James K Polk and Warren G Harding, two presidents both born on November 2nd.

Now there 365 possible days that a person could be born.

So out of just 44 presidents it's an amazing coincidence that two would share the same birthday, right?

Actually, no.

In the group of 44 people there's a 93% chance that two of them could share a cake.

Here's why.

[music playing] First, let's simplify the problem a bit.

Let's figure out how many people you'd have to get together in a room to do just better than a 50/50 chance of a birthday match, assuming none of them are twins, or triplets, or that you're not at a convention for people whose birthday on like April 7th or something.

You're the first person in the room.

There's a 100% chance that your birthday is your birthday.

What are the chances that the second person to walk in the room doesn't share your birthday?

Well they have 365 birthdays to choose from.

And 364 won't match yours.

So a measly 0.3% chance of a birthday in common.

When a third person walks in, 363 possible birthdays left that don't match any birthdays already in the room.

But how do we put these together?

When we combine the odds of independent choices together, we multiply their probabilities.

That let's-- whoa, whoa, whoa.

Don't run away from the math.

It's not that scary.

Look, a kitten.

[music playing] So the chance that you, person number two, and person number three have a unique combination of birthdays with no one sharing is about 99.2%.

With every new person we add to the room there's one fewer birthday available.

And we continue to multiply the combinations.

23 people, that's all the people we need before you have a greater than 50% chance of two sharing a birthday.

This is a birthday paradox.

It goes against our intuitions, because our brains are bad at figuring out the power of chance.

Sure, in that room there's only 22 possible combinations of your birthday someone else's.

But there's 253 combinations of everyone's birthdays.

Our brains have trouble imagining these combinations, estimating things that grow exponentially.

For example, how many times do you think you'd have to fold a piece of paper in half before it's stacked height reached the moon?

Well, we fold it once, two sheets.

We fold it again, we have four, where we once had one.

Now if I keep folding this piece of paper indefinitely by 41 folds we'd reach over halfway to the moon.

We only need one more fold to cover all of that remaining distance, 42 folds.

42, that number does seem to come up a lot.

OK so now you understand how few people it takes to get one common birthday.

But let's make it personal.

What size group would we need to get, say, a 90% chance of one of them sharing your specific birthday?

Now-- whoa, whoa, whoa.

Hold on.

The kitten is back.

And this time it's in a box.

[music playing] In this case every new person that we add to the room has the same chance of not sharing your birthday.

The chances of not sharing combine and combine with every new person we add to the room.

The chances of them having your exact birthday?

Ah, you're catching on.

And we want to figure what that group size is, that number n. So we can do some rearrangement.

We take the log of both sides.

To have a 90% chance of one person having your birthday you need 840 people.

I've been noticing on Facebook lately that I seem to have at least one friend with a birthday every single day.

But on closer examination, it's not actually every single day.

I've got most birthdays filled.

But a few are still open.

So how many would I have to have to be able to type happy birthday on a friend's wall every single day?

Now hypothetically, you'd only need 365, one for every day.

But as we've seen the chances of that happening are pretty small.

Now the first person you add will have a unique birthday.

The second person probably checks another birthday off the list, although there is a tiny chance they share the first person's birthday.

The third person could be a new birthday or the same as one or two.

As we go on and on adding new friends we'll have fewer open birthday slots to fill.

And many of our birthdays we'll start to have two, three, maybe five people per day.

We check off a large number of birthdays very quickly.

But the last few will take way longer than we expect.

If everyone on Facebook added friends until they filled every birthday the average number needed would be 2,153.

Of course, any individual person may need to add way more than that.

To have a 90% chance of hitting every birthday you'd have to add more than 21,000 friends.

So get to clicking.

It's our third birthday here at "It's OK to be Smart."

And we've made a lot of new friends in that time, enough that it's safe to say we've got a reason to celebrate every single day.

I hope you enjoy these birthday-related mind benders.

Thanks for three great years.

And here's to many more shared with you.

Stay curious.