When a Poisson distribution approximates the binomial

Conditions. A binomial distribution B(n, p) can be replaced by a Poisson distribution with mean m = np when:

1. the number of trials n is large (in theory, n → ∞);
2. the probability of success on each trial p is small (p → 0), so the event is rare; and
3. the product np = m stays finite and moderate, being the average number of successes.

The trials must still be independent of one another, with the same p on each, just as in the binomial.

Why the approximation works. With p small, q = 1 − p is close to 1, so the binomial variance npq is almost equal to its mean np. The Poisson distribution has mean and variance both equal to m, so the two distributions then share their centre and spread, and the much simpler Poisson formula P(x) = e⁻ᵐ mˣ / x! can be used.

Illustration. Our own example: suppose 0.25% of the UPI payments made through a bank's app fail. In a batch of 800 payments, n = 800 and p = 0.0025, so the number of failed payments in the batch can be taken as Poisson with m = 800 × 0.0025 = 2, which spares us binomial powers such as (0.9975)⁷⁹⁸.