Problems with using a binomial distribution when n is large

Although the number of 'successes' in a random sample always has a binomial distribution, it is computationally difficult to obtain probabilities from a binomial distribution when n is large. In a large random sample of say n = 10,000 categorical values, probabilities of interest usually involve summing the probabilities for a large number of individual values for the number of successes.

P(X < 5,600)  =  P(X = 0) + P(X = 1) + ... + P(X = 5,599)

We next describe a way to approximate such probabilities without summing so many values.

Proportions and means

Binomial mean and standard deviation