## Tuesday, November 29, 2011

### The Tuesday boy problem, in under 300 words

I saw a post this morning about something called the "Tuesday Boy Problem". It illustrates a really interesting and counterintuitive topic in probability, but it was written up in a frustratingly verbose, confusing, and ambiguous way. So I'm going to try to restate it more clearly.

Let's start from the basics. If a person on the street says to you, "I have two children," what is the probability that they're both boys? 1/4, since there are four possibilities, one of which produces two boys:

Now, imagine he says, "I have two children, and at least one of them is a boy." What is the probability they are both boys? Strangely, 1/3: we eliminate one of the four possible universes from the previous diagram, and one of the remaining three has two boys:

Here's where it starts getting strange: Let's say he says, "I have two children, and at least one of them is a boy born in an even-numbered month." You'd think that the information about his birth-month would be irrelevant, but believe it or not, it actually matters. We cross off 9 of the 16 possible worlds, and of the remaining 7, there are 3 with two boys. To put it another way, the even-month information eliminated half of the previous table's GB and BG universes, but only a quarter of its BB ones. So the odds are 3/7:

Finally, to return to the original problem: If the man says, "I have two children, and at least one of them is a boy who was born on a Tuesday", the probability that both are boys is a bizarre 13/27:

Kinda looks like a Scandinavian flag.

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