Two numbers decide whether and how much to bet: the expected value of the bet, which says whether the price is better than fair, and the Kelly fraction, which says how much of your bankroll to risk given that edge. Both need the same inputs — a probability you believe and a price you are offered — and both are simple formulas that are easy to misuse. This guide gives the formulas, the worked examples, and the reasons to be careful.
Expected value
Expected value is the average profit per unit staked if you could make the same bet many times. With a fair probability p and decimal odds d:
EV = p × (d − 1) − (1 − p) × 1 = p × d − 1 p = 55%, d = 2.00: 0.55 × 2.00 − 1 = +0.10 (+10%) p =
50%, d = 1.91: 0.50 × 1.91 − 1 = −0.045 (−4.5%)
The second example is the standard -110 line bet at random: every such bet loses 4.5% of its stake on average. The first is a bet with a 10% edge, which is large; real edges are usually 1–5%. Expressed as a percentage, EV is often called the edge.
Only positive EV bets are worth making. A bet with negative EV loses money on average no matter how you stake it, and no staking system changes that. The entire problem of profitable betting is finding prices where your probability is higher than the price implies.
Where the probability comes from
The formula is only as good as p. Three common sources:
- A devigged sharp market. Take the full market at a low-margin bookmaker or an exchange, remove the overround, and read the fair probability. This is the standard for most bettors; the devigging guide covers the methods.
- A model. Anything from a rating system to a full simulation. Good models are rare, and calibrating one is a project in itself.
- Judgment. Cheap, and usually worse than the market.
The EV & Kelly calculator can devig a market you paste in and use the resulting probability directly, so the two steps happen together.
Commission
If the price is on an exchange, use the effective odds after commission. At 5% on net winnings, 2.10 becomes 2.045, and the EV changes with it. For turnover commission the calculators work with the win/lose pair instead of a single price, so the EV comes out right either way.
EV = p × (1 + (d − 1)(1 − c)) − 1 p = 52%, d = 2.10, c = 5%: 0.52 × 2.045 − 1 = +6.3%
The Kelly criterion
Kelly answers: given an edge, what fraction of my bankroll should I risk to make it grow fastest over the long run? For a bet that wins b units per unit staked with probability p and loses one unit with probability q = 1 − p:
f* = (p × b − q) / b = (p × d − 1) / (d − 1) p = 55%, d = 2.00: (1.10 − 1) / 1 = 10% p = 60%, d =
1.80: (1.08 − 1) / 0.80 = 10% p = 30%, d = 4.00: (1.20 − 1) / 3 = 6.7%
Notice that the numerator is the EV. Kelly is the edge divided by the net odds: the same 10% edge suggests a 10% stake at even money but only 6.7% at 4.00, because the longer price has more variance per unit of edge.
Kelly is zero or negative when EV is zero or negative, which is the formula's way of saying: do not bet.
Why fractional Kelly
Full Kelly maximises the growth rate of the bankroll only if p is exactly right. Real probability estimates have errors, and Kelly punishes overestimates severely: betting twice the Kelly fraction has an expected growth rate of zero, and anything above that shrinks the bankroll on average despite the positive edge. The variance at full Kelly is also brutal; a 50% drawdown is routine.
Most practitioners stake a fixed fraction of Kelly, usually a half or a quarter. Half Kelly gives up about a quarter of the growth rate for a large reduction in variance; quarter Kelly gives up about half the growth for a very smooth ride. The calculator reports full, half and quarter Kelly and multiplies by your bankroll.
Half Kelly on a 10% edge
p = 55%, d = 2.00, bankroll 1,000. Full Kelly: 10% → 100. Half: 50. Quarter: 25. A bettor unsure whether p is really 55% or "somewhere around 53–57%" should be at half or below.
Kelly with several simultaneous bets
Kelly assumes one bet at a time on a bankroll that updates between bets. With many bets open at once — a weekend of matches, say — the fractions interact, and the safe approach is to scale them down further, or to treat the sum of open stakes as the constraint. A rule of thumb: keep total open exposure under the full Kelly fraction of the largest edge, and stake each bet at a quarter or less.
