"I have two children, one of whom is a son born on a Tuesday. What is the probability that I have two boys?"
As always, the challenge is the assumptions intentionally hidden in the problem statement.
"I" - was your family chosen at random, and if so, from what set?
"two children" - exactly or at least?
"one of whom" - exactly or at least?
"son" - was the sex to say chosen at random, or did you pick a child and announce his/her sex?
"Tuesday" - was the day chosen at random, or did you pick a child and announce his.her birthday?
"What is the probability..." - Some parent you are! Don't you know the sex of your own children?
Simply and honestly reveal the assumptions and the math is straightforward.
"Given a family, chosen at random from the set of all families that have exactly two children and have at least one son born on a Tuesday, what is the probability that both children are boys?"
To make the math easier, let's start with 196 families with two children, with the expected mix of boys and girls. 49 (25%) have two boys and 98 (50%) have a boy and a girl. Of the 98 boy-girl families, 84 do not have a Tuesday-Boy, leaving 14 that do. Of the 49 boy-boy families, 36 do not have a Tuesday-Boy, leaving 13 that do. That leaves a total of 27 families, of which 13 have at least one son born on a Tuesday.
So the probability is 13/27.
Reveal different assumptions, and the answer changes.