The Law of Large Numbers, From Telemetry to the House Edge

Averages become reliable when you have enough independent samples. That single idea explains why engineers repeat measurements and why casinos can plan their income.

The law itself

The law of large numbers states that the average of the results from a large number of independent random samples converges to the true value, if it exists. Toss a fair coin ten times and you might get seven heads; toss it ten thousand times and the proportion of heads will almost certainly be very close to one half.

Note what the law does not say. It does not say that a run of tails will be "corrected" by extra heads. The coin has no memory. Early deviations are not cancelled; they are diluted by the sheer number of later tosses. Believing otherwise is the gambler's fallacy, covered in the note on randomness.

Why engineers repeat measurements

Every sensor reading on a spacecraft includes some noise. Averaging many readings reduces the effect of random noise, because the errors tend to cancel out in the mean. The same principle underlies testing campaigns: one successful test says little; many successful tests under varied conditions say much more. It also underlies Monte Carlo simulation, whose estimates become steadier as the number of simulated runs grows.

Small samples mislead

The flip side is that small samples are unreliable. A new launch vehicle with a short flight history can show a perfect record, or a poor one, largely by chance. Estimates built on few observations should be reported with wide uncertainty, and should change as data accumulates. Anyone presenting a percentage from a handful of events is presenting noise as if it were signal.

The house edge

Casinos are a textbook application of the law of large numbers. The house edge is the built-in average advantage a casino game gives the operator: on average, over many bets, the player loses a fraction of what is staked. Any single session can go either way. But across a very large number of bets the average result for the house converges towards its edge, which is why the operator can plan on it and the player cannot plan against it.

E(loss) = 0.02 × 1000 = 20 units

Promotions and the long run

Gaming sites advertise promotions to attract players, and they are usually described with conditions on how much must be played before any bonus can be withdrawn. Because each bet carries the house edge, those conditions matter more than the headline amount. For readers who want to see how such offers are presented, one Traditional Chinese page from a gaming brand is an overview of its current promotions (夜色娛樂城優惠). Read it with the law of large numbers in mind: the more play a condition requires, the closer your result is likely to come to the house's average.

The takeaway

Large numbers make averages dependable. That is good news for engineers who can repeat tests and simulations, and for anyone estimating a risk from plenty of data. It is not good news for anyone hoping that, with enough tries, a game with an edge against them will turn in their favour. Back to the start: what an orbit really is.

REV 2026-11-09 / Tobias Wren