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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!

By
Jason Marshall, PhD,
August 15, 2014
Episode #208

Page 1 of 2

Birthday BalloonsAs you may know, the quadrennial World Cup was recently played in Brazil. As you may not know, each of the 32 teams in this year's tournament had 23 players on their roster. This week's Math Dude episode has absolutely nothing to do with soccer or the World Cup, so why have I mentioned this fun fact?

Allow me to answer this question with a question: What are the odds that two players from one World Cup team share a birthday? It seems like the chances are pretty slim. After all, there are 365 days in a year, and only 23 players on a team—so surely there's not much chance of a shared birthday, right? Believe it or not, wrong.

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As we'll soon find out, there's actually a decent chance that many—perhaps even the majority—of World Cup teams will have a pair of players who share a birthday. How is that possible? Keep on reading to find out!

Probabilities and Dice

To help us understand the probabilities of shared birthdays, let's think a bit about the probabilities of rolling dice. To begin with, what's the probability of tossing 6 when you roll a single die? Since dice have 6 sides, the probability must be 1 out of 6, or 1/6.

Now let's imagine that we roll the same die twice. What's the probability that we get the same number—no matter what it is—twice? Whatever we get on the first roll, there's a 1 out of 6 chance that we get the same number again. So, once again, the probability is 1/6.

If we roll the same die three times, what's the probability of getting the same number three times in a row? As we've seen, there's a 1/6 chance that the first two rolls will give the same number. And whatever that number is, there's a 1/6 chance that the third roll will yield it again. So the probability of rolling the same number three times in a row is 1/6 x 1/6 = 1/36.

Probabilities and Three Dice

Let's now think about something that's a touch harder: If we toss three dice, what's the probability that at least two of them show the same number? One way to approach this problem is to figure out the probabilities of all the different ways that three dice can give two or more of the same number, and then to add these together to find the total probability. In this case, there are two outcomes that we need to consider:

  1. If the second roll matches the first, then the third roll is irrelevant, since you already have at least one pair. What are the chances of this happening? As we've seen, the probability of this outcome is 1/6.
  2. If the second roll does not match the first—which will happen 5/6 of the time—the only way to get a pair is for the third roll to match either the first or second roll. The probability of the third roll matching either the first or second is 2 out of 6. That means that the overall probability of this happening is 5/6 x 2/6 = 10/36, or 5/18.

To find the total probability of rolling two or more of the same number, we need to add these two probabilities together. So the total is 1/6 (from the probability that the second roll matches the first) plus 5/18 (from the probability that the third roll matches either the first or second), for a total of 1/6 + 5/18 = 3/18 + 5/18 = 8/18, or 4/9. Which says that there's about a 44.4% chance of rolling two or more of the same number.

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