sacstat mark: a grid of pockets with a single one filledsacstat edge x handle

sacstat › Long-run behaviour › the long run

the long run

edge x handle

The claim that the house wins over time is not a proverb. It is a theorem about averages applied to a fixed negative expectation, and it is worth stating precisely, because the precise version explains several things the loose version does not.

The law of large numbers

Given repeated independent trials with a fixed expectation, the average result converges to that expectation as the number of trials grows. Nothing forces any particular trial; the convergence comes from later results diluting the influence of earlier ones. This is why the law is compatible with any finite sequence of results whatsoever.

For a commercial game the expectation is negative for the bettor, so the average result converges to a loss. The operator experiences the same theorem from the other side and with far more trials, which is why an operator's revenue is predictable while an individual's results are not.

The margin applies to handle

This is the point at which intuition most often fails. A house edge is a percentage of the amount staked, not of the amount brought. Money that wins a round is staked again, so a given sum is exposed to the edge repeatedly, and the expected cost is the edge multiplied by the total staked — the handle — rather than by the initial bankroll.

It follows that time at the game is the real variable. Doubling the number of rounds doubles the expected cost while increasing the spread only by about forty percent, which is the same asymmetry as in the variance entry seen from the player's side.

100 units, staked repeatedly
bankroll            100 units
stake per round       5 units
house edge            2.70%

300 rounds:  handle = 1,500 units
             expected cost = 0.027 x 1,500 = 40.5 units

1,000 rounds: handle = 5,000 units
             expected cost = 0.027 x 5,000 = 135 units

the bankroll never changed; the exposure is set by turnover

Ruin

A finite bankroll facing an unlimited sequence of negatively-expected rounds has an absorbing state: zero. Once reached, play stops, and no subsequent favourable variance can undo it. The asymmetry between a bankroll that can be exhausted and an operator's float that effectively cannot is as important to the outcome as the edge itself.

This is also why staking systems that increase after losses fail in a specific and predictable way. They convert a moderate chance of a small loss into a high chance of a small win and a small chance of a very large one, and the large one arrives at the point where the bankroll or the table limit stops the sequence.

What the theorem does not say

Commonly misread. The law of large numbers does not say a losing player is due a correction, that a winning session will be given back, or that any individual will experience the average. It is a statement about the limit of a sequence, and every finite outcome remains possible on the way there. What it does say is that the aggregate is settled: over enough rounds, across enough players, the arithmetic that was fixed before play began is what shows up in the accounts.

Return to the Long-run behaviour index on the reference landing.

‹ variance and streaks history of games of chance ›

Last modified 17 August 2026.