Puzzling Again
Another look at TSM's puzzle:
I don't doubt or misunderstand his answer (see previous post, or his posting). The question is, why do we find that answer so counterintuitive. I think that it is because there is some ambiguity in the question. Consider a different but similar seeming problem:
What is the probability that Mike's sibling is a sister? Is this like TSM's puzzle, or like my Ace-Jack puzzle in my previous post?
Aaron Bergman has posted a comment pointing out that the second sentence ("We know that one of them is a boy.") of the puzzle is ambiguous. It could mean "exactly one of them is a boy," and that's actually the most likely interpretation of the sentence in isolation. In that case, the probability of a sister is 100%. Another interpretation is "at least one of them is a boy." That's the interpretation TSM uses, and, stated thusly, pretty obviously leads to the answer 2/3.
I claim that there is another, implicit interpretation, that leads to the counter intuitive feel of the answer. That interpretation is "You have learned that one of them is a boy." In that case, we are back to the situation of the Johnson kid, whose sibling is a sister with a probability of 50%.
Why so? By plausible assumption, your learning that the sex of one J kid was a male was a random event whose probability was 100% if the siblings were bb, 50% if bg or gb, and 0 if gg. Since you did first observe a boy, the probabilities of the initial distribution have to be corrected to bb 50%, bg and gb each 25%.
Your initial observation has "collapsed the wave function" (that's a joke kids).
If the problem is stated in the unambiguous and explicit way "at least one of them is a boy," the counter intuitive aspect goes bye-bye. At least for me.
A family has two children. We know that one of them is a boy. What is the probability that the other one is a girl?
I don't doubt or misunderstand his answer (see previous post, or his posting). The question is, why do we find that answer so counterintuitive. I think that it is because there is some ambiguity in the question. Consider a different but similar seeming problem:
You know that the Johnsons have two children, but you don't know their respective sexes. Your daughter comes home and says, "I talked the Johnson kid today. His name is Mike."
What is the probability that Mike's sibling is a sister? Is this like TSM's puzzle, or like my Ace-Jack puzzle in my previous post?
Aaron Bergman has posted a comment pointing out that the second sentence ("We know that one of them is a boy.") of the puzzle is ambiguous. It could mean "exactly one of them is a boy," and that's actually the most likely interpretation of the sentence in isolation. In that case, the probability of a sister is 100%. Another interpretation is "at least one of them is a boy." That's the interpretation TSM uses, and, stated thusly, pretty obviously leads to the answer 2/3.
I claim that there is another, implicit interpretation, that leads to the counter intuitive feel of the answer. That interpretation is "You have learned that one of them is a boy." In that case, we are back to the situation of the Johnson kid, whose sibling is a sister with a probability of 50%.
Why so? By plausible assumption, your learning that the sex of one J kid was a male was a random event whose probability was 100% if the siblings were bb, 50% if bg or gb, and 0 if gg. Since you did first observe a boy, the probabilities of the initial distribution have to be corrected to bb 50%, bg and gb each 25%.
Your initial observation has "collapsed the wave function" (that's a joke kids).
If the problem is stated in the unambiguous and explicit way "at least one of them is a boy," the counter intuitive aspect goes bye-bye. At least for me.
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