There’s this fairly popular social media site I frequent which has a small blue square logo with the letter “f” to one side. Over the past few days, I noticed that I’ve been getting a large number of notifications on this site (not that I normally don’t, given how undeniably popular I am, as you can tell from the way I’m sat here on a Sunday evening writing a blog post). Every morning, I’ll awake to a sharp ping on my phone, indicating that either someone wants to communicate with me in some way, or there’s a particular event which might take my interest. Grabbing my phone with vivid anticipation, I’m immediately disheartened to see that “Yanhao and 3 others have their birthdays today. Help them celebrate!” It’s not that I’m not appreciative of Yanhao’s date of birth, I just don’t want to be seeing the same thing every single day (Psst… don’t tell him, but I don’t actually know when his birthday is).
So, how likely is it that so many of my friends seem to be celebrating their life anniversaries on the same day? You might instinctively say, “probably not very, you clearly just have a lot of friends and hence a very large sample size”. Well let’s whip out our calculators and take a closer look, shall we?
Let’s assume there are 22 people in my friends list. Disregarding the fact that this is clearly an absurd premise *ahem*, how could we find out the probability of just two of me or my buddies sharing the same birthday? I think the first thing I’d do is consider my own birthday, and compare it with all of my friends’. From this alone, the probability is indeed going to be quite small (about 1 in 16), given that we are only making 22 comparisons. However, what we’ve foolishly neglected is the 231 other comparisons between everyone else.
Okay, so there are 253 comparisons for us to make between this tightly-knit group of friends, and all we need to find is one mutual birthday. We could sit here and write a list of all of the pairs and the ways they match, but that would be a waste of time… probably. What we can do instead is think a little outside the box, and do the converse. In order for there not to be a match, every single birthday must be on a different day. This is the only scenario we need to consider, and we’ll just take it away from the total probability, which is 1.
Since one friend only has one specific birthday, then there are 364 days which are not their birthday. So another friend needs to have their birthday on one of these days to not share a birthday with the first friend. The probability of this is:
364/365 = 0.997260
Seems pretty high right now, but just you wait. So, that is the probability that two friends don’t share a mutual birthday. To find the probability that both this is true, and our next pair has the same outcome, we can multiply the two probabilities together since they are independent of each other, known as the multiplication rule of probability. Repeating this 253 times for all possible comparisons, we get:
(364/365)253 = 0.499523
Taking this number away from 1, the maths shows us that there’s just over a 50% probability that two of me or my 22 friends share a birthday. Curiously surprising, huh? In fact, if I became slightly more popular and had 70 friends, there would be a 99.9% chance that two of them shared a birthday. When I become famous and acquire an immense 366 friends, there will be a 100% chance.
So, what have we learned today? Probably not much, apart from how useful and counter-intuitive statistics can be. Maybe those of you who have enough friends could try out this experiment, you know, for the sake of maths. In the middle of a party, get out a megaphone and gather everyone around. Tell them that you are sure at least two of the people in the room share a birthday. Get it right, and everyone will be blown away by your psychic powers. Get it wrong, and you’ll have made a fool of yourself and will probably lose a few friends in the process. It’s just 50/50, I suppose.