How to Solve the Famous Birthday Problem

What are the chances that two players on the same soccer team share a birthday? How about two students in the same algebra class? Both seem pretty unlikely, right? The answer might surprise you!

Jason Marshall, PhD
Episode #208

A Trick For Calculating Probability

But it turns out we don't actually have to do all of this work. Instead of figuring out the probabilities of all the ways of rolling two or more of the same number, we can instead figure out the probability of not rolling two or more of the same number. In other words, we can start by figuring out the probability that each of the three dice show a different number.

Why is this helpful? Because the probability of rolling two or more of the same number plus the probability of not rolling two or more of the same number must be equal to 1—that's how probabilities work. So instead of the probability calculation we just did (which was kind of a lot to keep track of!), we can instead calculate the probability of not rolling two or more of the same number (which we'll see is much easier), and then subtract it from 1.

In our problem, there's a 1/6 chance that the second roll will match whatever number you got on the first roll, and there's a 5/6 chance that it will not. Since we're interested in figuring out the probability of having three different numbers after three roles, we need to follow the latter path. Continuing on, there's a 4/6 chance that the third roll will also yield a number that matches neither. Which means that the probability of all three numbers being different is 5/6 x 4/6 = 20/36 = 5/9, or about 55.6%.

So what does this tell us about the probability of rolling two or more of the same number? It tells us that the probability of that happening must be 1 - 5/9 = 4/9. This is indeed the same answer we got before—but this time it was a lot less work! Pretty cool, right?

The Birthday Problem

Believe it or not, this trick is all we need to know to calculate the probability that two people on a 23 person World Cup soccer team share the same birthday. Here's the idea: Instead of calculating the probability that two people share the same birthday, let's start by calculating the probability that no two players on a 23 person team share a birthday. Once we know that number, finding the complementary probability is easy.

The probability that the first two players on a team don't share a birthday is 364/365. The probability that the third player also does not share a birthday is 363/365 (since there are already two "taken" birthdays.) The probability that the fourth player also does not share a birthday is 362/365, and so on, up to the 23rd player--when the probability is 343/365.

To find the probability that none of the players on the team share a birthday, we have to multiply all of these probabilities. If you do that, you'll find that 364/365 x 363/365 x 362/365 x ... x 343/365 is approximately equal to 0.49. This tells us that there is about a 49% chance that no two players will share a birthday. And it also means that there's a 1 - 0.49 = 0.51, or 51%, chance that two or more players will share a birthday!

How about an entire math class? If you do the calculation for a 30 student class, you'll find that the probability that two or more students share a birthday is about 71%.

It may be counterintuitive, but it's true—even on a relatively small soccer team or in a relatively small class, there's a pretty decent chance that two people will share the same birthday. Sometimes math can be very surprising!

Wrap Up

OK, that's all the math we have time for today.

Please be sure to check out my book, The Math Dude’s Quick and Dirty Guide to Algebra. And remember to become a fan of The Math Dude on Facebook, where you’ll find lots of great math posted throughout the week. If you’re on Twitter, please follow me there, too.

Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!.

Balloons, dice, and math calculations mages courtesy of Shutterstock.


About the Author

Jason Marshall, PhD
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