How to bet (and win)

An interactive explainer on the Kelly Criterion and gambler's ruin.

The opening question

A coin pays even money?If you bet £1 and win, you get £1 profit (plus your £1 back). If you lose, you're out £1. and lands heads 55% of the time. You have £1,000 in your account. How much do you bet?

£0

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:

edge = bp − q

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:

f * = edge / b

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.

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.

55.0%
1.0 : 1
Edge (per £1 staked)
+£0.10
Fair payout (b for zero edge)
0.82 : 1
the payout you'd need for a break-even bet at this probability
What % of your money to bet
10.00%
the Kelly fraction, f*
Suggested bet on £1,000 bankroll
£100

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.

scroll to see 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.

£1,000
500
55.0%
1.0 : 1
1x = full Kelly. 0.5x = half Kelly. Anything above 2x enters territory where even a positive edge goes bankrupt.
1.0x
Scenarios
Kelly strategies
Figure 1 - Theoretical growth rate by Kelly multiplier. Your current choice is marked. Above 2x, growth goes negative: even a positive-edge bet loses money on average.
Figure 2 - Median bankroll trajectory, with 10th to 90th percentile band.
Reset

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.

scroll to see why ↓

Full Kelly vs half Kelly

Coming soon.

The warning: Gambler's Ruin

Coming soon.

Beyond the casino

Coming soon.