Is it necessary to have infinite(or near infinite) population size for a binomial distribution?

Question

I was reading about single sampling plan in the book "Introduction to statistical quality control", by DOUGLAS C. MONTGOMERY.

The author has mentioned that under the assumption that lot size is infinite(or near infinite), the distribution of number of defects can be assumed to be a binomial distribution.

Suppose that the lot size N is large (theoretically infinite). Under this condition,the distribution of the number of defectives d in a random sample of n items is binomial with parameters n and p, where p is the fraction of defective items in the lot

But is this assumption necessary?

I tried to look on web whether infinite lot size is a necessary condition for establishing a binomial distribution. I couldn't find anything conclusive.

I am fairly new to quality management and statistics in general.

Can anyone please help?

Thank you.

Answer 1

If you sample with replacement, the resulting distribution will be binomial. If you sample without replacement, the distribution is hypergeometric. In particular, the variance of the hypergeometric distribution is less than the variance of the corresponding binomial (except in trivial cases).