How to bet (and win)
An interactive explainer on the Kelly Criterion and gambler's ruin.
The opening question
A coin pays even money? and lands heads 55% of the time. You have £1,000 in your account. How much do you bet?
Building intuition: edge
Every bet has two numbers: the probability you win, and how much you win if you do. Neither alone tells you whether the bet is worth taking. A 20% chance to win at 10:1 payout is a great bet; a 20% chance at 2:1 is a terrible one. We need a single number that combines both.
That number is edge: how much money you expect to make, on average, for every £1 you stake.
For a bet with win probability p, loss probability q = 1 − p, and net payout b : 1 (meaning: stake £1, win £b profit if successful, lose your £1 if not), edge is:
Positive edge means you expect to make money. Zero edge means you break even on average. Negative edge means you expect to lose (which is why casinos exist).
The Kelly Criterion takes edge and tells you what fraction of your bankroll to bet:
The formula captures the full picture: bet more when edge is large, and bet less when the payout b is large. A bigger b means rarer wins, so you can go through long losing runs before the payoff arrives. Kelly shrinks your bet to keep you solvent through those runs.
The formula is named after John Kelly Jr., a Bell Labs researcher who derived it in 1956 in a paper on information theory. He wasn't trying to solve a gambling problem - he was trying to understand how much information a noisy communication channel could carry, and betting happened to be the cleanest way to frame the maths. Kelly himself never used the formula to make money. He died at 41, in 1965, without publishing anything else on the topic.
Try it below. Drag the probability and payout sliders, and watch edge, fair odds, and the Kelly fraction update. Then click one of the reference chips below the calculator: each applies the parameters of a real-world bet so you can see how edge behaves across orders of magnitude.
Notice how the pre-seed VC's fraction is only 5.5%, despite the enormous edge. Notice how even a 55/45 coin has you only betting 10%. The formula is conservative for a reason. The next section shows you why.
The main event
Kelly gives you a number. The simulator lets you disobey it. Pick your bet, pick how much of Kelly to actually wager, and run it a thousand times to see what you'd have ended up with.
Try the over-Kelly preset. Same coin, same edge. The only difference is bet size. Look at the bankruptcy rate.
Most professional gamblers bet half Kelly, not full. The next section explains why.
Full Kelly vs Half Kelly
Every Kelly calculation depends on two numbers: the probability you win, and how much you win if you do. The simulator takes these as given. Real life doesn't.
In practice, your edge is an estimate. A card counter thinks the count is favourable, but how favourable exactly? A pre-seed VC thinks 10% of investments will return 20×, but maybe it's 8%, or 13%. A sharp sports gambler thinks they're hitting 54%, but the true rate won't be revealed for thousands of bets. Every real-world Kelly calculation rests on numbers that are slightly off.
This wouldn't matter if Kelly were symmetric around its peak. It isn't. Betting 20% less than optimal barely dents your long-run growth rate. Betting 20% more can push you into negative territory, where every additional round erodes your bankroll. The punishment for overshooting is much worse than the reward for hitting the mark.
The chart below shows what happens to your growth rate if your edge estimate is wrong. Move the slider to vary how far off your estimate is, and watch where full Kelly lands compared to half Kelly.
Notice the asymmetry. When your edge estimate is spot on, full Kelly beats half Kelly by roughly 25%. When you're 20% over-confident - a realistic amount - full Kelly gives back almost all of that advantage, and it keeps falling from there. Half Kelly's curve is much flatter: it rewards you when you're right, and it barely punishes you when you're wrong.
This is the case for half Kelly. You sacrifice a quarter of your theoretical growth rate in exchange for a simulation that doesn't explode when your inputs are slightly off. Given that your inputs are always slightly off, this is a trade professionals make without thinking.
Everything so far has assumed you had an edge in the first place. The next section is about what happens when you don't.
The warning: Gambler's Ruin
A coin pays even money? and lands heads 50% of the time. You have £1,000 in your account. How much do you bet?
Gambler's Ruin Simulator
You have £1,000. Target: £2,000. How do you get there?
Notice: at every setting, your expected profit is exactly zero. Gambler's ruin isn't about losing more than you win - it's about losing everything, which is permanent, while gains only compound while you're still in the game.
This is what a casino is, mathematically. A small negative edge, repeated until you run out of money. The house doesn't need to beat you on any single round - it just needs to stay in the game longer than you can.
Beyond the casino
Three sentences, if you remember nothing else:
Only bet when you have an edge. That edge can come from either side of the equation - better-than-fair odds, or better-than-fair payout for the odds you face. And even with a real edge, bet less than you think: half of what the maths says is about right.
Why fund managers diversify
A pre-seed VC with genuine edge faces a Kelly fraction of around 5%. Most partners physically can't deploy 5% per cheque - their LPs demand 30+ portfolio companies, which forces each position down to 2-3%. What looks like caution is really half-Kelly dressed up as diversification policy. Kelly explains the shape of venture portfolio construction in one number.
Why one life isn't 500 lives
The mean final wealth across 500 parallel simulations can look great even if almost every individual simulation ended in ruin. That's because the mean is dominated by a few lucky paths. You don't get to live the average; you live one path. This is called ergodicity economics, and it's the deepest argument for betting sub-Kelly.
This isn't only about gambling
Kelly shows up wherever compounding and uncertainty meet. Portfolio sizing. Business reinvestment. AI compute budgets. Whether to build out capacity before you know demand. Any repeated decision with positive expected value and variance is, secretly, a Kelly problem. The maths never stops being relevant - it just changes costume.
Further reading
- Kelly's original 1956 paper: "A New Interpretation of Information Rate"
- William Poundstone, Fortune's Formula - the popular book on Kelly, Shannon, Thorp, and the MIT blackjack team
- Ed Thorp, A Man for All Markets - the autobiography of the man who applied Kelly to beat both blackjack and the stock market
- Ole Peters, The ergodicity problem in economics - the deeper mathematical case for half-Kelly and sub-Kelly strategies