The Birthday Paradox!

Here’s something wacky: In a room of 23 people, there is a 50% chance that two of those people have the same birthday. In a room of 75 people, there is a 99.9% of there being a set of birthday twins. You might think I’m crazy, but this is very true. This is an example of the birthday paradox.

The birthday paradox is special because it doesn’t really make sense in your brain. It is not a paradox, but more of a brain teaser. The math behind it makes sense, but we usually think of probability in a linear… like the probability of me pulling the only blue marble out of a bag of 16 marbles is 1:16. With the birthday paradox, the probability of there being a set of birthday twins is exponential.

Let’s break the birthday paradox down. The easiest way to prove this paradox is to prove the opposite case. What is the probability that in a room of 23 people that there are no birthday twins? What about in a room of 2 people?

Well, there are 365 possible birthdays (excluding February 29th, because that is a special case), and remember that we are trying to find the probability of there not being a birthday twin, there are 364 other birthdays for the second person to have. That’s equivalent to 364/365 or a 99.7% chance that they have different birthdays.

*If math scares the crap out of you, just skip the next paragraph. Probability can be so weird.*

In a room of 3 people, the probability of everyone having a unique birthday is (the probability of there being all unique birthdays in a room of two people) 364/365 * 363/365. In a room of 4 people, the probability of everyone having a unique birthday is (the probability of there being all unique birthdays in a room of two people) 364/365 * (the probability of there being all unique birthdays in a room of three people) 363/365 * 362/365. If you keep doing that, the probability will even out to about a 49.3% chance of all birthdays being unique. On the flip side, that’s a 50.7% chance of there being a set of birthday twins in a room of 23 people.

Why might you ask? I dunno. I don’t know much about probability, but if you think about it *really hard* it kind of makes sense. Think about how in a room of 366 people, there is a 0% chance of there being all unique birthdays, but in a room of 1 people there is a 100% chance of the birthday being unique. The graph that all of this probability would make is exponential.

If you want more information for this to make a bit more sense, there are several youtube videos of real mathematicians explaining it. Here are my sources:

Also, *Happy Birthday* to Mrs. Cook and Isabelle Wilson who are birthday twins in a room of 23 newspaper people. 🙂

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