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Variance and the long run

Field: Mathematics. This page was last modified on 17 August 2026.

Expected value gives the direction of a game. Variance gives the width of the road. Together they explain the single most misunderstood feature of games of chance: that short sequences of results can point almost anywhere, while long ones cannot.

Measuring scatter

Variance is the average of the squared distances between each possible result and the expected value. Squaring keeps positive and negative deviations from cancelling and gives large deviations more weight. Standard deviation is the square root of the variance, which returns the measure to the units of the original wager and makes it directly comparable to a stake.

The practical use of the number is that results cluster within a small number of standard deviations of the expectation. A wager repeated many times will, in the large majority of cases, finish within about two standard deviations of where the expected value says it should be. That band is wide early and narrow later, in a specific and calculable way.

Why the long run wins

Consider a game with a fixed edge played n times at a constant stake. The expected loss grows in proportion to n. The standard deviation grows in proportion to sqrt(n). Because n outgrows sqrt(n), the drift eventually dominates the scatter, and it does so without any change in the rules, the equipment or the player.

Some numbers make the ratio concrete. After one hundred wagers, the drift is one hundred times the per-wager edge while the scatter is ten times the per-wager standard deviation. After ten thousand wagers, the drift is ten thousand times the edge while the scatter is only one hundred times the standard deviation. The scatter has grown, but its share of the total has fallen by a factor of ten.

This is the law of large numbers stated in the units of a game. It was proved in general form in the early eighteenth century, in Jacob Bernoulli's Ars Conjectandi, published in 1713, and it is the theorem that turns a per-wager edge into a business model.

The gambler's fallacy and its mirror image

The law of large numbers says that the average converges. It does not say that deviations are repaid. A run of one colour on a wheel does not make the other colour more likely on the next spin; the wheel has no memory and no obligation. What actually happens is that the early deviation is diluted by the volume of later results rather than reversed by them.

The mirror error is equally common: treating a run of wins as evidence that a mechanism is favourable. Given enough sessions, streaks of every length occur exactly as often as chance requires. Distinguishing a genuinely favourable mechanism from a lucky sequence needs far more data than a person accumulates by playing, which is why claims of that kind are settled by testing the mechanism rather than by watching results.

High and low variance games

Two games with the same edge can have very different variance. A wager that pays even money resolves close to its expectation quickly. A wager that pays several hundred to one produces long droughts punctuated by occasional large wins, and can run far from its expectation for a long time in either direction.

Designers exploit this deliberately. High variance sustains the possibility of a large result, which is the main thing a low-frequency, high-prize game is selling; low variance sustains a longer session at a given stake. The design article covers how paytables are tuned to sit at a chosen point on that scale while leaving the edge untouched.