My brother-in-law knows six people with the same birthday today. But it turns out he’s wrong to think that noteworthy.

Many readers will no doubt be aware of the birthday paradox, by which it only takes a group of 23 people before there’s a 50 percent chance of two people in the group sharing a birthday. As both mathematicians looking for a party trick and teachers trying to freak out a class know, this is extremely counter-intuitive. The problem is that most people misinterpret the question as referring either to the odds of a specific pair of people sharing a birthday, or to the odds of at least one person in the group sharing their own birthday.

But what happens when you increase the group size. Well, my brother-in-law today noted that of his 454 Facebook “friends”, six have their birthday today, even though he expected that with an even distribution the figure would be closer to one.

His mistake was having an actuary for a brother who pointed out that “You’re going to look a right tit if you go round regaling your friends with tales of… amazing friend-birthday coincidences when in fact they amount to statistical likelihoods.”

It turns out that it’s just the same birthday paradox with the numbers scaled up. The actuary pointed out that the chances of a specific set of six friends sharing a birthday is an admittedly unlikely one in 6.5 trillion.

However, when you look for any combination of six friends among the 454 on the list, things are very different. According to the actuary, there are 11.8 trillion different possible sets of six people from the list.

A quick glance would reveal that if you have 11.8 trillion sets, each with a one in 6.5 trillion chance of a match, it should be more likely than not that you’ll get a match. In fact, the probability lies at 83.6 percent.

Furthermore, while having seven matches is less likely than not, it would still be unremarkable at a 28.5 percent probability. It’s not until you get to the chances of eight people having a match where said match really becomes worthy of mention at a five percent probability.

Of course, while it might have been a bit churlish for my brother-in-law to simply reply with this image:

there are some social elements beyond the numbers. For example, regardless of the probability, the fact is that for most people it does feel quite unusual to have six friends share a birthday. However, that may simply be down to the Facebook effect. The site makes it much easier to keep track of people you have some form of social relationship with: without such a tool, it would be difficult to estimate this figure, let alone list 454 people from memory.

The real difference, though, is that this list likely includes many people who we “know” but not well enough to remember and celebrate their birthday. With Facebook’s prompts we can see the birthday of every “friend” unless they are in the rare group who’ve delved into the privacy settings. So the chances are that for as long as people have had wide social circles, they’ve been friends with several people who shared a birthday, but were not aware of the fact.

And of course there’s a big difference between the pure maths behind a group of friends sharing a birthday and the varying chances of that match happening on a specific date. And let’s just say that for people who, like my brother-in-law, live in the United Kingdom, nine months before September 30 is a period when many people are experiencing some combination of vacation, intoxication, boredom and cold.