Let’s make a couple of assumptions. First, let’s assume that birthdays are randomly distributed—given enough people, you’ll have roughly the same number born on, say, December 13 as you will on November 22 or April 14. (As it turns out, this isn’t quite true.) Second, let’s assume that February 29—Leap Day—doesn’t exist. (Also untrue.) Finally, let’s assume that everyone uses the 365-day Gregorian calendar. (Mostly true.) Got it? Nothing too controversial.
Say you walk into an empty auditorium. A minute or so later, someone else walks in. Given the previous assumptions, there’s a 1 in 365 chance (0.27 percent) that this person shares your birthday. A second person walks in a minute or two later. The odds of you sharing a birthday with either jump to about 0.55 percent. A third and a fourth and—you get the idea. Only when the 253rd person walks in do you have a 50 percent chance of having the same birthday as someone else in the room. It isn’t person 182 or 183, because some of the first 200-something people may share birthdays with each other. So a birthday shared with person 254 (including you) should make intuitive sense—or, at least, not be terribly shocking.
But let’s look at it a different way. Again, you start off in an empty auditorium and again, every few minutes, someone new comes into the room. Instead of wondering if you share a birthday with anyone else in the room, let’s make this about everyone in the room. Let’s ask: “Do any two people in the room share a birthday?” The math starts off the same—with two people, there’s still a 1-in-365 chance. The third person? The odds aren’t 0.55 percent anymore—now, there’s a 0.82 percent chance that anyone in the group matches someone else. Yes, you could share a birthday with either of the other two people—that’s the 0.55 percent—but they could share a birthday with each other, too. That’s where the extra percentage boost comes from.
How many people before we hit a 50 percent chance that any two share a birthday? Twenty-three. Not 230. Twenty-three. Here’s a graph:
Probability of Two or More Sharing a Birthday
The odds go up very quickly because each new person can match every other person in the room, and as the number of people in the room grows, the gains are huge. At fifty-seven people, there is just over a 99 percent chance of any two people in the room sharing a birthday. And, in case you are wondering, at 124 people, there is less than a 0.0000000001 percent chance of there not being a match. That’s one in one hundred trillion.
Of course, birthdays aren’t random. Some months have more births than others, for reasons one can imagine (and you’d almost certainly be correct). The days of the week should be random—but, at least in the United States, they aren’t. Why not? As BabyCenter explains, Tuesdays and then Mondays have the highest number of births, because hospitals try not to schedule C-sections or induce labor on the weekends.