## MatheMUSEments

Articles for kids about math in everyday life, written by Ivars Peterson for Muse magazine.

## May 9, 2007

### What a Coincidence!

What do you think the chances are that two or more kids in your class have the same birthday? A year has 365 days, so to have a 50-50 chance that two kids have the same birthday, you'd need to have 180 kids. Right?

Wrong. It turns out that you'd need far fewer.

Suppose there are only two kids in the class. Allen's birthday is July 4. One day is used up, so if Brandi's birthday is different, it can fall on any of 364 days. That means the probability that Allen and Brandi have different birthdays is 364/365. Now Carol enters the room. There are only 363 unused birthdays, so the probability that her birthday is different from the other two is 363/365. The probability that all three birthdays are different is 364/365 times 363/365.

For four children, you mulitply 364/365 times 363/365 times 362/365. Amazingly, by the time you have 23 kids in the room, the chance that two of them have the same birthday is higher than 50-50. Or to put it another way, if you checked all classrooms with 23 kids, about half of the classes would have a duplicate birthday.

Try it out on your class. If there are more than 23 kids, the chances are even better that you'll find matching birthdays. In fact, because birthdays aren't really spread evenly throughout the year, you're likely to find a duplicate birthday in about half of all classes with as few as 20 children.

If this seems surprising, consider this twist: The chance that two people have the same birthday is pretty high, but the chance that someone else has your birthday is pretty low. Those probabilities are trickier than they seem.

Muse, March 2002, p. 43.