The Birthday Surprise: Why 23 Is Enough

Put 23 people in a room. The chance that at least two share a birthday is a little over 50 percent.

Most people guess you would need about 180. The mistake is imagining your own birthday being matched. The question is about any pair at all, and 23 people make 253 different pairs.

The calculation

It is easier to work out the chance that nobody matches. The second person avoids the first with probability 364/365. The third avoids both with 363/365, and so on down to 343/365 for the twenty-third person.

Multiply all of those together and you get about 0.493. So the chance of at least one match is about 0.507, or 50.7 percent. We ignore leap days and assume birthdays are evenly spread, which is close enough.

It climbs quickly

• 23 people: about 51 percent
• 70 people: about 99.9 percent
• 366 people: guaranteed, since there are only 366 possible days

The lesson

When something seems like an unbelievable coincidence, ask how many chances it had to happen. Large numbers of possible pairs turn surprising events into ordinary ones. That covers a lot of spooky stories about running into someone you know on vacation.