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( a ≤ X ≤ b ) = 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.