Use of the normal approximation to the binomial distribution

When the sample size, n , is large, the probability of the number of successes, X, being within an interval may involve addition of many individual small binomial probabilities.

P( aXb )   =   P( X = a )  +  P( X = a + 1 )  +  ...  +  P( X = b )

This sum can be difficult to evaluate by hand and rounding errors can lead to inaccuracies. Even on a computer, such summations are unnecessarily difficult.

An alternative is to use a normal approximation. Its accuracy depends on the value of n being large enough. A common rule-of-thumb for using a normal approximation is when

nπ > 5    and     n(1-π) > 5

The examples below use a normal approximation to evaluate binomial probabilities.