Finding a good bet is one problem. Deciding how much to put on it is another, and it gets far less attention. Bet too little and a real edge barely moves your bankroll. Bet too much and a normal losing streak can wipe out months of good decisions.
The Kelly criterion is a formula for that second problem. It tells you what fraction of your bankroll to stake on a bet, based on the price and your estimated chance of winning, so that your bankroll grows as fast as possible over the long run.
The formula
f = (bp − q) ÷ b
Where:
- f is the fraction of your bankroll to bet.
- b is the profit per $1 staked, which is decimal odds − 1. At +120, b = 1.20. At -110, b = 0.909.
- p is your estimated chance of winning.
- q is the chance of losing, 1 − p.
The top half, bp − q, is just the bet's expected value per $1. Dividing by b scales it to the price. So Kelly bets more when the edge is bigger and less when the payout is longer (because long shots swing more).
Worked examples
+120 at a true 50%. b = 1.20, p = 0.50, q = 0.50.
f = (1.20 × 0.50 − 0.50) ÷ 1.20 = 0.10 ÷ 1.20 = 8.33% of bankroll.
-110 at a true 55%. b = 0.909, p = 0.55, q = 0.45.
f = (0.909 × 0.55 − 0.45) ÷ 0.909 = 0.05 ÷ 0.909 = 5.5% of bankroll.
+150 at a true 45%. b = 1.50.
f = (1.50 × 0.45 − 0.55) ÷ 1.50 = 0.125 ÷ 1.50 = 8.33%.
-150 at a true 55%. b = 0.667.
f = (0.667 × 0.55 − 0.45) ÷ 0.667 = −12.5%.
When Kelly says zero
A negative result means the bet has negative expected value. Kelly's answer is simple: don't bet. The same goes for a result of exactly zero, which is a break-even bet with nothing to gain.
That's one of the most useful things about the formula. It won't let you size up a bet you like but can't justify on price. A -110 bet at a true 50% comes out to −5%. No amount of confidence in the pick changes that.
If you're not sure whether a bet is positive in the first place, run the price and your probability through the EV calculator first.
What it looks like with a real bankroll
Using the -110 at 55% example (full Kelly 5.5%) and the +120 at 50% example (full Kelly 8.33%):
| Bankroll | -110 at 55%, full | -110 at 55%, ¼ | +120 at 50%, full | +120 at 50%, ¼ |
|---|---|---|---|---|
| $1,000 | $55.00 | $13.75 | $83.33 | $20.83 |
| $5,000 | $275.00 | $68.75 | $416.67 | $104.17 |
Kelly also resizes as you go. After a win, your bankroll is bigger and the next bet is bigger. After a loss, it shrinks. That's what protects you from going broke: you never bet the whole roll, and you bet less when you're down.
Why most bettors use fractional Kelly
Full Kelly maximizes long-run growth, but only if your probability estimate is exactly right. In practice it never is, and full Kelly is a rough ride even when it is.
Ten straight losses at full Kelly on the 5.5% example leaves you with about 56.8% of your starting bankroll. At quarter Kelly, the same streak leaves about 87.1%. Ten straight losses on 55% bets is rare, but over a long betting life, streaks well beyond what feels reasonable do happen.
Fractional Kelly gives up surprisingly little growth for that protection. Measured by long-run growth rate on the -110 at 55% example:
- Half Kelly keeps about 75% of full Kelly's growth.
- Quarter Kelly keeps about 44%.
- Double Kelly grows at slightly below zero. Betting twice the Kelly amount on a genuinely good bet is a long-run losing strategy.
That last point is why betting too much is so much worse than betting too little. Under-betting costs you some growth. Over-betting can cost you everything.
The real risk: overestimating your edge
The formula is only as good as your p. Suppose you believe a -110 prop wins 55% of the time, but the truth is 52%.
- What you think: +5.0% EV, Kelly says 5.5%.
- The truth: −0.73% EV, Kelly says don't bet.
Here's the long-run growth rate per bet at each sizing, if the true chance is 55% versus 52%:
| Sizing | True 55% | True 52% |
|---|---|---|
| ¼ Kelly | +0.060% | −0.019% |
| ½ Kelly | +0.103% | −0.054% |
| Full Kelly | +0.138% | −0.178% |
| 2× Kelly | −0.002% | −0.637% |
When the edge isn't real, every sizing loses. But the bigger the fraction, the faster it loses. Quarter Kelly turns a three-point overestimate into a slow leak. Full Kelly turns it into a real problem.
This is the strongest argument for fractional Kelly. You will overestimate some edges. A smaller fraction is a margin of safety against your own mistakes.
Practical rules
- Start at quarter Kelly or less. Move up only once you have a long record showing your probabilities are well calibrated.
- Cap any single bet. Many bettors set a hard ceiling, like 2–3% of bankroll, regardless of what the formula says.
- Use fair probabilities, not hopes. De-vigged prices from sharp books are a more honest starting point than gut feel.
- Account for simultaneous bets. Kelly assumes one bet at a time. If you have ten bets running at once, sizing each at full Kelly overstates your total exposure.
The practical takeaway
Kelly turns sizing from guesswork into a rule: bet in proportion to your edge, and bet nothing when there isn't one. Use a fraction of it, because your edge estimates will be wrong sometimes and the cost of over-betting is far higher than the cost of under-betting.
Before sizing any bet, check that it's positive in the first place with the EV calculator. If it isn't, Kelly has already given you the answer.