Think you understand Monty Hall? Try the Tuesday boy problem.

Doesn't this rest on the simple ambiguity in the phrasing?

> I have two children and one is a son born on a Tuesday.

If by that is meant:

> I have two children. Here is some information about one of them: son, born on Tuesday.

Then the probability of the other child being a son is 1/2.

If on the other hand we mean:

> I have two children. One or more is a son. Exactly one of them was born on a Tuesday.

Then we get the 13/27 probability.

In fact it doesn't seem reasonable at all to assume that only one was born on Tuesday, while at least one is a son. One single interpretation of 'one of them' must be applied to both the gender and day of birth. Otherwise we're picking and choosing our interpretation on a whim.

edit: Colin appears* to think that what I've said here is incorrect, and I'd like to know why. I'm not a maths/stats person at all so am very keen to be re-educated on this matter.

* based on his now-deleted reply to ars which said "no, it doesn't, and no, you're not"

Important Edit Two:

If I'm reading this right, I think the defender of the 13/27 solution would say:

No. We don't discount the possibility of both being tuesday-boys (TB), we just adjust the calculation so that it doesn't count (eldest=TB, youngest=TB) and (youngest=TB, eldest=TB) as two separate possibilities.

To which I respond:

Right, so it's not down to ambiguity. But shouldn't you also discount every other symmetrical pair such as (eldest=TB, youngest=WB) and (youngest=TB, eldest=WB) and thus return the odds to 1/2? Or does that not return the odds to 1/2?

To answer the question posed by the title: no, I didn't think I understood Monty Hall, but thanks to this article I'm pretty sure I do now. This is a good explanation and analysis.

    Let's try a simpler problem. Suppose we know that a 
    certain man has two children and we also know that the
    older one is a boy. In this case we would say that the
    probability that the other child is a boy is 1/2. After
    all, the sex of one child is independent of the sex of
    the other child. That the older child is a boy has no
    bearing on the sex of the younger child.

    Now suppose we know simply that a man has two children
    and that one of them is a son. This time we would reason
    that there is no possibility that the person has two
    girls. It follows that the sexes of his two children,
    ordered from oldest to youngest, are either BB, BG or GB.
    Since these cases are equally likely, and since only one
    of them involves having two boys, we would say the
    probability that the man has two boys is 1/3.

Maybe I'm missing something, but this seems fundamentally wrong. Pr(two boys | older child = boy) is not equal to Pr(two boys | one+ child = boy)? Why is time so special? Could we not order them based on their height, and assert Pr(two boys | taller child = boy) = 1/2. Or, if there were a scale of masculinity, order them based on that and assert Pr(two boys | manlier child = boy) = 1/2, where manlier child = boy one+ child = boy, so Pr(two boys | manlier child = boy) = Pr(two boys | one+ child = boy) (contradiction).

I stopped caring about 'paradoxes' ever since I understood there's no isomorphism between pure mathematical concepts and human language.