Bob has one sibling. What is the probability that he has a brother?

Question

I've been thinking about this for at least a solid 30 minutes. I can't understand! I was looking at the 2 kids paradox (Tom has 2 children. At least one of them is a boy. What is the probability that the other child is a girl?) and thinking it's too easy. Of course, it's 2/3! But then I started thinking about the question in the title. It seems 1/2, but I couldn't differentiate it from the other one. Maybe it's 2/3 too but I can't see it. Helps are appreciated :)

Answer 1

In these cases usually you should look if the statement provides some information that fixes one of the kids as which. In the given statement, Bob is fixed (we don't care about their gender), and now the other sibling has equal probability of being a boy/girl = $1/2$.

When you say atleast one then the information is not about any one kid. Ways of providing info: they name the kid/tell elder/younger kid/see the parent with one kid.

Answer 2

These two events should have the same probability because of symmetry

"Bob has a brother | Bob has one sibling"

"Bob has a sister | Bob has one sibling"

And they should sum up to 1, so each has a probability of 0.5

Answer 3

Tom has 2 children. At least one of them is a boy. What is the probability that the other child is a girl?

Of course, it's 2/3!

Before I explain what is up with that answer, let me offer a variation.

• I have written a gender inside a sealed envelope. Tim has 2 children, and at least one of them is that gender. What is the probability that Tim has a boy and a girl?

You know nothing about the genders of Tim's two children, so the answer must be the same as the expected proportion of mixed-gender families of two; that is, 1/2.

But were I to open the envelope, and show you that I had written "Boy," it is the same as your question about Tom. Of course, the answer is 2/3! But if I show you that it says "girl," the answer should just as obviously be "2/3", right? Since those are the only two things I could write, do I actually need to open the envelope? The fact that I wrote a gender seems to have already changed the answer from 1/2 to 2/3.

This is known as Bertrand's Box Paradox, from published by Joseph Bertrand in 1889. In modern times, that name is used for the problem itself. But Bertrand used it to describe this apparent change in probability, when no information is conveyed that could change the probability. And in fact, if we add a fourth box to Bertrand's Problem, it is identical to the one about Tim. And Bertrand's methodology says the answer is 1/2, not 2/3.

In fact, your version was first published by Martin Gardner in 1959. And, while he originally said that the answer was 2/3, he later withdrew that and said the question itself was ambiguous. Why do we know this peculiar fact?

1. If somebody had asked Tom "Is at least one of your two children a boy?", then with three of the four possible combinations he would answer "yes." Since in two of those three he has a boy and a girl, the answer is 2/3.
2. But if Tom volunteered the information, in those two cases he could have said either "at least one is a boy" or "at least one is a girl." If he chose randomly between these statements, we can only count half of the mixed-gender families as learning "at least one is a boy." This makes the answer 1/2.
3. According to Gardner, the problem is ambiguous because it does not provide a reason to pick either #1 or #2.
4. According to Bertrand, because the problem does not provide a reason to pick either #1 or #2, using any solution except #2 leads to a paradox.

I'll point out that there is another famous problem called the Monty Hall Problem where Bertrand's solution is generally accepted, although it is seldom fully explained. It is usually stated that your original chance, 1/3, can't change when you see a door opened. The truth is that it can, but only if we assume a bias. Like if Monty Hall always opens door #3 if he can, much like how Tom #1 always answers "yes" when he has a boy. It is only when we assume that the chances must be the same whichever door (other than the contestant's) he opens, that the accepted solution is correct.