The Stats Behind a Birthday Match

As part of an icebreaking event at the latest meeting of my local social media enthusiasts group Brrism, the 30 or so people on hand were asked to compare birthdays. It turned out that three pairs of people each shared a birthday.

As unlikely as that may sound, nobody is calling Ripley’s Believe It Or Not? to report an incredible freaky event.

Why not? Because it’s an example of the birthday paradox, a classic example of how probability theory can be difficult to understand. And while three matches is certainly unusual, having one match was more likely than not to happen.

Many people asked the question “What are the chances of two people in this group sharing a birthday?” assume it is very remote. That’s because they use a false premise to solve the question: they take the idea of one particular birthday and assess the likelihood of somebody else sharing that birthday, which is of course around 1 in 365 (or a little higher with leap years). While this is partly down to confusion over probability, it’s also possibly down to people instinctively personalizing the question and approaching it as if it were “What is the probability of a particular person sharing my birthday?”

The actual answer will depend on the size of the group, but what surprises people is that it only takes a group of 23 people before there’s a 50% probability of a match. (Put another way, in most school classes, it’s more likely than not that there will be a match.)

Why exactly the number is 23 involves the type of mathematics that requires both apostrophes and exclamation marks. But put in a more simplified way:

  • any time you compare a pair of people, there’s a roughly 1 in 365 chance that the first birthday matches the second birthday; and
  • comparing one person’s birthday to the rest of the group creates 22 possibilities of a match; but
  • comparing every person’s birthday to the rest of the group creates 253 possibilities of a match (23 individual people, multiplied by 22 others in the group, divided by two to avoid double-counting each pairing).

If you imagine having 253 chances to attempt something with a 1 in 365 possibility of success, you can see how it suddenly becomes much more likely.

(There are some limitations to this explanation: this logic should mean it only takes 20 people to create a 50-50 chance of a match as 20 people creates 190 pairings, which is more than half of 365. That I don’t entirely understand why this isn’t the case is why I took an arts degree…)

While it would, of course, take 366 people to be absolutely certain of a match, the likelihood rapidly increases as a group gets larger. While a group of 23 is a 50-50 shot, once you get to a group of 50 the likelihood of a match is around 97%. With a group of 100, there’s barely a one in three million chance of not having a match

Perhaps the most surprising element of this mathematical puzzle is that matches actually happen more often in reality than on practice. That’s because births are not evenly spread out over the year. For example, in many Northern hemisphere countries there are more births in September. That’s not down to chance: without wishing to be crude, cold weather means many couples have to opt for indoor leisure activities in December, while Christmas and New Year is the time with the highest number of couples spending more time together than usual. An uneven distribution of birthdays thus increases the chance of a match.

And if you’re wondering which particular date is the most likely to be the subject of a match, well according to one site which analyzed birth certificates, it’s October 5. Why? Well, it might just be that that on average a child born on that date would have been conceived on December 31.

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7 Responses to The Stats Behind a Birthday Match

  1. Glad you enjoyed the icebreaker, John. I was really surprised that there were THREE pairs in the group (which consisted of 33 people). When I planned the icebreaker I was worried that there wouldn't even be ONE pair. What a pessimistic geek I am…

  2. Glad you enjoyed the icebreaker, John. I was really surprised that there were THREE pairs in the group (which consisted of 33 people). When I planned the icebreaker I was worried that there wouldn’t even be ONE pair. What a pessimistic geek I am…