The Birthday Problem

A well-known problem asks for the smallest number of people (N) who must be in a room before it is more likely than not that two share the same birthday. The answer, surprising to most people who have not heard the problem before, is N = 23.

I thought it would be interesting to modify the problem where we ask for people who share that same day of the month for their birthday. While the answer is not as surprising as the original problem, the computation is much easier. Direct computation for the first problem using factorials will result in overflow on scientific calculators such as the TI-83. Also, the answer to the day-of-month problem (N = 7) is more suitable for empirical testing in small classes. Simply ask each student for their birth day (1 - 31) and record on a large month calendar. For N = 11 the probability of a match increases to almost 88%.

The formula for the probability of one or more matches amongst a group of N people is
Prob = 1 - (31)(30)...(32 - N)/31^N 
= 1 - 31! /[(31 - N)! * 31^N]