September 24, 2009 2 Comments
In a group of 23 people, the probability that two or more people will share their birthdays is greater than 50%. For a group of 57 people, this probability increases to 99%.
This may seem counter-intuitive at first sight. There are 365 possible days on which a birthday can occur. So it may seem odd that in a group of 57, there is 99% probability of a shared birthday.
Question: In a group of ‘n’ people, what is the probability that any of those ‘n’ persons shares a birthday with any of the others in the group?
This is very simple.
The answer is simply:
1 – (364/365) (363/365) … ((365 – n + 1 )/365 )
= 365 ! / [ (365^n) (365 – n)! ].
Approximating this function, and plugging in values of n, we get the probability of a group of those many people having a common birthday in their midst.
See http://en.wikipedia.org/wiki/Birthday_paradox for more on this.