The Birthday Problem

Cool little bit of probability that would be fun to pull out at parties, if you go to the right kind of parties — or wrong kind of parties, probably (get it? probably? eh? ehhh?):

In a room full of fifty people, what is the chance that two people have the same birthday?

Think about it for a second.  There are 365 days in the year (this is an idealized mathematical world, so leap years don’t exist), and only fifty people in the room — not nearly enough to ensure that every day is covered, right?

Well, right, but that’s not what the question is asking.  We want to know whether two people share a birthday, not whether two people are born on a certain day.  Assuming a uniform distribution of birthdays over the year (which isn’t really totally true, but whatever), let’s find the chance that two people in the room don’t share a birthday.  To do that, we start with the second person in the room.  The chance that the second person shares a birthday with the first person in the room is one out of 365 — but the chance they don’t share a birthday is 364/365.  The chance the third person does not share a birthday with the first or the second person is 363/365.  The fourth person has a 362/365 chance of not sharing a birthday, and so on, until you’ve gotten to the fiftieth person, who has a 316/365 chance of being born on a unique day.

Since everyone’s birthdate is independent of one another’s (for simplicity we’re assuming no weird parental pregnancy pacts between the parents of people in the room), to find the total probability that two people don’t share a birthday we have to multiply all these individual probabilities together.  Imagine flipping a coin, which is the archetypal independent event — the chance of getting heads is 1/2, but the chance of getting two heads in a row is 1/2 x 1/2 = 1/4, and so on.  That means we have:

Or, using the factorial operator and rearranging:

Since probabilities always have to add up to 100%, the probability that at least two people share a birthday is one minus this result — which works out to 97% (!).  That means that, in a random assortment of just fifty people, having two people share a birthday is almost a certainty.

In fact, at only 23 people in a room, there’s over a 50% chance that two of them will share birthdays.  Math!

Unfortunately, applying probability formulas to poker is less surprising.