win 55% of the time and are paid 1-to-1. Then b = 1, p = 0.55, q = 0.45, so f = (1 times 0.55 minus 0.45) divided by 1 = 0.10. Kelly says stake 10% of your bankroll. With a 1,000 SUI roll, bet 100 SUI; if you win, the next bet is 110; if you lose, 90. Notice the discipline built in: bets shrink automatically after losses, which is why full ruin is mathematically impossible under Kelly (you always bet a fraction, never the whole roll). Now change the payout: same 55% win rate but paid 2-to-1. Then f = (2 times 0.55 minus 0.45) divided by 2 = 0.325. Better odds justify a much larger fraction. Run it the other way — a 25% shot paid 2-to-1 — and f = (2 times 0.25 minus 0.75) divided by 2 = negative 0.125. Negative Kelly. The formula is telling you something important, which brings us to the uncomfortable part.
The honest part: Kelly says bet zero on casino games
Every standard casino game has a negative expected value for the player — that is what house edge means, and no betting system rearranges it. Llamabet is upfront about its numbers: dice carries a flat 1% edge with payout equal to 99 divided by your win chance, single-zero roulette gives up 2.70%, double-zero 5.26%, and even well-played blackjack retains a small house edge despite the 3:2 natural payout. Plug any of these into Kelly: p and b are set so that bp minus q is negative, so f is negative, and since you cannot bet a negative fraction, the Kelly-optimal stake on every house game is exactly zero. There is no loophole. Martingale, Fibonacci, streak-chasing — under Kelly's growth lens they are all just different speeds of the same decline. Any article that teaches Kelly as a way to size your roulette bets for profit is selling you a misreading of the formula it claims to explain.
So what is Kelly's real lesson for casino players?
If the optimal wager is zero, why should a casino player care? Because Kelly reframes the correct question. Casino play is entertainment with a known, disclosed price — expected cost equals edge times total amount wagered — and Kelly's logic about geometric ruin still governs how fast variance can take you out. The practical translation: size your entertainment bankroll as money you have fully written off, then size bets so a normal losing streak cannot end your session early. Kelly's shrink-after-losses discipline is worth stealing even without an edge: betting a fixed small fraction of your remaining roll, rather than a fixed amount or an escalating chase, maximizes time at the table for a given expected cost and makes catastrophic sessions rare. On a 1% edge dice game, 100 bets of 1 SUI has an expected cost of about 1 SUI — a knowable price for an evening. That is the honest deal a fair casino offers, and provably-fair games let you verify the edge you are paying is the one advertised.
Fractional Kelly: why pros bet half the formula
Even in genuinely positive-EV settings, full Kelly is wild. The bankroll path under full Kelly is extremely volatile — long, deep drawdowns are routine, and the formula's output is exquisitely sensitive to your estimate of p. Overestimate your win probability even slightly and you are unknowingly betting beyond full Kelly, which is provably worse than betting less: growth falls and risk rises simultaneously. That asymmetry is why serious advantage players and quantitative investors bet fractional Kelly, typically half. Half Kelly delivers roughly three-quarters of the optimal growth rate with dramatically smoother swings, and it builds in a margin of safety against the estimation error that plagues every real-world edge. The rule of thumb: your edge estimate is always softer than you think, so the fraction should be too. If a betting plan only works when your probability estimate is exactly right, it is not a plan — it is a leveraged opinion.
Where Kelly genuinely applies, and how it gets misused
Kelly earns its keep wherever a quantifiable positive edge exists: card counting and other advantage-play situations, sports bettors whose models beat the closing line, poker players choosing what bankroll a stake level requires, and long-horizon investors sizing concentrated positions — Edward Thorp famously used it in both blackjack and markets. Note that poker is the one casino-adjacent game where the logic can apply, because you play against other players rather than the house; Llamabet's multiplayer Texas Hold'em tables, including the AI bot tables, are peer-versus-peer in exactly this sense. The common misuses: applying Kelly to negative-edge games (covered above), feeding it hoped-for probabilities instead of measured ones, betting full Kelly on noisy estimates, and treating it as a get-rich formula rather than what it is — a ruin-avoidance constraint on money you already have good reason to bet. Kelly does not create edges. It only tells you the truth about the ones you have, including when that truth is zero.
Frequently asked questions
What is the Kelly criterion formula in simple terms?
Bet the fraction of your bankroll equal to your edge divided by the odds: f = (bp minus q) / b, where b is the profit per unit staked, p is your win probability and q is your loss probability. For a 55% chance at even money, that is (0.55 minus 0.45) / 1 = 10% of your bankroll.
Does the Kelly criterion work for casino games like roulette or dice?
No — and Kelly itself says so. House games have negative expected value (2.70% edge on single-zero roulette, 1% on Llamabet dice), which makes the Kelly fraction negative, meaning the growth-optimal bet is zero. Kelly's useful lesson for casino players is bankroll discipline: bet small fixed fractions of an entertainment budget you have already written off.
What is fractional Kelly and why use it?
Fractional Kelly means betting a fixed fraction — usually half — of the full Kelly stake. It sacrifices a modest amount of theoretical growth (half Kelly keeps roughly 75%) in exchange for far smaller drawdowns and protection against overestimating your edge, which in practice almost everyone does.