## Poisson Approximation to the Binomial

When the value of n in a binomial distribution is large and the value of p is very small, the binomial distribution can be approximated by a Poisson distribution. If n > 20 and np < 5 OR nq < 5 then the Poisson is a good approximation.

The Binomial distribution tables given with most examinations only have n values up to 10 and values of p from 0 to 0.5

The similarities between the two distributions can be seen in the vertical line graph below. The black graph is a binomial distibution with n = 10 and p = 0.2 The red graph is a Poisson distribution with λ = 2

The value of the mean needed for the Poisson approximation is λ = np

To summarise:

 For large values of n and small values of p, the Poisson distribution approximates the binomial distribution Test n > 20, np < 5 OR nq < 5 New parameters λ = np

Example

A factory puts biscuits into boxes of 100. The probability that a biscuit is broken is 0.03. Find the probability that a box contains 2 broken biscuits

Solution

This is a binomial distribution with n = 100 and p = 0.03.

These values are outside the range of the tables and involve lengthy calculations.

Using the Poisson approximation (test: np = 100 x 0.3 = 3, which is less than 5)

Let X be the random variable of the number of broken biscuits

The mean λ = np = 100 × 0.3 = 3

P(X = 2) = 0.224 (from tables)

The probability that a box contains two broken biscuits is 0.224.