The birthday paradox

Put 70 people in a room and there’s a 99.9% chance that two of them share a birthday. Why?

[Today is this website’s 2nd anniversary. Happy birthday to us!]

There are 365 days in a year, 366 in a leap year. But if you put 70 people into a room it’s almost certain that two of them will have the same birthday. In fact, if you put just 23 people into a room there’s at least a 50% chance that two of them will have the same birthday.

This is known as the birthday problem, or the birthday paradox. (It’s not an actual paradox, just a surprising and counterintuitive result, like the kidney stone paradox or the false positive paradox.) The math behind it is a little opaque, but one intuitive way of explaining it is that the number of potential combinations of people in these groups is large – 70 people equals 2415 different pairings – and any one of those pairings could be a matching birthday. To actually prove this, we’ll need to dig into the math. To make it easier we’re going to skip over the question of leap years, and we’re going to calculate the odds by approaching this from the opposite direction: the chances that two people don’t share a birthday.

So, if two people are in a room together, what are the chances that they do not share a birthday? Of the 365 possible days that one person could have a birthday, 364 of them would not be the same as the other person. So the chance that these two don’t share a birthday is 364/365, or roughly 99.726%. The chance of the opposite, that they do share a birthday, is that remaining 0.274%.

Now we’re going to add a third person. What are the chances that their birthday is different from the first two? Well, there are two days already claimed by someone else, so the chance that the third person’s birthday is new is 363/365, or 99.452%.

To recap, in this group of three people the chance that the second person has a different birthday than the first is 99.726%. The chance that the third person has a different birthday than the first or the second is 99.452%. We multiply those two together to find the overall chance that no-one has any birthdays in common: 99.726% x 99.452% = 99.180%. And, finally, the opposite – that one of these three have the same birthday – is the leftover chance, now 0.820%. Add one person, and the odds have already tripled. Add a fourth person, and the odds double again.

As you add more and more people the number of possible pairs increase, and so the chance that one of those pairs will have matching birthdays also increases at an alarming rate. At 15 people there’s already a 1 in 4 chance that there’s a matching birthday; at 18 people it’s 1 in 3. The 1 in 2 chance (50% chance) hits at 23 people, and by 70 people the chance is up to 99.916%.

Here’s the thing about the birthday paradox: this is the minimum chance, and assumes that birthdays are evenly distributed across the year. In the real world, some birthdays are more popular than others. In Western countries, for example, there’s always a small jump in the number of birthdays in September and October… nine months after the Christmas and New Year holidays. So the actual chance of matching birthdays is even higher than the calculations above.

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