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Argument by Gary Foshee

"My solution was based on set theory. Look at the entire set of all families with two children. Then look at a subset: those with two boys. Then look at another subset: those with a boy born on Tuesday. If you look at it that way, then 13/27 is the correct answer."

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13/27 is the correct probability if we got the answer "I have two children, one of whom is a son born on a Tuesday" by dialing random numbers from the phone book and asking "could you confirm if you have to children, one of whom is a son born on Tuesday?" Which might or might not be equivalent to the actual data generation process.

That's one way to look at it, but not the only one. Look at the entire set of children. Then the subset of those born on Tuesday. Then the subset who have exactly one sibling. In that case the probability is 50%.

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