Find the least positive integer 'N' so that, in a room containing 'N' people, the odds that two of the people have the same birthday (not the year, just the day) are more than 50%.

[Note: Forget about Leap Year. Assume that a year has 365 days and that people are born randomly throughout the year].

The math works out easier when considering N people and the odds that NO two people have the same birthday. For 2 people, the odds are 364/365. For three people, the 2-person scenario must be true first, and for the third person not to share a b-day with either, the odds are 363/365. Using this logic, the odds of no two people sharing a birthday is:

2 people = 364/365
3 people = 364/365 * 363/365
4 people = 364/365 * 363/365 * 362/365
etc.

If you keep multiplying these fractions, you will find that 22 people have a 52% chance of no common birthdays. 23 people have a 0.49% chance of no common birthdays or, inversely, a 51% chance that at least two people share the same birthday. The answer is N = 23 people.

Posted by Bryan on 2003-03-27 08:40:02