6.1.4—Binomial distribution

Bin(n,p) has a fixed number n of independent trials with success probability p; Po(λ) models a count in an interval with a rate. They answer different sampling questions.

Use binomial when trials are explicit and bounded; use Poisson when exposure and event rate are natural. A Poisson approximation to binomial needs large n, small p and λ=np.

Ten quality checks with defect probability 0.02 are binomial; defects over a long production interval may be modelled Poisson with the measured rate.

A small p alone does not justify Poisson approximation; n and the product np also matter.