The birthdays trick

In a group of 48 people, there is a 95% chance that two will have the same birthday (eg two born on Jan 1).  If you don't believe me, you might like this simple demo.  Have a look at the following explanation, then click the link at the bottom of the page to start the demo.  The particular proof that follows is the one I found; there are other equally accurate ways of explaning it.

The demo's display is a bit like a calendar.  On the right hand side is the Pick a Day button.  When you click this button the computer will pick a random date (Feb 29 is not included).  There is absolutely no trick here - the date is simply generated using a standard pseudo-random function.  When the computer detects that it has picked a date for the second time, a message appears.  Click Reset to have another go.

The key to the counter-intuitive probabilities is shown on the right hand side of the calendar.  It has to do with the pairing of the days.  If you have n days, then you have n(n-1)/2 unique pairs of days.  For example, suppose you have four days: March 28, April 3, May 22 and June 7.  There are 4(4-1)/2 = 6 pairs:

• March 28 and April 3
• March 28 and May 22
• March 28 and June 7
• April 3 and May 22
• April 3 and June 7
• May 22 and June 7

So the number of pairs is proportional to the square of n.  And every one of those pairs is independent of all the others: in general, just because you know that A is not equal to B and B is not equal to C, this does not tell you whether A is equal to C or not.  The probability of a single pair of days being different is of course very high - 364 out of 365.  But the probability that none of the pairs of dates match is 364/365 to the power of the number of pairs.  When you multiply by a number less than one, you get a smaller number.  So when you multiply 364/365 by itself repeatedly the result will shrink towards zero.  There are so many pairs that the probability shrinks surprisingly quickly.  Conversely, the probability that at least one of the date pairs matches will rise quickly (it is simply one minus the shrinking number).  With 48 days we have (364/365)^48(48-1)/2, which is approximately 0.05, leaving the 0.95 (ie 95%) probability of a matching pair.

I guarantee¹ you will not have to pick too many days before the computer announces that it has picked the same day twice.  The highest I have gone without getting a match is 65 days.

Click here to start the demo. 

¹ _{Not a guarantee.}