The Tuesday Birthday Problem - why does the probability change when the father specifies the birthday of a son? [duplicate]

Solution 1:

As Jason Rosenhouse points out in the blog post to which you linked, the correct answer depends very much on the assumptions made about the sample space. Specifically, it depends on what the speaker would say if he had a different set of children. Look at the three scenarios provided by Tanya Khovanova: in one of them the correct answer is $1/2$, in another it’s $1/3$, and in the third it’s $13/27$. In particular, if you assume that the speaker was randomly chosen from the pool of all men who could honestly say ‘I have two children, and one is a son born on a Tuesday’, $13/27$ is the right answer.

If the man says simply ‘I have two children, at least one of whom is a son’, the probability that the other child is a boy again depends on the sample space $-$ on the assumptions made about how the speaker was chosen. If he was chosen at random from the pool of all men who could honestly say ‘I have two children, and one is a son born on a Tuesday’, but simply made the weaker statement ‘I have two children, at least one of whom is a son’, the correct answer is $13/27$, as before. If, however, he was chosen at random from the pool of all men who could honestly say ‘I have two children, at least one of whom is a son’, the correct answer is $1/3$. And if he was picked at random from the pool of all fathers of two children and told you the sex of one of his children picked at random, then the correct answer is $1/2$. (These are, in reverse order, Tanya Khovanova’s three scenarios, modified for the revised statement by the father.)

These are bad puzzles, in the sense that they can’t be answered without making assumptions that go beyond what’s actually stated in the problem. Thus, there really is no single right answer. Rather, there are several answers that are right for different background assumptions.

For your final question, note that even with twins you normally have an elder and a younger child, even if it’s only by a very small time interval, so the elder/younger argument still works (in the settings in which it’s the appropriate interpretation).