Everybody loves birthdays. You become older and people like to sing to you. What’s not to like? Sometimes you even meet someone with the same birthday and you share a special connection with them. After all, there are 366 possible days on which to have your birthday, including February 29^th. Two people sharing the same one must be very rare! Or is it? If we do the math, we can figure out just how likely it is to share a birthday. For the rest of this post I am going to ignore leap days because it makes the math a tad easier and doesn’t impact the end result too much.

So let’s say you meet a random person on the street. What’s the probability that they share your birthday? It’s just 1/365, or about 0.274 percent. This is because there are 365 days upon which their birthday could fall, but only one of those days, your birthday, would make you two have the same birthday. This is quite unlikely, so it seems that the event is quite rare. But what if you are with multiple people, each with their own birthday. What does the situation look like now?

Let’s say there are ten people gathered. What’s the probability of any two of them sharing a birthday? The easiest way to calculate this is to in fact calculate the opposite and then subtract it from 1, or 100 percent. The opposite is the probability that nobody shares a birthday, in other words, the probability that everyone has different birthdays. So the first person can have any birthday. The second person can’t have person one’s birthday, but they can have any of the other 364 days, so the probability that they don’t share person one’s birthday is 364/365. Now person three has 363 days left, so they have a probability of 363/365. This pattern continues for everyone in the group, so person ten will have 356 possible days, and a probability of 356/365. Multiplying all of the probabilities together gives approximately 0.883, or 88.3 percent. This is still pretty high. 100% – 88.3% = 11.7%. So the probability of two people sharing the same birthday in a group of ten is slightly more than one in ten. However, while low, this number is much greater than our original 0.274%. What’s fascinating is that this number grows quickly, such that given a group of 23 people, the probability of two people sharing the same birthday is more than 50%, and is in fact approximately 50.73%. This seems astonishing, as 23 is so much smaller than 365.

The key here is that there aren’t just 23 possible pairings of people. Given 23 people, there are 253 ways to pick two of them, and so there are that many pairings to consider when checking to see if any two people share the same birthday. In fact, this number, 253, appears in a similar problem as well. We have been discussing the probability of any two people having the same birthday, but what if you want to be in that birthday pair? You have a specific birthday and so it requires more people to be sure that someone will match your birthday. The probability that someone in a group of n people will share your birthday is 1-((365-n)/365)^n, and this becomes more than 50% when n=253, or when there are 253 comparisons you are making.

I think this is a really cool probability application because the mathematics makes sense, but the answer seems surprising. It shows the power of mathematical thinking.