the binomial-to-Poisson limit

This is the precise mathematical statement that ties the binomial and the Poisson together — the rigorous backbone under both the law of rare events and the Poisson approximation. It says: take a sequence of binomial distributions where the number of trials n marches off to infinity while the success probability p shrinks toward zero, arranged so that the product np stays pinned at a fixed value lambda. Then the binomial pmf converges, term by term, to the Poisson pmf with parameter lambda.

You can watch it happen with the algebra. Write the binomial probability as C(n, k) p^k (1 - p)^(n-k) and substitute p = lambda/n. The combinatorial factor C(n, k) divided by n^k tends to 1/k! as n grows (the product n(n-1)...(n-k+1) is about n^k). The term lambda^k stays put. And the survival factor (1 - lambda/n)^(n-k) tends to e^(-lambda), using the classic limit that (1 - x/n)^n approaches e^(-x). Multiply the three limiting pieces and you land on exactly e^(-lambda) lambda^k / k!, the Poisson pmf. The two parameters of the binomial have fused into one: only their product lambda survives.

Why does this deserve its own name beyond the approximation? Because as a limit theorem it is the discrete-event analogue of the central limit theorem, and it explains a structural fact about the world: whenever a count arises from very many nearly-independent chances each individually negligible, the Poisson appears, no matter the fine details. It is a small instance of universality. The honest boundary is the same as always — the trials must stay independent with a fixed total rate; clustered or contagious events fall outside the theorem's reach.