The Birthday Paradox

by Katie Bowell, Curator of Cultural Interpretation

Today is my birthday. In this I am far from alone. It turns out that, at least in the Western world, October 5th is the most common day to be born. “What are the odds?,” you might exclaim. Well, count back the requisite number of months for human gestation from October 5th and you end up pretty close to New Year’s Eve. Mystery solved.

Thinking about how many people in the world share my birthday lead me to think about The Birthday Paradox, and how shared birthdays are more common than you might think. Get ready for some math!

Paraphrased, The Birthday Paradox states that: The probability of two people in a room sharing the same birthday is bigger than you’d probably guess.

The first thing you need to know? The Birthday Paradox isn’t actually a paradox. A paradox is a statement or situation that leads to a contradiction. The Birthday Paradox isn’t a contradiction, but it is so immediately counter-intuitive that it leaves every scratching their heads and so we call it a paradox.

Here’s why it stumps so many smartypants: If you meet someone randomly and ask them what their birthday is, the chance of the  two of you having the same birthday is 1/365 (or 1/366 if you’re counting Leap Years). Those are pretty small odds. And even if you go and ask a bunch of people, say 20 or 30, the probability of one of them having the same birthday as you is still very small – around 5%. So what if you put a bunch of people in a room together? How many people would you need to put into the room before there is a birthday match?

Most people assume that since the odds of finding their own birthday match are so low, the only way to ensure that there’s any birthday match at all is to put 367 (the number of days in a Leap Year +1) people into the room. However, this is where the math steps in and reminds us that we’re not thinking the problem through or showing all our work.

The question is not: Does anyone in the room have the same birthday as you? Rather, the question is: Does anyone in the room have the same birthday as anyone else in the room? Once that’s the question, the numbers change.

Instead of it just being a comparison of your birthday to everyone else’s, it’s a comparison of everyone’s birthday to everyone else’s. Put 10 people in a room (you being one of them), and you have 9 chances that you’ll share a birthday with someone. Put 10 people in a room and have them all compare their birthdays with each other? 45 chances.

When you sit down and do the math (which I’m not going to do, since it’s my birthday and I’d rather not), what you find is that as soon as you have 23 people in a room, the probability of two of them sharing the same birthday is 50%. Up the number of people to 57? 99% probability of a shared birthday.

This is what the math looks like

So, internet, want to try an experiment? (You have to do it, it’s my birthday). Enter your birthday in the comments and lets see if we find any matches.


2 Responses to “The Birthday Paradox”

  1. 1 Suzy Riding October 5, 2010 at 3:00 pm

    Hi Katie! Happy happy birthday! What a cool thing that it is your special day. My birthday is…October…seventeen!

    Have a wonderful day!

  2. 2 chris March 18, 2011 at 1:58 pm

    Mine is october 1st

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