In a room of 23 people, the chance that there are two people who share the same birthday is 50%. In a room of 75, that chance of overlapping birthdays increases to 99.9%!

But there are 365 possible birthdays. How is this possible?

But first…

How is it a Paradox?

There are many different types of paradoxes. Most paradoxes are based on contradictions, but some paradoxes, such as the Birthday Paradox, are only called paradoxes because they defy common sense. The Monty Hall problem, which I explain in depth with wonderful illustrations over in another post, could also be called a paradox for the same reason. …But both paradoxes are completely true.

From People to Pairs

We know that there are 23 people in the group that we’re testing. But we didn’t ask for the chances of them matching you, we wanted to know the chance of any person having the same birthday as any other person.

So if we had a group of 4, containing You, Alice, Bob, and Carol. If we just see who has the same birthday as you, there are only three matches to test: You-Alice, You-Bob, and You-Carol. Bt with the actual problem matching any person to any other person, we have 6 matches to test: You-Alice, You-Bob, You-Carol, Alice-Bob, Alice-Carol, and Bob-Carol.

If you do the math, any group of size x will have x choose 2, or _xC₂, both of which is the Binomial coefficient function, or more commonly a “combination”. “x choose 2″ means that if we have x things, how many different ways can we combine 2 of them?

For example, with our previous group of 4, we could combine them in 6 different ways. That was “4 choose 2″. Note that “You and Alice” and “Alice and You” are the same combination.

For 23 people, we’re finding way more than 23 pairs:

…253 pairs.

A Sprinkle of Probability

Now, what’s the chance of two different people having the same birthday? Well, ignoring leap years, there are 365 possible birthdays. That means that if, say, Alice has the same birthday, than the chance of Carol’s birthday matching Alice’s birthday is 1/365.

So how do we check the chances for every single person?

Well, we know from statistics that 1 – [the chance of the event not happening] is the chance of the event happening. For example, if rolling a 3 on a 6-sided die happens 1/6 of the time, the chance of it not happening is 5/6. 1 – (5/6) is 1/6.

And we also know that you can multiply events to see the chance of both of them happening. If flipping a coin and getting heads happens 1/2 of the time and rolling the 3 happens 1/6 of the time, then getting heads and rolling a 3 at the same time will happen (1/2)(1/6) of the time, or 1/12 of the time.

The last trick we’ll need is exponentiation. If we wanted to know the chance of the same event happening several times in a row, we would have to multiply it a lot. For example, the chance of flipping 5 coins and getting 5 heads is (1/2)(1/2)(1/2)(1/2)(1/2), which is (1/2)^5, or one-half to the fifth power. (It evaluates to 1/32.)

Putting it All Together

So we have a group of 23 people. We know that’s 253 pairs.

The chance of any pair not sharing a birthday is 364/365. That’s 99.73%, so it’s very likely that any pair will not share a birthday. However, the real action gets in when we factor for all 253 pairs. The chance of all 253 pairs not sharing a birthday is (364/365)^253.

That’s 49.95%. And since 1 – [the chance of the event not happening] is the chance of the event happening, the chance of a group of 23 people having one shared birthday is 1 – 49.95%, or 50.05%.

What About the Group of 75?

Well, for the group of 75, we just do the same thing.

First, we find the number of pairs:

Then we do the probability for all 2775 people:

Despite starting out with a 1/365 chance for a single pair, groups make for lots of pairs, and the exponentiation works out quickly.

So the next time your in a classroom of 20 people, think of how many of them might share birthdays with each other! Or don’t, because that would be excessively nerdy…

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