Birthdays in a Room

There’s this imaginary person who makes their money by going around various rooms, placing little bets with people. Let’s call this person the Gambler.

The Gambler enters a room on a given day with a certain number of people in it. Then, they go around placing bets with anyone who will take the odds.

The funny thing is that the Gambler wins most of the time, but never has much trouble finding people to accept these bets. The Gambler is fabulously rich from all this gambling.

The bet is always the same, but you’ve noticed an important clue: the Gambler only places these bets when they can find a room with more than 23 people in it.

Oh, I almost forgot: the bet is whether 2 people in the room have the same birthday.

How does the Gambler do it? Let’s break it down a bit.

First, imagine that there are only 2 people in the room. What are the odds that they share the same birthday? That seems like some kind of cosmic coincidence, right?

Well, no. It’s about one in 365 (let’s ignore leap year for simplicity’s sake, just to make the point much more quickly). You don’t both have to have birthdays on a particular day—just the same day.

It’s easier to think about this a little differently: there is a 364/365 chance that the second person will have a different birthday as the first. This part should be very intuitive for us humans! We think in linear steps like this all the time, and it comes naturally for us.

What happens when a third person enters the room? Here, things get a tad wonky. By wonky, I mean counterintuitive.

If we imagine that those two folks have birthdays on different days, then there are now two days that are potential candidates to be on the same day as this new person’s birthday. We have to admit that the new person coming into the room now has a chance of two in 365 to find the same birthday as one of the others in the room.

If you’re ready to say, well, duh! and conclude that 23 people in a room means a 23 in 365 chance of there being two birthdays in a room, hold on to your butts.

There’s also the possibility that the first two people have birthdays on the same day, even before that third person enters. We already know there’s a one in 365 chance of that happening, and we can’t forget that distinct possibility when we’re figuring this whole thing out.

This is exactly what makes this birthday paradox so interesting, and so counterintuitive. We have to take that original probability into account with the new probability being introduced with the third person.

The first two folks (1/365) and the third person entering (2/365) don’t combine to give you 3/365 odds, though—that’s not at all how probabilities work. If it was, then you could just flip a coin twice and have it land on heads one of those two times. After all, 1/2 + 1/2 = 1, or 100% probability.

We can see why this can’t be the case if we think about two potential outcomes. If flipping the coin twice always results in heads (since 1/2 + 1/2 = 1), then you would always have a heads result every time. However, if you calculate your odds of getting tails during one of those flips, you’d also come up with 100% probability. You can’t have both heads and tails every time.

The birthday paradox is just like this. When you add another person into the room and you still have to take into account the previous folks in the room, this isn’t an example of linear growth, but instead of exponential growth. That’s because the leftovers from the previous effort continue to pile up, and those little piles grow bigger and bigger with each step.

As you add that fourth and fifth person into the room, each new person creates a potential birthday pairing with every person already in the room. By the time you get to 23 people and add up all of these potential pairings, you end up with just over a 50% chance that this room now has two people in it with the same birthday.

That’s why the Gambler waits for the number 23.

I mentioned that we tend to think linearly, mainly because that has been conductive to our survival in nature. We see threats and respond in an instant, in the here and now, where exponential growth doesn’t really come into play. Survival and deep or long-term thinking don’t always tend to go together well, and dreamers of the ancient past who focused too much on thinking and not enough on surviving… didn’t.

However, in this modern world, it pays to think exponentially. Our population has grown along these lines, and so has our technology. Money grows along these lines, too, and that goes a long way to explain how rich people tend to get richer over time: money begets more money, and that new money begets more money still.

This is another one of those spots where we need to slow our thinking down and resist our instincts to think linearly. We need to consider just how nonintuitive the world around us really is.