The Tuesday Boy Variant

"I have two children. One is a boy born on a Tuesday. What's the probability I have two boys?" This is a variant of the well-known "one is a boy, what's the probability both are boys" puzzle, and the Tuesday detail — which sounds like irrelevant noise — actually shifts the answer, which is the entire point of the puzzle.

Without the Tuesday fact, "at least one is a boy" (among four equally likely two-child combinations: BB, BG, GB, GG) leaves three cases consistent with the statement, one of which is BB, giving probability $1/3$ for two boys — not $1/2$, because "boy, boy" is only one of three surviving equally-likely cases, not one of two. Adding the day of the week changes the sample space: among 14 possible boy/day combinations per child (7 days × boy-or-girl doesn't quite apply — instead there are 14 equally likely (gender, day) pairs per child), the condition "at least one child is a boy born on Tuesday" is satisfied by more of the mixed-gender cases than you'd naively guess relative to the two-boys cases, and careful counting gives a probability of $13/27$ for two boys — closer to $1/2$ than the $1/3$ from the plain version, because the specific Tuesday detail makes it less likely that two boys were both coincidentally the one described.

The puzzle's real lesson is that these answers depend entirely on the (usually unstated) procedure that generated the sentence — if a parent was always going to mention a boy and a day whenever at least one existed, the math above applies; if the sentence was generated some other way, the answer changes again. Interviewers use it to test whether a candidate notices that the sampling procedure, not just the raw fact, determines a conditional probability.