The Kelly Criterion: The Math of How Much to Bet

The Kelly Criterion is a formula telling you what fraction of your bankroll to stake on a favorable bet so your wealth grows fastest over the long run. For odds-style bets it is f = (bp - q) / b, where p is your win probability, q = 1 - p, and b is the net odds.

A formula born from information theory, not a casino

The criterion came from an unlikely place: a 1956 paper by John Larry Kelly Jr., a scientist at Bell Telephone Laboratories, titled "A New Interpretation of Information Rate" and published in the Bell System Technical Journal. Kelly was working in the shadow of his colleague Claude Shannon, whose information theory described how much signal could be pushed through a noisy communication channel.

Kelly's leap was to notice that a gambler with inside knowledge faces a mathematically identical problem to a communication channel. If you receive "side information" about a race or a hand that the posted odds do not fully reflect, the rate at which you can grow your money is bounded by exactly the same quantity Shannon used to measure information: the channel capacity. Bet optimally on that information and your capital compounds exponentially at a rate equal to the rate of information transmitted.

According to the popular history of the formula, the paper's title was a corporate compromise. Kelly is said to have first called it "Information Theory and Gambling," but AT&T executives feared headlines suggesting Bell Labs was helping bookmakers, so the word "gambling" was scrubbed from the title. The math underneath did not change.

The formula, decoded

The classic odds form is compact. For a bet where you win b dollars per dollar staked, with win probability p:

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

Read it as a ratio of edge to odds. The numerator bp - q is your expected profit per dollar wagered, your edge. Dividing by b scales that edge down by how generous the payout is. A huge edge on a low-odds bet still warrants a large stake; a thin edge on a long shot does not. Three features fall out immediately:

• If your edge is zero or negative (bp <= q), Kelly says bet nothing. It never recommends a bet you expect to lose.
• The output is a fraction of your current bankroll, so the dollar amount shrinks after losses and grows after wins automatically. You cannot go broke betting a fixed fraction of what remains.
• For an even-money bet (b = 1), the formula collapses to a handy shortcut: f = 2p - 1, simply twice your win chance minus one.

A worked example: the 60% coin

Imagine a biased coin that lands in your favor 60% of the time, paying even money. So p = 0.6, q = 0.4, b = 1. The shortcut gives f = 2(0.6) - 1 = 0.20. Kelly says stake exactly 20% of your bankroll on every flip, no more, no less. On a $1,000 bankroll the first bet is $200. Win and you have $1,200 and bet $240 next; lose and you have $800 and bet $160 next. The wager rides your wealth up and down.

Why 20% and not more, given such a strong edge? Because Kelly maximizes the expected logarithm of wealth, the geometric growth rate g:

g(f) = p * ln(1 + f) + q * ln(1 - f)

At f = 0.20 this gives g = 0.6 * ln(1.2) + 0.4 * ln(0.8) = 0.0201, about 2.01% growth per flip. That is the peak. Crucially it is far below the naive arithmetic edge of 4% per flip (0.2 * (0.6 - 0.4)), because compounding punishes losses asymmetrically: a 50% drawdown needs a 100% gain to recover. Push the stake to triple Kelly (f = 0.60) and the growth rate turns sharply negative, to roughly -8.5% per flip. You lose money fast in the long run despite winning 60% of every individual bet.

Why it maximizes long-run growth

The deep reason Kelly works is the difference between averaging and compounding. If you care about the average of many parallel universes, you would bet everything every time, the arithmetic-maximizing choice. But you live in one universe where outcomes multiply sequentially. The quantity that compounds is the product of your per-round multipliers, and the long-run growth rate of a product is governed by the average of the logarithms, the geometric mean. Kelly is simply the bet size that sits at the top of the log-wealth curve. Maximize expected log wealth and, by the law of large numbers, you provably end up with more money than any other strategy over a long enough sequence. This is the same compounding logic behind tools like our compound interest calculator, where the order and consistency of growth, not just the average, determine where you finish.

The honest catch: variance and bad estimates

Full Kelly is mathematically optimal and emotionally brutal. Even with a healthy edge, simulations routinely show drawdowns exceeding 50-60% along the way to the optimum. Sitting at the very peak of the growth curve means tolerating wild swings that wipe out most human discipline. Worse, the peak is dangerously asymmetric: the cost of overbetting is far steeper than the cost of underbetting, because the log function falls off a cliff as your fraction approaches total ruin.

This is why practitioners almost never use full Kelly. They use fractional Kelly, typically half or quarter. The trade-off is remarkably favorable. Betting half Kelly keeps roughly 75% of the maximum growth rate while cutting the variance of growth by about 75%. Quarter Kelly is gentler still. You give up a little speed to buy a much smoother ride.

The second reason to shrink is sharper than mere nerves: the formula assumes you know p. You never do. In the real world p is an estimate, and Kelly is hyper-sensitive to it. Because overbetting is punished so much harder than underbetting, an optimistic probability estimate can push you past the peak into negative-growth territory without warning. Betting half of what your estimate suggests is partly an admission that your estimate is probably too rosy. A 2017 study by Victor Haghani and Richard Dewey drove this home: 61 finance-literate participants bet on a coin they were told was 60% biased, and despite the strong, known edge, about 28% went bust and only 21% reached the payout cap, with many betting on the wrong side entirely. Knowing the formula is not the same as having the discipline to follow a scaled-down version of it.

Where it shows up

Kelly stayed largely theoretical until mathematician Edward Thorp used it to size bets while card-counting blackjack, then carried it to Wall Street to run a long-running profitable hedge fund. Its logic threads through serious quantitative investing, professional sports betting, and poker bankroll management, anywhere a participant has a genuine, estimable edge and faces repeated bets. The same instinct, find a measurable edge and size your commitment to it, appears in competitive ranking systems too; our Elo rating calculator turns win probabilities into rating adjustments using a related logarithmic feel for how much a single result should move you.

The takeaway is not "bet 20% of your money." It is a way of thinking: your edge and your odds together dictate a single rational stake, growth compounds geometrically rather than arithmetically, and the safe error is always to bet less than the math allows. You can experiment with the inputs directly in our Kelly Criterion calculator to see how sensitive the optimal fraction is to small changes in your assumed probability.

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