Probability terms behind the house edge and variance

Probability explains why casino games make money for the house over the long run even though players can win in a short session. The house edge comes from the gap between true odds and payout odds, while variance describes how far your results stray from the average. The eight terms below use European roulette (37 pockets) as the example, so every figure can be checked.

Edition 20 October 2026 · compiled by the glossary editors · 8 terms

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ProbabilityChance; likelihood: A measure of how likely an outcome is, written as a number between 0 (impossible) and 1 (certain), or as a percentage. In a game where every outcome is equally likely, the probability is the number of outcomes you want divided by the number of possible outcomes. The probability of two independent events both happening is their individual probabilities multiplied together.; Example: One die shows a six with probability 1/6, about 16.7%, while three Sic Bo dice show three sixes at once with probability 1/216, about 0.46%.
True odds vs payout oddsFair odds; paid odds: True odds reflect the real probability of an outcome, while payout odds are what the game actually pays when that outcome lands. The gap between the two is where the house edge comes from. If a game paid exactly at true odds, its house edge would be zero. At a sportsbook the same gap is known as the bookmaker margin. Payout odds are usually written in the paytable as a ratio such as 35:1 or 1:1.; Example: A single number in European roulette has true odds of 36 to 1 but pays 35 to 1, and that one-unit gap across 37 spins is the 2.70% house edge.
Expected value (EV)Expectation; long-run average: The average profit or loss per bet if the same bet were repeated many times. You get it by multiplying each possible outcome by its probability and adding the results together. A negative EV means the bet wears down a balance on average. For a slot, the EV of a spin is the stake times (RTP minus 1). EV does not tell you the result of one round, only the direction of the long-run average.; Example: RM10 on red in European roulette gives (18/37 × RM10) minus (19/37 × RM10), about −RM0.27, an average loss of 2.70% per bet.
VarianceSpread of results: A statistical measure of how widely results are scattered around the expected value, calculated as the average squared distance of each outcome from the mean. Two bets can share the same EV yet have very different variance. High variance means a bumpier session, with bigger but less frequent wins and losses. In slots the same idea is presented as volatility. It is measured in squared ringgit.; Example: RM1 on red and RM1 on a single number both have an EV of about −RM0.027, but the variance of the red bet is close to 1 while the single number's is about 34.
Standard deviationSD; spread: The square root of the variance, which puts it in the same unit as the stake, here ringgit. For a run of independent bets the standard deviation grows with the square root of the number of bets, not in direct proportion: four times as many bets only doubles it. As a rough guide, about 95% of sessions land within two standard deviations of the expected value.; Example: After 100 bets of RM1 on red, the expected loss is RM2.70 and the standard deviation about RM10, so anything from a RM23 loss to a RM17 profit is an ordinary result.
Law of large numbersThe long run: The principle that the average result per bet moves closer to the expected value as the number of bets grows. It does not mean past results get balanced out; an early loss is not won back, it simply becomes a smaller share of the overall total. For a house taking millions of bets, this law turns a small edge into income it can predict quite closely.; Example: After 100 bets of RM1 the RM10 standard deviation dwarfs the RM2.70 expected loss, but after 10,000 bets the RM270 expected loss is already larger than the standard deviation of about RM100.
Independent eventsIndependent rounds; no memory: Events whose outcome is not affected by any other event. Roulette spins, dice rolls, RNG slot spins and crash rounds are independent, because the wheel, the dice or the software has no memory of earlier results. Cards dealt from a blackjack or baccarat shoe are not fully independent, since cards already dealt change what is left in the shoe, though the effect is small and does not remove the house edge.; Example: The probability of red on the next spin stays at 18/37, about 48.6%, whether the last spin was red, black or zero.
Gambler's fallacyMonte Carlo fallacy; 'it's due' error: The mistaken belief that an outcome which has not appeared lately is now due, or that a streak has to reverse. It comes from confusing the probability of a whole streak with the probability of the next round. The fallacy pushes people to raise their stakes after losing, and it sits close to chasing losses and to doubling systems such as the Martingale.; Example: Five reds in a row happen only about 2.7% of the time, yet once they have happened the chance of black on the sixth spin is still 18/37, not higher.

These terms support other entries in the glossary. RTP is a slot's expected value written as a percentage, and volatility is variance in the language of slot providers. To see how the odds gap is worked out at a sportsbook, read the entry on the bookmaker margin. If you want to work out the house edge of a particular game with your own numbers, use the House Edge Calculator for Casino Games and Slots. No betting system turns a negative expected value into a positive one.