Birthday Party Problem

Here is another popular statistical/mathematical brain teaser.

Imagine that you come to a house party with 23 people. What, in your opinion, is the probability that two of the people have their birthday on the exact same day (same day and month, not year)?

The answer is: just over 50%.

Not surprisingly, not many people give this answer when they are presented with the riddle. Most common response is 5-10%!

What is the reason for the probability of this event being so high?

This is the proper calculation:

23 people make 253 possible pairs of people;

23*22/2=253

The probability of finding a match at the party is

.69; 253/365days in a year=.69

Now, this is a simple version of solving the task. As you can see, it gives us 70% rather than 50%. To be fair, we originally asked about TWO PEOPLE ONLY having the same birthday on the exact same day. The probability of 70% includes cases when more than two people have birthday on the same day.

As a condition to get 50% we have to use the following formula:

P(A)= 1 - 364*363*362...(365-n+1) / 365^n

If we plot our 23 people into this formula we will get 50% (you have to believe me :-) ). Voila!

Rosenthal, J. S. (2005). Struck by the Lightning. London: Granta Books