Non-Recursive Computation of the Probability of More than Two People Having the Same Birthday

We address a well known problem of computer science, the problem of computing the probability that a given number of people m > 1 have the same birthday from among the members of a larger set of cardinality n >= m. The solution to this problem for m = 2 is well known and is usually referred to as the 'birthday surprise probability'. A solution for m = 3 is also known and appears in the 2004 paper by DasGupta [The matching, birthday and the strong birthday problem: a contemporary review, Journal of Statistical Planning and Inference]. Further approximations to the solution of the related problem of computing the minimum number of people to interview until m people with the same birthday are found are presented in the seminal work by Klamkin and Newman [Extensions on the birthday surprise, Journal of Combinatorial Theory, 1967]. In this paper we present a new non-recursive approximation for the birthday probability applicable to any value of m > 1, which yields results that are experimentally proven accurate under the assumption that the number of birthdays is significantly larger than the number of people. Our expression is easy to compute, non-recursive, and applicable to values of m that can be arbitrarily larger than 2 or 3. We verify the validity of our result computing the birthday probability for different values of m, over billions of sets of random values generated using the Intel (R) RDRAND hardware random number generation instruction. Our solution is based on a novel tree-based description of the event space which, if used, allows for the computation of the birthday probability efficiently and without involving recursions or multinomial distributions.