The Kelly Criterion vs Fractional Kelly — Formula, Worked Example and the Risk of Full Kelly
Researched and drafted with AI assistance from published sources; published by LokeNessiSport Editorial Team · Last updated: 16 August 2026
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The Kelly criterion tells you what fraction of your bankroll to stake on a bet with a positive expected value: f* = (b × p − q) ÷ b, where b is the decimal odds minus 1, p your probability of winning and q = 1 − p. Full Kelly maximises long-run growth only if your probabilities are right, and it produces deep drawdowns even when they are. Fractional Kelly stakes a fixed share of f*: half Kelly keeps about three quarters of the growth rate while making large drawdowns far less likely. This guide works through the formula, the maths of over-betting and how to pick a fraction.
Where the formula comes from
John L. Kelly Jr., a researcher at Bell Labs, published "A New Interpretation of Information Rate" in 1956. The idea was later applied to blackjack, sports betting and investing, most famously by Edward O. Thorp, whose paper "The Kelly Criterion in Blackjack, Sports Betting, and the Stock Market" (presented in 1997; table of contents and section PDFs) is the standard practical reference for the fractional-Kelly figures below.
Kelly's aim is to maximise the expected logarithm of wealth, which is the same as maximising the long-run growth rate of the bankroll. Unlike flat staking, which bets the same amount whatever the edge, Kelly stakes more when the edge is bigger and nothing when there is no edge.
The formula, step by step
f* = (b × p − q) ÷ b, which can also be written as (p × odds − 1) ÷ (odds − 1).
- f* is the fraction of your current bankroll to stake
- b is the decimal odds minus 1, the profit per unit staked
- p is your estimated probability of winning, as a decimal
- q is 1 − p, your estimated probability of losing
Worked example: a bookmaker offers 2.10, so b = 1.10. Your own estimate of the probability is 54%, so p = 0.54 and q = 0.46.
- Numerator: (1.10 × 0.54) − 0.46 = 0.594 − 0.460 = 0.134
- Divide by b: 0.134 ÷ 1.10 = 0.1218
Full Kelly says stake 12.18% of your bankroll. Notice what that estimate implies: an expected value of 0.54 × 2.10 − 1 = +13.4% per bet, which is a very large edge. Real edges are usually much smaller, and a probability estimate that is a few points too high changes the answer completely, which is the problem the rest of this guide deals with.
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Fractions of Kelly: what you give up and what you gain
If you stake a fixed fraction c of the full Kelly amount, the long-run growth rate is roughly c × (2 − c) of the full-Kelly growth rate. This is the usual small-edge approximation, and it matches Thorp's statement that half Kelly gives about three quarters of the growth rate.
Thorp also gives, in the same continuous approximation, the probability that the bankroll ever falls to a fraction x of its starting value: x^((2 ÷ c) − 1). For full Kelly that is simply x.
Putting both together, with the probability of ever halving the bankroll (x = 0.5) as the risk measure:
- Full Kelly (c = 1): 100% of the maximum growth rate; probability of ever halving 0.5, so one chance in two
- Half Kelly (c = 0.5): about 75% of the growth rate; probability of ever halving 0.5³ = 12.5%
- Quarter Kelly (c = 0.25): about 44% of the growth rate (0.25 × 1.75); probability of ever halving 0.5⁷, about 0.8%
- Eighth Kelly (c = 0.125): about 23% of the growth rate (0.125 × 1.875); probability of ever halving 0.5¹⁵, about 0.003%
These figures assume your probability estimates are correct. If they are too optimistic, your real Kelly fraction is smaller than you think, and the stake you choose is in effect a bigger fraction of the true Kelly amount.
The trap of over-betting
The growth approximation c × (2 − c) shows why over-betting is so costly:
- At 2 × Kelly, expected growth is roughly zero: 2 × (2 − 2) = 0
- Beyond 2 × Kelly, Thorp notes that expected logarithmic growth turns negative, so the bankroll is expected to shrink over time even though every bet has a positive expected value
Estimation error makes this a real risk rather than a theoretical one. In the example above, if the true probability is 49% rather than 54%, the bet at 2.10 has an expected value of 0.49 × 2.10 − 1 = +2.9%, and the true full-Kelly stake is (1.10 × 0.49 − 0.51) ÷ 1.10 = 2.6% of the bankroll. Staking the 12.18% you calculated would be more than four times the true Kelly amount.
