This morning, at a meeting with 70 participants, a manager called out someone’s birthday. The thing is, today is also my birthday! What are the odds of that? Well, actually pretty good.
It is one of the more well known “surprising statistics facts” that once you have a group of 23 people, the odds are better than 1 in 2 that there’ll be a shared birthday. The reason this seems so shocking is that when we hear the question “what are the chances that two people will share a birthday” what we tend to hear is “what are the chances that someone else will have my birthday.” But that’s not the question anymore than the odds of having a pair of aces are the same as the odds of having any pair.
But other than our tendency to make probability questions excessively specific (and personal), how does this work?
Ok, imagine you’ve got a pool of people with all kinds of random birthdays. You are going to let them in to a room until you get a matching pair of birthdays. To keep things simple, we’ll use a Julian calendar (just numbering dates rather than assigning them month/day values). We’ll also ignore leap years and just make it a 365 day year.
Suppose the first person you let in was born on Day 1 (January 1st). Obviously no matching birthday yet.
The second person comes in…there’s a 1/365 chance that they have a matching birthday. Well suppose they were born on Day 2. So no match.
But, and here’s the crucial thing, as the third person comes in, their odds of matching someone’s birthday has gone way up because they could match TWO dates, Day 1 or Day 2. So it is 2/365. Let’s say they were born on Day 3. No match
Now it keeps going – the fourth person could match three dates (Day 1, 2, or 3). If they don’t, then the fifth person could match four dates (Day 1, 2, 3, or 4). In short, every time you add someone to the room who doesn’t have a matching birthday, you expand the available pool of matching dates for the next person to enter.
If we actually do the math, it works out to a shade over 50% in combined probabilities by the time the 23rd person walks in. If anyone wants to see this worked out in some more detail, check out Dr. Math, including a good explanation of why it is easier to calculate this by finding the odds that no birthdays will match. But as for me, I’ll leave it here at the fun conceptual level, just a quick probability lesson to celebrate my birthday.
This kind of thing is why I love statistics and probability. When it is counter-intuitive at first grasp, there’s usually a really good thought experiment that’ll make it all clear. And that makes for a fun “aha!” moment!