and this doesn't change. When we were drawing without replacement, the proportions of successes would change, depending on the result of previous draws. For example, if we were to obtain a ``success'' on the first draw, then the proportion of ``successes'' for the second draw would be (M-1)/(N-1), whereas if we were to obtain a ``failure'' on the first draw the proportion of successes for the second draw would be M/(N-1).
The random variable Y is defined as the number of ``successes'' in the sample, when we are drawing with replacement. Then Y is a binomial random variable:
The probability mass function for Y is
If the population size in
such a way that the proportion of successes ,and n is held constant,
then the hypergeometric probability mass function approaches the binomial
probability mass function:
In practice, this means that we can approximate the hypergeometric probabilities with binomial probabilities, provided .As a rule of thumb, if the population size is more than 20 times the sample size (N > 20 n), then we may use binomial probabilities in place of hypergeometric probabilities.
We next illustrate this approximation in some examples.