Guide · 7 min read
The Kelly criterion for betting: formula, worked example and why fractions are safer
The Kelly criterion tells you what share of your bankroll to stake on a bet with positive expected value. Here is the formula, a calculation you can follow line by line, a stake table, and the catch that matters most: Kelly is only as good as your probability estimate, and you never know that exactly.
Updated 22 October 2026
Contents
Kelly calculator
Enter the decimal odds, your own probability estimate and your betting bankroll to see the edge, the full Kelly share and the stake for the fraction you choose; the example values are the worked example below.
The Kelly formula
In 1956 John L. Kelly Jr., a researcher at Bell Labs, showed which stake makes wealth grow fastest in the long run when you have an edge. For a simple bet with two results, win or lose, the formula is:
f* = (b × p − q) / b
f* = share of your bankroll to stake
b = net odds, i.e. decimal odds − 1
p = your estimated probability of winning
q = 1 − p, the probability of losing
An equivalent form is often handier: f* = (p × odds − 1) / (odds − 1), which reads as edge divided by net odds. If the top line is zero or negative, you have no edge and Kelly says: don't bet. Kelly is not a way to find value; it only sizes a stake once you already believe you have it. How to judge that is covered in expected value in betting and value betting.
Worked example: odds 2.05, probability 52%
You estimate the true probability of an outcome at 52%, and a bookmaker offers 2.05. Your betting bankroll is €2,000.
- Net odds: b = 2.05 − 1 = 1.05.
- Check the edge: p × odds − 1 = 0.52 × 2.05 − 1 = 1.066 − 1 = +6.6% expected value. Positive, so carry on.
- Kelly share: f* = (1.05 × 0.52 − 0.48) / 1.05 = (0.546 − 0.48) / 1.05 = 0.066 / 1.05 = 6.29%.
- In euros: full Kelly is 6.29% of €2,000 = €125.71. Half Kelly is €62.86. Quarter Kelly is 1.57% = €31.43.
The odds of 2.05 imply a probability of 1 / 2.05 = 48.8%. Your edge exists only because you think the real chance is higher than that. If you want to convert odds into probabilities in other formats, see implied probability.
Stake table: Kelly at odds 2.05
The same bet with different probability estimates, bankroll €2,000. For other odds or bankrolls, use the Kelly calculator at the top of the page.
| Your p | EV | Full Kelly | Quarter Kelly | Quarter Kelly on €2,000 |
|---|---|---|---|---|
| 48.8% | 0.0% | 0% | 0% | €0 |
| 50% | +2.5% | 2.38% | 0.60% | €11.90 |
| 51% | +4.6% | 4.33% | 1.08% | €21.67 |
| 52% | +6.6% | 6.29% | 1.57% | €31.43 |
| 53% | +8.7% | 8.24% | 2.06% | €41.19 |
Amounts rounded to the cent. For other odds, plug your numbers into f* = (p × odds − 1) / (odds − 1), multiply by your fraction and then by your bankroll.
Why half or quarter Kelly
Full Kelly maximises long-run growth, but the ride is rough. Edward Thorp, who applied Kelly to blackjack and markets, notes that a full-Kelly bettor has a 50% chance of seeing the bankroll fall to half its value at some point. Fractions give up a little growth and remove a lot of the swings. Using Kelly's growth formula g = p × ln(1 + b × f) + q × ln(1 − f) per bet for our example:
| Stake | Share of bankroll | Growth per bet if p = 52% is right | … if p is really 50% |
|---|---|---|---|
| Double Kelly | 12.57% | −0.001% | −0.518% |
| Full Kelly | 6.29% | +0.207% | −0.050% |
| Half Kelly | 3.14% | +0.156% | +0.027% |
| Quarter Kelly | 1.57% | +0.091% | +0.026% |
Half Kelly keeps about three quarters of the growth at half the stake. Quarter Kelly still grows even when your estimate was two percentage points too optimistic. And double Kelly earns nothing at all on average, even when your estimate is exactly right: that is what over-betting looks like.