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Worked example: Kelly at a realistic edge
Take a more typical case: odds of 1.95 (b = 0.95) and an honest estimate of 53%.
- Full Kelly: (0.95 × 0.53 − 0.47) ÷ 0.95 = 0.0335 ÷ 0.95 = 3.53% of the bankroll — £35 on £1,000.
- Half Kelly: 1.76%, about £18. Quarter Kelly: 0.88%, about £9.
- Expected value per bet: 0.53 × 1.95 − 1 = +3.35%.
Now move the estimate by two points. If the true probability is 51%, the expected value is 0.51 × 1.95 − 1 = −0.55% and the Kelly formula returns a negative number: there is no bet at all. A two-point error — well within the normal uncertainty of any forecast — turns a 3.5% Kelly stake into a losing one. That is the practical case for fractions: at quarter Kelly the £9 stake on a bet that turns out to be slightly negative costs very little, while full Kelly's £35 compounds the mistake across every similar bet.
The c × (2 − c) rule also follows from the arithmetic. For small edges, growth per bet when staking a fraction f of the bankroll is roughly f × edge − f² × variance ÷ 2, which is largest at the Kelly stake. Stake c times that amount and the growth is (c − c² ÷ 2) of twice the maximum, which simplifies to c × (2 − c) of the full-Kelly rate: 0.75 at half Kelly, 0 at 2 × Kelly.
Choosing a fraction
There is no universal correct fraction, and claims that professionals all use one particular number are not something anyone can verify. The honest way to choose:
- Start from how much you trust your probabilities. If you have no long record showing your estimates beat the closing price, assume they are optimistic and use a small fraction, or flat stakes.
- Decide how deep a drawdown you can sit through without changing your plan. The halving probabilities above are a guide.
- Cap any single stake. A fixed ceiling, for example 2-3% of the bankroll, protects you when the formula produces a large number from a large but probably wrong edge.
- Recalculate the bankroll on a fixed schedule, for example weekly, rather than after every result.
- Compare prices before staking. A better price raises b and changes f*. Check more than one book (Thunderpick is one we hold a partner account with), and remember that every operator sets its own stake limits and account terms.
If you are unsure, our guides to betting unit size and staking plans compared cover flat and percentage staking, which do not depend on a probability estimate at all.
Key takeaway: Kelly is a formula for the stake that maximises long-run growth when your probabilities are right. Because they never are exactly, use a fraction of it: half Kelly keeps about three quarters of the growth with a much lower chance of a deep drawdown, and anything above twice Kelly is expected to lose ground.
Staking maths does not make betting safe. Only bet money you can afford to lose; our responsible gambling page lists tools and free support. 18+.
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Related reading
Sources
- table of contents and section PDFs (sites.oxy.edu)
Frequently Asked Questions
What is the Kelly criterion formula?
f* = (b x p - q) / b, where b is the decimal odds minus 1, p your probability of winning and q = 1 - p. At odds of 1.95 and an estimate of 53%, f* = (0.95 x 0.53 - 0.47) / 0.95 = 3.53% of your bankroll. A negative result means do not bet.
Why do bettors use half Kelly or quarter Kelly?
Because probability estimates are never exact. Staking a fraction c of Kelly keeps roughly c x (2 - c) of the growth rate: half Kelly keeps about 75% of it, while Thorp's approximation puts the chance of ever halving the bankroll at 12.5% instead of 50% for full Kelly.
What happens if I bet more than the Kelly stake?
Growth falls. At twice the Kelly stake expected long-run growth is roughly zero, and beyond that the bankroll is expected to shrink even though each bet has positive expected value. Overestimating your probability has the same effect as over-betting.
Is the Kelly criterion good for beginners?
Only with a small fraction and a stake cap, or not at all. Kelly depends entirely on your probability estimates; a two-point error at odds of 1.95 can turn a 3.5% stake into a bet with negative expected value. Flat staking of 1-2% needs no probability estimate.