the Poisson law of rare events

The law of rare events is the idea that gives the Poisson distribution its meaning and its name. Imagine a huge number of independent opportunities for something to happen, where each individual opportunity almost never succeeds. There are millions of drivers, but any one of them crashes on a given day with tiny probability. There are billions of base pairs, but any one mutates rarely. Add up these many tiny chances and the total count of successes — accidents today, mutations in this gene — follows a Poisson distribution. Rare events, aggregated over many chances, become Poisson.

The precise content is a limit. Take a binomial with n trials and per-trial success probability p, and let n grow very large while p shrinks very small, keeping the expected number of successes np = lambda fixed. In that limit the binomial pmf converges to the Poisson pmf e^(-lambda) lambda^k / k!. The intuition: with so many trials at such low odds, the count no longer remembers either n or p individually — only their product, the average rate lambda, survives. This is why a Poisson needs just one parameter where the binomial needed two.

The honest caveat lives in the word 'independent and at a steady rate'. The classic illustration, the number of Prussian soldiers killed by horse kicks per year, fits beautifully because kicks are independent and rare. But the law fails when rare events cluster — earthquakes trigger aftershocks, a viral post triggers more shares — because then one event changes the chance of the next, violating independence. In such cases the counts are overdispersed and the Poisson underestimates the spread.