Here’s a brain teaser for you: Imagine we’ve got a room full of people. We’re trying to figure if any two people in the room have the same birthday.
For us to reach a fifty-percent probability that there are two people in the room with the exact same birthday, how many people need to be in the room?
I told you this was a brain teaser, so suffice to say that the answer -- to how many people need to be in a room for there to be a fifty-percent probability that two people have the exact same birthday -- is not what you would intuitively expect.
The “birthday problem” tells a lot about how we fail to see hidden complexity For the sake of this puzzle, let’s assume there are no twins, no leap year birthdays, and there are no seasonal variations. No spike in birthdays nine months after Christmas or some big snowstorm.
Most people start with a rough calculation like this: There’s 365 days in a year, so for there to be two people in the room with the same birthday, take 365, divide it by two -- you’ve got about 180, give or take. With 180 people in a room it seems you’d have about a fifty-percent chance that two of them have the same birthday.
This intuitive calculation is wrong. It’s very wrong. If you had 180 people in a room, the chances that two of them will have the same birthday is damn close to 100%. Even if there were only 100 people in the room, rather than 180, the chances that two of them would have the same birthday would be 99.99997%.
The actual answer is fun to know, but it also tells us a lot about our minds. It tells us a lot about how bad we are at understanding complexity. It tells us a lot about how complexity tends to get out of hand, and weigh us down, and cause us to stagnate. Complexity creep.
If we know the answer to what is known as the birthday problem, maybe -- just maybe -- we can fight against complexity creep: That insidious tendency for us to make things more complex and more complex and more complex, until we find ourselves paralyzed.
And there’s a flip side. If you can understand complexity creep -- if you can understand how things that seem simple are actually complex, you can also use that to your advantage.
Each “one thing” interacts with every other thing So how do you actually find the answer to the birthday problem? Let me start by saying that if you have trouble following the next minute or so, don’t worry about it. That’s the point. Our brains aren’t wired to intuitively understand this.
On a basic level, you wouldn’t just calculate based upon the total number of people in the room and the total number of potential birthdays.
In actuality, you would calculate based upon potential interactions amongst the birthdays of every person within the room.
Like this: If there’s only one person in the room, there’s a 365 out of 365 -- 100% -- chance that person does not share a birthday with another person in the room. There are no other people in the room, after all.
Add a second person, and there’s a 364 out of 365 chance that person does not share a birthday with the first person in the room.
With each person you add, you take away one from the numerator of that fraction. With the third person, instead of 364, it’s a 363 out of 365 chance that person does not share a birthday with either of the first two people in the room.
So on and on, that numerator gets lower -- from 363 to 362 to 361 -- with each additional person in the room. So far, there’s five people in the room, and a 361 out of 365 chance that fifth person does not share a birthday with any of the other four people in that room.
That’s a 98.9% chance of no match. A merely 1.1% chance that this fifth person shares a birthday with one of the other four people in the room.
But wait. If there are five people in a room, the chances that any two of them share the same birthday is not