The Birthday Problem

 

Not a Coincidence!                      

By Larry Tesler                      

May 25, 1998                      

In "What a coincidence!" (Mathematical Recreations, Scientific American, June 1998, p. 95), columnist Ian Stewart discussed the well-known coincident birthday problem.                       

If there are at least 23 people at a party, more likely than not, two share the same birthday. Citing an article by Robert Matthews, Stewart derives this result by iteratively multiplying  
until the product is less than p= .5; the number of factors required is k= 23. To convey an intuition about why so few people are needed, he points out that there are a lot of ways to pair 23 people, specifically, 
.   
Stewart similarly demonstrates that if you are in a room with at least 253 other people, more likely than not, one of them shares your birthday. He obtains this result by multiplying  
until the product is less than p= .5; the number of factors required is b= 253.    

The author cautions:                      

“Incidentally, the fact that the answer to the second problem is the same as the number of pairings in the first problem (253 pairings for 23 people) seems not to have any mathematical significance. It seems to be a coincidence.”

I will show that is not much of a coincidence.

  1. A Generalization

  2. The Proof