Why the long run favours the house

The law of large numbers says that as the number of independent trials grows, the observed average result converges towards the expected value. It is a theorem, not a tendency, and it is the mathematical foundation of every commercial game of chance. Applied to a game with a fixed negative expectation per unit staked, it says that the observed return per unit staked converges towards that negative figure, and that the convergence is one-directional because the expected value does not move.

The theorem is often misread as a promise about individual sequences. It promises nothing about any particular run. What it establishes is that the proportion of runs that stay far from the expected value shrinks as runs get longer, and shrinks without limit. There is no length at which the drift reverses, because nothing in the mechanism ever changes.

This is the decisive asymmetry, and it can be stated in one line. Over n independent unit stakes at a fixed edge e, the expected loss grows as n times e, while the standard deviation of the result grows as the square root of n times the per-play standard deviation. One term is linear in n and the other is proportional to its square root, so however small e is and however large the scatter is, there is a value of n beyond which the linear term dominates.

Everything that people describe as luck lives inside the square-root term. It is real, it is often large, and over a hundred plays it can easily swamp the drift. Over a hundred thousand plays it cannot. This is why the same mathematics that makes an individual session genuinely unpredictable makes an operator's aggregate result highly predictable, and why the two facts are not in tension: they are the same equation read at two different values of n.

An operator does not need to win any particular decision. It needs a large number of decisions at a positive expectation for itself. This is the same structure that underlies insurance underwriting, and it works for the same reason: many independent small exposures at a known margin aggregate into a narrow, reliable distribution. The margin per decision can be tiny provided the count of decisions is large.

It follows that the design levers that matter most are not the size of the edge but the rate of decisions and the fraction of returned money that is staked again. A faster game with a smaller edge can be worth considerably more per hour than a slower game with a larger one. Anyone reading a game's design should therefore look at its speed and its recycling behaviour alongside its stated margin, because those three quantities multiply together.

Put together, the arithmetic supports a single conclusion that this board states plainly rather than hedging. In a game where the paytable pays less than the true odds, the expected result of continued play is a loss proportional to the total amount staked, and no pattern of stake sizing, timing, selection or persistence changes that proportion. The mechanism has no memory to exploit and the margin is applied to every decision equally.

This is not a moral claim and it is not a warning; it is the arithmetic the rest of this board sets out. Games of chance are worth understanding as designed objects, as a mathematical subject with an unusually clean structure, and as a long and genuinely interesting strand of economic and legal history. They are not a mechanism for producing money, and the reason is a theorem rather than an opinion.