Suppose the friend just says “One of them is a boy.” and I ask, “What day of the week is he born?” and I get a day, do I have more data for the problem??? The boy is certainly born a day of the week! (Except if he was born on Joshua’s missing day!)

So, I don’t see how this is different from the classic probability problem, to which the answer is 1/3 (i.e., one of 3 possible cases boy-girl, girl-boy, boy-boy). ]]>

If I have a probability distribution in my head of what a person might say in different situations, I can calculate the final answer. If my friend says “I have a son born on a Tuesday” and nothing else, I might think that the other child is very probably a daughter, because otherwise he would have probably said: “One of my sons is born on a Tuesday”.

]]>One thought – if you assume each of the three ways in which the information could have been obtained has the same probability (the default in the light of no information?), shouldn’t you end up with a final, overall probability for the man being a BB? ]]>

Your analysis on https://blog.tanyakhovanova.com/?p=221 was really great. It articulated much better than I was able to the problem with the standard 13/27 answer and why it might seem unintuitive (because it is sometimes not the right answer). I had a look at other blogs/discussions and yours was the first that properly made sense. Much obliged.

Francis

]]>The one-boy fathers are very over represented in the group who volunteer “Tuesday” for the day a son was born. Half of the two-boy fathers with a son born on Tuesday will volunteer the birthday of the other son.

13/27 doesn’t apply, therefore. ]]>

The 13/27 answer, though perfectly correct, is the answer to a different problem from the one stated.

It is the probability within the “Boy born on Tuesday” subset.

We are asked for the probability within the whole set (2 children, one of which is a boy), not the probability within the subset “2 children, one of which is a boy born on a specific day of the week”

]]>So 0 of 98 fathers with two daughters will say “I have a son born on Tuesday.” 14 of 98 fathers with a son and a daughter will have a son born on Tuesday but only 7 of them will say it, and by the same logic 7 of 98 fathers with a daughter and a son will make the statement. Of 98 fathers with two sons, 26 will have at least one son born on a Tuesday. 2 of those must say “I have a son born on a Tuesday” as that describes both children. Of the remaining 24, 12 of them will talk about the son born on Tuesday while the other 12 will talk about the son born on a different day.

Therefore, of 392 fathers of two children, 28 will make the statement “I have a son born on Tuesday.” Of these, 14 have a son and a daughter, and the other 14 have two sons.

So with no information about why the father made this unsolicited statement, the answer must be 1/2, as Misha points out.

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