[zaik]

If you accept the Bayes theorem, the answer is that the likelihood of "At least one boy is born on a Tuesday" is not the same for different numbers of boys. The more boys the more likely the statement is true. Therefore this information is indicative of how many boys Mrs. Chance has.

[jammaloo]

It seems to me that it comes down to how the day of the week was picked.

If they picked a random day of the week, and there was only one boy, then there is only a 1/7 chance of a boy being born on that day.

If they have one boy, who was born on a Tuesday, and that is why they picked the day, then there is a 100% chance of a boy being born on that day, so no additional information is conferred.

[fenomas]

Puzzles like that one have always seemed dishonest to me. It only makes sense if you start from the conclusion that it's meant to illustrate Bayes rule, and then work backwards to the assumption that the predicate "boy born on a Tuesday" is supposed to be independent of who's being asked about.

But in plain English, ".. At least one of them is a boy born on Tuesday" suggests the speaker is giving a fact that was chosen because it's true of the person spoken about - like if the kids were both chosen on Thursday then that would be the day named. And read that way, the Bayes illustration doesn't stand and the "correct" answer makes no sense.

To make it honest, it should really be worded like: "Mrs. Chance has two children of different ages. You ask whether at least one of them is a boy born on Tuesday, and you are told yes. What is the probability that both of them are boys?" Or am I missing something?

[kgwgk]

> If you accept the Bayes theorem

That doesn’t make a lot of sense. A theorem is just a theorem. It’s proved, and in this case the proof is trivial.

The question is whether you accept that the description of the problem in terms of conditional probabilities is adequate, and then whether you accept that the values assigned to those conditional probabilities are appropriate.