Variance, the long run and the law of large numbers

Two numbers, not one

Expected value is an average, and an average on its own is a poor description of a gambling result. Two games can share an identical edge and feel nothing alike, because they differ in how widely their outcomes scatter. That scatter is measured by variance, and by its square root, the standard deviation, which has the convenience of being expressed in the same units as the wager itself.

Compare two wagers on the thirty-seven pocket wheel. Both carry an edge of 2.70 per cent. The even-money wager wins a little under half the time and moves the balance by one unit each spin. The single-number wager wins about once in thirty-seven spins and moves the balance by thirty-five units when it does. Over a hundred spins the first produces a familiar, slowly drifting line; the second produces a long decline punctuated by occasional jumps, and whether the hundred spins end up or down depends almost entirely on how many jumps happened to fall inside them.

What the law of large numbers actually says

The law of large numbers is the reason the arithmetic is dependable, and it is routinely misquoted. It does not say that results even out. It says that the average result per wager converges on the expected value as the number of wagers grows. The total distance from expectation is free to keep growing; it simply grows more slowly than the number of wagers, so the ratio between them shrinks toward zero.

The arithmetic behind that is worth stating plainly. For independent wagers, total variance adds, so the standard deviation of a total across n wagers grows in proportion to the square root of n. The expected loss, meanwhile, grows in proportion to n itself. Multiply the number of wagers by a hundred and the expected loss multiplies by a hundred while the typical scatter multiplies by ten. The edge outruns the noise, and it does so at a predictable rate.

Read the last column downward and the whole commercial structure of the industry appears. At a hundred wagers the scatter dominates the edge by several times over, so an individual session is genuinely unpredictable and a winning one is unremarkable. At a million wagers the scatter is a few per cent of the expected loss, so the outcome is effectively determined. The player occupies the top row and the operator occupies the bottom one.

Why a win proves nothing

This asymmetry explains why individual results carry almost no information. A player who finishes a session ahead has observed one sample from a distribution centred slightly below zero and spread widely around it, and a single sample from such a distribution lands above zero quite often. The result is entirely consistent with the edge; it is what an edge of a few per cent looks like at small sample sizes.

The same reasoning applies to any run of results offered as evidence that a method works. Testing a staking method against a few hundred wagers cannot distinguish a genuine effect of a few per cent from ordinary scatter, because the scatter at that sample size is several times larger than the effect being looked for. The number of wagers needed to resolve a difference of that size runs into the tens of thousands, and by that point the expected loss has accumulated in full.

Risk of ruin

Variance has one further consequence that pure expectation conceals. A player has a finite bankroll and cannot continue below zero, so a run of bad results can end the sequence permanently. This is called risk of ruin, and it rises as the stake grows relative to the bankroll and as the variance of the chosen wager grows.

It also runs in one direction only, which is the point. The operator faces the same variance but with a far larger reserve relative to any single wager and a far larger number of wagers, so its position sits near the bottom row of the table above while every individual player sits near the top. Two parties can face the same arithmetic and be in completely different situations purely because of scale.