Counting Rare Events

How likely are exactly 6 support tickets in an hour when the average is 4? The Poisson distribution answers questions like this with one parameter and one formula — the workhorse model for counts of independent, rare events over time or space.

Exact, Cumulative, and Beyond

P(X = k) gives the exact probability; P(X ≤ k) sums everything up to k; P(X ≥ k) covers the tail that matters for planning — "what's the chance we see at least 6 failures?" The chart shows the whole mass function so you can see where the distribution sits and how it spreads.

Know Its Limits

The Poisson assumes a constant rate and independent events. Customer arrivals at a lunch rush, network packets, and radioactive decay fit beautifully; clustered or trending phenomena do not — if your variance looks far larger than the mean, reach for a different model.