A generalized version of the birthday problem is as follows. Suppose each member of a population independently receives a number randomly selected from \ 1,2,3, ⋯ ,x \, and a random sample of size n is to be taken. If $0 < p < 1$, what is the smallest value of n so that the probability that at least two of the sample have the same number is at least p? Both empirical modeling and approximation techniques are used to determine n as a function of x when p is fixed. An error analysis of the approximation is presented.
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Frank H. Mathis (1991) studied this question.