Sample Binomial Distribution

Below is a single random sample, n = 30, of a binomial distribution.

Below are 100 random samples, n = 30, from the same sequence of numbers (X). You can see that the distribution begins to approach normal

Below are 100 random samples of the distribution’s minimum value, n = 30, from the same sequence of numbers (X). You can see that the distribution does not approach normal. In fact sample minimum distribution’s do not approach normal (see source below)

Source: Simulation of Binomial Random Variables

Source: The Distribution of Sample Minimums as a Function of Sample Size: Not All Sample Statistics Approximate a Normal Distribution as Sample Size Increases