In a village of 2000 people, what is the chance of finding a day in the year which is not the birthday of someone? How do you solve this problem? The solution to this problem makes use of the Poisson approximation for r entities trying to occupy n empty cells.
When r and n are high so that lambda = n exp (-r/n) is bounded, the probability of m empty cells becomes the Poisson distribution function, p(m; lambda).
In or case, lambda = 365 x exp(-2000/365) = 1.52. The required probability may be obtained by subtracting the chance of seeing no empty day from one.
1 – dpois(0,1.52) = 0.78 or 78%
