the Poisson approximation to the binomial

This is the practical, computational side of the law of rare events. Suppose you face a binomial with a large n and a small p — say, the number of defective chips in a batch of 5000 when each is defective with probability 0.001. Computing C(5000, k) (0.001)^k (0.999)^(5000-k) directly means wrestling with astronomically large binomial coefficients and astronomically small powers. The Poisson approximation lets you sidestep all of that.

The recipe is one line: set lambda = np, then approximate P(X = k) by the Poisson value e^(-lambda) lambda^k / k!. In the example, lambda = 5000 times 0.001 = 5, so the chance of exactly 3 defectives is roughly e^(-5) 5^3 / 3! = e^(-5) times 125 / 6 which is about 0.140 — a calculation you can do on a basic calculator, with no giant factorials in sight. The approximation is good precisely when n is large and p is small, which is exactly when the exact binomial is most painful to compute, so the trade is all upside.

How good is it? The error is controlled by p itself: roughly, the total discrepancy across all k is bounded by something on the order of p (more precisely n p^2, or p per event). So the smaller p is, the better the fit, regardless of how large lambda is. A common working guideline is n at least 20 with p at most 0.05, or n at least 100 with np at most 10. Note this is a different limit from the Normal approximation: the Normal approximation wants np large and the distribution symmetric, whereas the Poisson approximation wants p small and embraces the skew. Use the Poisson when successes are rare, the Normal when they are common.