To Replace or Not to Replace — That is the Question
The College Mathematics Journal
In this paper, the authors consider the following urn model. An urn contains R red balls and W white balls. n balls are drawn successively at random according to the following sampling rule: if a red ball is selected on a single trial, then the ball is put back into the urn; if a white ball is selected, then it is removed from the urn. Let X denote the number of white balls in the whole sample of n balls. The authors derive an expression for the probability mass function of X and compare their result with those for the classical models of samplings with and without replacement (leading to the binomial and hypergeometric distributions, respectively).
Engbers, John and Hammett, Adam J., "To Replace or Not to Replace — That is the Question" (2020). Science and Mathematics Faculty Publications. 407.