the Poisson limit theorem (law of rare events)
/ pwah-SON /
Consider something that has many, many chances to happen but only a tiny probability on each chance — a typo on any one character of a long book, a radioactive atom decaying in any one second, a customer arriving in any one millisecond. The total count of such rare-but-numerous events follows a beautifully simple pattern. The Poisson limit theorem says that count is approximately Poisson, governed by a single number: the average number of events expected.
Precisely: take a binomial count of n independent trials each succeeding with probability p, and let n grow large while p shrinks so that the mean n*p stays fixed at lambda. Then the binomial probabilities converge to Poisson probabilities: P(X = k) approaches e^(-lambda) * lambda^k / k! for each k. The mechanism is a limit you can almost see in the formula — the binomial coefficient times p^k times (1-p)^(n-k), with n large and p = lambda/n, collapses into the Poisson form, because (1 - lambda/n)^n approaches e^(-lambda). Many independent rare chances, summed, forget their binomial details and remember only their total rate lambda.
This is why the Poisson distribution is the universal model for counts of rare independent events: calls to a switchboard, mutations in a genome, goals in a match, particles hitting a detector. The honesty to keep: the approximation needs the events to be roughly INDEPENDENT and individually RARE (small p) while numerous (large n). If events cluster, or any single one is not rare, the Poisson shape can be wrong — and unlike the CLT, this is a limit of a DISCRETE count toward another discrete law, not a march toward the bell.
A book has n = 100000 characters, each mistyped independently with probability p = 0.00003, so lambda = n*p = 3. The number of typos is binomial, but with n huge and p tiny it is essentially Poisson(3): P(0 typos) ≈ e^(-3) ≈ 0.050, P(exactly 3) ≈ e^(-3) * 3^3/6 ≈ 0.224 — computed from lambda = 3 alone, never needing n or p separately.
Many rare independent chances aggregate into a Poisson count governed only by the mean lambda = n*p.
The approximation needs events roughly independent and individually rare while numerous. If events cluster or any single one is not rare, the Poisson shape can fail.