Math Guy: The Birthday Problem

Keith Devlin

Keith Devlin, Stanford University professor, makes regular appearances on 'Weekend Edition Saturday.' Bruce Cook hide caption

itoggle caption Bruce Cook

The Weekend Edition Saturday Math Guy, Stanford professor Keith Devlin, has a problem. In fact, he has more than one... which he's happy to share with Scott Simon.

Up for a Challenge?

If you think the two examples at right are tricky, take a look at two of the most notorious probability puzzles of all:

What is the probability that in a room filled with 23 people at least two of them have the same birthday? (It's more than half!)

Devlin explains:

The birthday problem asks how many people you need to have at a party so that there is a better-than-even chance that two of them will share the same birthday. Most people think the answer is 183, the smallest whole number larger than 365/2. In fact, you need just 23. The answer 183 is the correct answer to a very different question: How many people do you need to have at a party so that there is a better-than-even chance that one of them will share YOUR birthday? If there is no restriction on which two people will share a birthday, it makes an enormous difference. With 23 people in a room, there are 253 different ways of pairing two people together, and that gives a lot of possibilities of finding a pair with the same birthday.

Here is the precise calculation. To figure out the exact probability of finding two people with the same birthday in a given group, it turns out to be easier to ask the opposite question: what is the probability that NO two will share a birthday, i.e., that they will all have different birthdays? With just two people, the probability that they have different birthdays is 364/365, or about .997. If a third person joins them, the probability that this new person has a different birthday from those two (i.e., the probability that all three will have different birthdays) is (364/365) x (363/365), about .992. With a fourth person, the probability that all four have different birthdays is (364/365) x (363/365) x (362/365), which comes out at around .983. And so on. The answers to these multiplications get steadily smaller. When a twenty-third person enters the room, the final fraction that you multiply by is 343/365, and the answer you get drops below .5 for the first time, being approximately .493. This is the probability that all 23 people have a different birthday. So, the probability that at least two people share a birthday is 1 - .493 = .507, just greater than 1/2.

The Children Puzzle

I tell you that a couple has two children and that (at least) one of them is a boy. I ask you what is the probability that their other child is a boy. Most people think the answer is 1/2, arguing that it is equally likely that the other child is a boy or a girl. But that's not the right answer for the question I have asked you. Here's why. In terms of order of birth, there are four possibilities for the couple's children: BB, BG, GB, GG. When I tell you that at least one child is a boy, I rule out the possibility GG. That leaves three possibilities: BB, BG, GB. With two of these, the other child is a girl; so the probability of the other child being a girl is 2/3. Leaving the probability of the other child being a boy at 1/3.

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