The Matching Birthday Question

I believe I've written about this before on this blog, but I couldn't find that post so if this is a repeat, I apologize.  The other day in astronomy class we talked about statistics for a bit and I presented them with the matching birthday question.  What are the odds that any two people in the room have the same birthday?  Consider two students in the room, Mary and Bob.  What are the odds that Bob has the same birthday as Mary?  There are 365 days in the year so the odds Bob has the same birthday as Mary are 1:365.  Pretty simple.

What are the odds Mary and Bob have the same birthday and their birthday is on March 22?  Those odds are lower and are given by:

(1/365)(1/365) = 0.00075%

That's pretty low.  Not impossible, but very unlikely.  But the original question asked is what are the odds that ANY two people in the room have the same birthday.  In this particular class there are 11 students and myself, so a total of 12 people.  Students usually argue the odds are very low, but are they really very low?  Before we calculate the odds we go around the room as a test and see.  In this class, person 3 and person 11 had the same birthday!  Upon learning this there were several shocked faces in the room, but should there be shocked faces?  Let's see what the math tells us.  

What makes the odds of this happening higher is the fact that we are not selecting two specific students, but only asking about ANY two students.  Therefore you have to look at each possible pairing in the room.

(364/365)^(12*11/2) = 83.5%

The 12*11/2 looks at the number of possible pairings in the room and the 83.5% is the chance that there are NO matching birthdays.  To find the number of matching birthdays, just subtract from 100% to get 16.5%.  Therefore there was a 16.5% chance two people in our room had matching birthdays.  Low, but far, far from impossible.  Things with a 16.5% chance of happening happen every day!  Increase the number of people in the room and the odds of a matching birthday increase.  It only takes 23 people to give a 50% chance of any two people having a matching birthday.  

I love this problem because at first glance it goes against common sense.  The number 23 is far less than 365, so it seems impossible that there is a 50% chance of two people having a matching birthday.  But the math tells us otherwise!