Variance is a measure of how far actual results scatter from the expected value – a quantitative assessment of the dispersion of real outcomes around the theoretical mean.
Expected value and house advantage are only achieved over a long horizon. Actual results approach their theoretical values only after a large number of repetitions of the random event. In practice, over a short playing session, actual results always differ from theoretical values to some degree. Variance is precisely that measure of dispersion – how far actual results deviate from the mathematical expectation. The variance of a single bet shows the extent to which the actual outcome can differ from the mathematical average.
Variance underscores the probabilistic nature of gambling. It is variance that allows a player to win in a game with a negative expected value – thanks to volatility and the unpredictability of each individual random event.
Roulette is a game with a stable house advantage – the casino’s edge is identical on every bet. It makes no difference which position the player bets on. The casino retains the same fixed percentage from every wager placed on the table, regardless of bet type. However, different bets on Roulette carry different payout ratios, and as a result, differ significantly in variance.
Variance Formula
Variance of a random variable is the mathematical expectation of the squared deviation of the random variable from its expected value.
Expected value – the probability-weighted average of all possible outcomes:
The probabilities of all possible outcomes must sum to 1:
Variance – each outcome’s deviation from the mean is squared and weighted by its probability:
Where Xi is the i-th value of the random variable, Pi is the probability that the random variable takes the value Xi, and n is the number of possible values of the random variable.
Standard deviation – the square root of variance:
Variance is expressed in squared units of the random variable. Standard deviation is expressed in the same units as the random variable itself, which is why standard deviation is often more practical for characterizing the behavior of a random variable.
Why Variance Matters
For the player, variance is the opportunity to win. Every player knows that in the long run, the casino always wins – as long as a house advantage exists. Over a short horizon, however, the player has a chance to win precisely because of variance. The higher the variance, the more dramatically actual results can deviate from the theoretical average.
A player can therefore choose a higher-variance game or higher- or lower-variance bets within the same game. Variance reflects the degree of risk and allows the player to select a game with the desired risk level. A player with a limited bankroll, choosing a game and a playing strategy (bet size, number of hands or spins), can predict with high probability the range in which the results of their session will fall.
For the casino, understanding variance is essential for analyzing actual results. The casino needs to know the acceptable range within which actual results should fall – and what level of deviation is considered normal. In a game with large bets and high variance, the casino faces the risk of a significant loss that could threaten its solvency. Understanding variance allows the casino to calculate the probability of such a negative event.
