- Fact-checked by Josh Miller
- Edited by Diana Westfield
Gambling maths formulas sound like a textbook, but there are only a handful that matter. Strip any page on this site back, roulette edges, acca insurance, bankroll advice, and you’ll find the same seven underneath: the expected value formula, house edge, implied probability, overround, variance, Kelly and risk of ruin. We’ve put them on one page, each with a quick example using real numbers and a link to the free CasinoCalcs calculator that does the arithmetic.
Expected value formula
EV = p × win − (1 − p) × stake. Here p is your chance of winning and “win” is the profit if it comes in. Say you put £10 on something at 2.2 that you reckon is a coin flip. Half the time you’re £12 up, half the time £10 down, so the bet is worth +£1 on average. Above zero, it makes money in the long run. Below zero, it costs you. Try your own numbers in the expected value calculator.
House edge and RTP
House edge = −EV per £1 staked and RTP = 1 − house edge. Red at a single-zero wheel wins 18 spins in 37 and pays evens, so you’re 1/37 behind per pound: a 2.70% edge, 97.30% RTP. One thing trips people up. The loss comes from what you stake in total, not what you deposit. Put £20 in and keep re-spinning until you’ve staked £500, and the expected loss at 2.7% is about £13.50. The RTP and house edge calculator works it out.
Implied probability and overround
Implied probability = 1 ÷ decimal odds, or b ÷ (a + b) for fractional odds a/b. So 4.0 and 3/1 both mean 25%. Add up every outcome in a market and anything over 100% is the overround. A tennis match at 1.90 each way round makes 105.3%, which means the book keeps about 5% of each bet. The implied probability calculator and the bookmaker margin calculator do both jobs.
Accumulators and the binomial formula
Multiply the odds of the legs for the acca price, and multiply their chances for how often it lands. Four coin flips together is 0.54, so 6.25%. Margins pile up the same way: four legs at 5% leave you with 0.954, 81% of fair value. When you want “at least k wins out of n”, reach for the binomial formula, C(n, k) × pk × (1 − p)n − k, and add up the terms. Or let the binomial calculator do it.
Variance and standard deviation
If a bet wins w with chance p and otherwise loses the stake s, then variance = p × w² + (1 − p) × s² − EV². The square root of that is the standard deviation. Why care? Because over n bets your expected result grows in step with n, but the spread only grows with √n. Short runs are mostly luck; long runs are mostly edge. A £10 evens bet on a true 50% shot swings by £10 a bet, and by £100 over a hundred of them. The betting variance calculator shows the range.
Kelly criterion and risk of ruin
Kelly stake = (b × p − q) ÷ b of your bankroll, with b as the decimal odds minus 1, p your chance and q = 1 − p. Our 2.2 coin flip from earlier gives (1.2 × 0.5 − 0.5) ÷ 1.2, so 8.3% of the bank. Most sensible bettors use a half or a quarter of that. For risk of ruin on even-money bets with an edge e and N units in the bank, a good rough guide is ((1 − e) ÷ (1 + e))N. A 1% edge and 100 units still leaves around a 13% chance of going bust. The Kelly criterion calculator and the risk of ruin calculator handle both properly. Stuck on a term? The glossary explains it in plain English.
Maths explains the game; it doesn’t beat it. BeGambleAware.org has free support if gambling stops being fun.