The Birthday Paradox Revisited

Size Matters

The Birthday Paradox

Here’s a question.

How many people need to be in a room before it is more likely than not that at least two of them share a birthday?

A hundred? Two hundred?

The answer is 23.

That seems remarkably low. So how can it possibly be right?

Let’s assume, to keep things simple, that every day of the year is equally likely to be somebody’s birthday and forget about 29 February.

Start with two people, Julia and Julian.

Suppose Julia was born on 1 May. The chance that Julian was also born on 1 May is just 1 in 365.

Nothing very surprising there.

Now add a third person. The probability that Julian has a different birthday from Julia is 364/365. The probability that the third person then has a birthday different from both of them is 363/365.

So the probability that all three birthdays are different is:

(364/365) × (363/365).

Add another person and we multiply again, this time by 362/365.

And so it goes on.

The interesting thing is how quickly the probability of a shared birthday begins to rise.

With 5 people, the chance is only 2.7%.

With 10, it’s 11.7%.

With 16, it’s 28.1%.

But with 23 people it reaches 50.7%.

By the time we get to 32 people it’s around 75%, and with 40 it’s close to 90%.

So what’s going on?

How many pairs?

The trick is that we’re not asking whether anyone shares your birthday. We’re asking whether any two people share a birthday.

And that creates many more possibilities than we might at first imagine.

Put 23 people in a room and there aren’t just 23 opportunities for a match. There are 253 different pairs of people.

Julia can be paired with Julian and with everyone else. Julian can then be paired with everyone except Julia, whose pairing we have already counted. And so on.

By the time we have worked through the room, we have 253 different pairs.

Suddenly, a shared birthday doesn’t seem quite so unlikely.

There is another way of looking at it.

There are 365 open boxes and we drop 23 balls into the boxes at random. Now, what’s the chance at least two of the balls end up in the same box? It’s the same problem looked at in a different way, and the solution is identical. The chance is just over 50-50. 

What about your birthday?

Now change the question. What’s the chance that somebody in the room shares your particular birthday? With 23 people in the room, the chance now is only about 6%. Even with 366 people, the probability is still only about 63%. And that gets to the heart of the paradox.

We think of 23 people and consider the probability that someone will have a particular birthday. But the question is whether any birthday matches any other birthday.

There are a lot more ways for that to happen. 

Now here’s an experiment you can do. At the start of a football match there are 22 players on the pitch – 23 including the referee. Now check their birthdays. It’s actually a bit more likely than not that there will be at least one shared birthday. Size really does matter, but not in the way most people intuitively think!