Kelly when your p is wrong
The right-hand column shows the real problem. If the true probability is 50% rather than 52%, the bet is still slightly positive (+2.5% EV). Yet your bankroll shrinks in the long run if you stake full Kelly for 52%. The reason: growth turns negative at around twice the true Kelly share, and 6.29% is more than double the true Kelly share of 2.38%.
Sports betting estimates are rarely better than a point or two, so over-betting is the typical mistake, and it is invisible until a losing run arrives. A quarter Kelly with a cap of 2% to 3% of the bankroll per bet is a common, cautious choice. Even with a correct estimate, results swing a lot around the expected value; a single 52% bet still loses almost half the time.
Kelly vs fixed stakes
- Fixed stake (for example always 1% of the starting bankroll): simple and easy to audit, but it ignores how large the edge is.
- Fractional Kelly: stakes more when the edge is bigger and follows your current bankroll, so stakes shrink automatically after losses.
If your probabilities come from a rough model, a small fixed stake is often more honest than a Kelly calculation that only pretends to be precise. Either way, what matters is that stakes fit your bankroll. More on structuring the money itself in bankroll management.
Does Kelly apply to sure bets?
Hardly. Kelly is built for bets with an uncertain result. With a sure bet, the outcome of the match barely matters as long as every leg is accepted at the shown odds and stands. On paper Kelly would then suggest a very large stake. In practice the stake is limited by quite different things: the balance in each account, the bookmakers' maximum stakes, the risk of a price change or a voided leg, and account limits.
Kelly becomes relevant for value bets. Those are not yet available in ArbVantage (coming soon); the live product is sure bets, free-bet conversion and middles.
Austria: build the 5% betting fee into Kelly
If a bookmaker passes Austria's 5% Wettgebühr on to you, calculate Kelly with the odds after the deduction, not the displayed ones. For our example:
Effective odds = 2.05 × 0.95 = 1.9475.
EV = 0.52 × 1.9475 − 1 = +1.3% instead of +6.6%.
f* = 0.0127 / 0.9475 = 1.34%; quarter Kelly is 0.34%, or €6.70 on a €2,000 bankroll.
The deduction cuts the suggested stake to about a fifth. If your estimate is only just above the implied probability, there is often no edge left at all. Whether a bookmaker passes on the fee shows on the bet slip. The same rule applies to any handling fee a bookmaker deducts from stakes or winnings, wherever you bet: always size the stake on the net odds.
Which bankroll counts
Use only money set aside for betting that you could lose entirely, never your income or savings. Recalculate the bankroll regularly, ideally daily, so stakes get smaller in a losing run. When several bets run at the same time, stake less than each one's Kelly share, because the risks add up.
FAQ
Is full Kelly too risky?
For sports betting, usually yes. Full Kelly assumes your probability is right. If it is only two percentage points too optimistic, your bankroll can shrink over time. Half or quarter Kelly gives up little growth and copes far better with estimation errors.
Does the Kelly criterion apply to sure bets?
Hardly. With a sure bet the result barely depends on the match outcome as long as every leg stands. Stakes there are limited by account balances, maximum stakes, price changes and limits, not by the Kelly formula.
Which bankroll should I use as the base?
Only money you have set aside purely for betting and could afford to lose. Recalculate it regularly so that stakes shrink after losses.
What if Kelly gives a negative number?
Then, by your own estimate, the bet has no edge and you stake nothing. A negative Kelly share means the odds are below your fair odds.
Sources
- J. L. Kelly Jr.: A New Interpretation of Information Rate, Bell System Technical Journal 35 (1956), pp. 917–926
- Edward O. Thorp: The Kelly Criterion in Blackjack, Sports Betting, and the Stock Market
- Unternehmensserviceportal (USP), Austria: Wettgebührenabrechnung
As of October 2026. Worked examples by ArbVantage, rounded.
Betting carries risk, sure bets included: odds change, bets get voided and accounts get limited. Only stake money you can afford to lose. Our responsible gambling page lists free, confidential help in Austria and Germany.
Not betting, legal or tax advice. 18+. As of October 2026.