Tuesday Boy

By Oliver Hawkins

Every year, maths fans get together to celebrate the work of Martin Gardner, who popularised mathematical puzzles in his weekly columns for Scientific American.

It's called the Gathering 4 Gardner. And at last year's Gathering, puzzle-maker Gary Foshee got on stage and proposed a problem that has had mathematicians arguing ever since.

It goes like this:

"I have two children. One is a boy born on a Tuesday. What is the probability I have two boys?"

"Your first impression is: what does Tuesday have to do with it?" says Gary, "And you might think that it doesn't. But in fact Tuesday has everything to do with it. And the actual answer to the problem is 13/27."

If you're scratching your head, don't worry. All will become (reasonably) clear.

But to understand what's going on here you first need to know that Tuesday Boy is a variation of a much older, simpler puzzle that often catches people out. And once you understand it, the answer to Tuesday Boy becomes easier to understand. So let's start there.

The Two Boy Puzzle

You meet someone who tells you, "I have two children. One of them is a boy. What is the probability I have two boys?"

On hearing this for the first time many people think the answer must be 1/2. There are two children. One is a boy. So the other must be a boy or a girl with equal probability.

But the mathematical trap lurking in the question is that we don't know which of the two children is a boy. And this makes a big difference.

For any two children there are four equally likely pairs:

Girl, Girl

Girl, Boy

Boy, Girl

Boy, Boy

We know that at least one of the two children is a boy, so that rules out the possibility of two girls. There are three pairs left, and only of them is a pair with two boys. So the probability of both children being boys, given that one child is a boy, is 1/3.

The Tuesday Boy Puzzle

The Tuesday Boy puzzle is like the two boy puzzle, except now we're not just told that one of the children is a boy, we're told one of them is a boy born on a Tuesday.

So how does this change the probability from 1/3 to 13/27?

We've established that there are four equally likely combinations of two children and if we eliminate the girl plus girl possibility we're left with the probability of two boys at 1/3.

Now we can do the same thing as before: list all the equally likely possibilities of children, but this time include the days of the week they are born on. These are the possible combinations:

When the first child is a Tuesday Boy and the second is a girl born on any day of the week, there are seven different possibilities.

When the first child is a girl born on any day of the week and the second is a Tuesday Boy, there are another seven different possibilities.

When the first child is a Tuesday Boy and the second is a boy born on any day of the week, again there are seven different possibilities.

Finally, there is the situation in which the first child is a boy born on any day of the week and the second child is a Tuesday Boy - there are seven different possibilities but one of them - when both boys are born on a Tuesday - has already been counted when we considered the first to be a Tuesday Boy and the second to be a boy born on any day of the week. So, there are six possibilities here.

Add them up and there are 27 different equally likely combinations of children with a given gender and day of birth. And 13 of these combinations are two boys. So the answer is 13/27.

But …

The puzzle was first put to More or Less listeners by the writer Alex Bellos and that's how he explains the answer. And Gary Foshee, who put the puzzle to the Gathering 4 Gardner delegates, agrees.

But not all our listeners think that's right. For example Andrew Coulson and Steve Morris both point out that the answer all depends on how you find the parent with a son born on a Tuesday.

This is how Steve suggests we think about it. Imagine we have a massive hall full of people sitting in rows. We first ask anyone to stand up who has two children. Then we ask people to remain standing if at least one of their children is a boy. Then we ask them to remain standing if one of their boys is born on a Tuesday. Out of those people, 13 out of 27 will have two boys.

So far, so good.

But imagine we do the same thing again: ask people to stand up if they have two children, ask them to stay standing if at least one of their children is a boy - but now, instead of asking whether one of their children is a boy born on a Tuesday, we randomly pick one of the people standing and ask them to name the day of the week that one of their sons was born.

That person can say: I have two children, one of whom is a boy born on this particular day of the week. But now, knowing the day of the week doesn't change the probability from where we started, before we introduced the additional information about the day of the week. It's still 1/3.

So what does Gary Foshee make of that?

"There is definitely an argument to be made based on choice. My solution was based on set theory. Look at the entire set of all families with two children. Then look at a subset: those with two boys. Then look at another subset: those with a boy born on Tuesday. If you look at it that way, then 13/27 is the correct answer.

"If you start putting in factors about how the children were chosen, from which set, then yes there is an argument the answer could be different. It's a very tricky and controversial subject."