Birth, Death, Coincidences and Occupancies: Solutions and Applications of Generalized Birthday and Occupancy Problems

Abstract

We present exact solutions to the birthday and generalized occupancy problems using multiple approaches. We start with two alternative ways to solve the classical birthday problem for uniform distributions by iterating over people and iterating over days, respectively. The iterate-over-day method is used to find the probabilities of not only the birthday problem but also several variants, including the exact number of people sharing a birthday and the strong birthday problem, for non-uniform distributions. This method decomposes multinomial based problems as a recursive sum of binomial distributions and leads readily to a unifying solution for generalized occupancy and coincidence problems for any probability distribution and arbitrary sets of occupancies. The birthday, strong birthday and exact number of coincidences with arbitrary probability distribution and varying number of coincidences/occupancies can all be found from the generalized solution. The generalized solution also allows us to study the details of specific coincidences. We define a z-value, which is similar to the commonly used p-value but is calculated from the generalized occupancy model, to test the conformity of two distributions and show that z-value is more effective than p-value in hypothesis testing when one category (out of five) is an outlier. Several examples are used to illustrate the practical applications of the generalized occupancy model in real-world situations.