Common mistakes
- Staking on negative EV. No formula helps.
- Using the bookmaker's implied probability as p. That probability contains the margin; it will say every bet has negative EV. Devig first.
- Trusting p to two decimals. A 55% estimate is really a range. Fractional Kelly is the correction.
- Full Kelly on longshots. The variance at long odds is extreme; even half Kelly can be too much when the win rate is low.
- Confusing the stake with the liability. For a lay, Kelly applies to the liability.
Worked example, start to finish
From a sharp market to a stake
A sharp book prices a tennis match at 1.45 / 2.90. Implied: 69.0% + 34.5% = 103.4%. Multiplicative devig: 66.7% / 33.3%. Fair prices: 1.50 / 3.00. A recreational book offers 3.20 on the underdog. EV: 0.333 × 3.20 − 1 = +6.7%. Kelly: 0.067 / 2.20 = 3.0% of bankroll. Half Kelly on a 2,000 bankroll: 30. Run it in the
calculator
.
Kelly for lays and exchange bets
On an exchange the bet's payoff is asymmetric in a different way. A back bet at effective odds d has the same Kelly formula as above. A lay at odds L is a back on the complement at L/(L−1), so its Kelly fraction uses the complement's probability (1 − p) and the converted odds; the fraction then applies to the liability, not the backer's stake. Laying a 1.20 favourite you believe is only 75% likely to win is backing "not this outcome" at 6.00 with a 25% probability: EV = 0.25 × 6 − 1 = +50%, Kelly = 0.50 / 5 = 10% of bankroll as liability, which corresponds to a backer's stake of half that. The commission comes off the lay's winnings, so use the effective converted price before computing either number. The commission calculator in lay mode gives the effective figures to feed in.
Expected value versus expected growth
Expected value is linear: two bets with +5% EV each have +10% EV together, whatever their size. Growth is not. A bankroll that alternates between doubling and halving has an average return of +25% per bet and a growth rate of zero, because it ends where it started. Kelly maximises the growth rate, which is the average of the logarithm of the bankroll ratio, and that is why it caps the stake even on bets with large positive EV: beyond the Kelly fraction, extra stake adds variance faster than it adds growth. The practical reading is that the goal of staking is not to maximise the expected profit on the next bet but to maximise the rate at which the bankroll compounds, and the two disagree whenever variance is large. Fractional Kelly sits on the safe side of that disagreement.
Closing line value
Because true probabilities are never observed, bettors need a proxy to check their p. The most common is closing line value: compare the price you took with the price the sharpest market showed just before the event started. If you consistently beat the closing price, your probabilities are better than the market's at the time you bet, and your EV estimates are probably real. If you consistently do not, the edge is imaginary, however the results have gone so far. Results are noisy; a hundred bets tell you little. Closing line value shows up in a few dozen bets. Record both the price taken and the closing price, devig them the same way, and compare the implied probabilities; the difference is a direct estimate of the EV you are actually capturing.
Kelly when the odds are long
At long prices the Kelly fraction is small even for large edges, and the variance is still high. A +900 shot you believe is a true 12% is a +20% EV bet — a huge edge — and the Kelly fraction is 0.20 / 9 = 2.2%. At a quarter Kelly that is about half a percent of bankroll, which feels like nothing and is correct: the bet wins one time in eight, and a run of twenty losses is routine. Bettors who stake longshots at the sizes they use for favourites are the ones Kelly is protecting against. The rule of thumb that falls out of the formula is that stake size should scale with the edge divided by the net odds, so a favourite with a 3% edge at 1.50 (Kelly 6%) deserves several times the stake of a longshot with a 10% edge at 6.00 (Kelly 2%).
Summary
EV = p × d − 1 tells you whether to bet. Kelly = EV / (d − 1) tells you the growth-maximising stake, which is too aggressive in practice; use a half or a quarter of it. Both depend entirely on p, and p should come from a devigged sharp market or a calibrated model, never from the price you are about to take.