RE: [math-fun] Re: Error in Scientific American math puzzle.
-----Original Message----- From: math-fun-bounces+andy.latto=pobox.com@mailman.xmission.com On Behalf Of James Propp Sent: Monday, June 12, 2006 11:11 PM To: math-fun@mailman.xmission.com Subject: [math-fun] Re: Error in Scientific American math puzzle.
The way I usually teach this topic in introductory probability classes is that (under assumptions that need to be made explicit but are very reasonable), the question
"A family has two children, at least one of whom is a boy; what is the probability that both are boys?"
and
"A family has two children, at least one of whom is a boy named Bartholomew; what is the probability that both are boys?"
have different answers.
That's a really nice version of the puzzle. The (very slight) flaw is that in the second case, the answer is not "1/2", but "very slightly less than 1/2". It's important that you present the problem with an unusual name, like Bartholemew, since the more common the name, the greater the deviation from 1/2. Since I only know of one person named Bartholemew (and he has two sisters, Maggie and Lisa, so he doesn't affect the odds in this problem), the approximation is pretty good in this case. Andy Latto andy.latto@pobox.com
I learned this gimmick (giving the boy a name) from Charles Grinstead or Peter Doyle (I'm not sure which).
Jim Propp
"A family has two children, at least one of whom is a boy; what is the probability that both are boys?"
and
"A family has two children, at least one of whom is a boy named Bartholomew; what is the probability that both are boys?"
have different answers.
It depends a great deal on how you extracted the information in the latter case. If you found this out in answer to the question, "Do you have a boy named Bart?" you are quite correct. If, however, you found this out in answer to the question, "Tell me the name of one of just one of your boy(s)," (to which the answer is Bart) then I assert you've been given no additional information.
In your posted analysis, those families with two boys, one of whom is named Bart, only half of them would answer Bart to the second question, while all would answer yes to the first. The problem here is what assumptions you are likely to make having heard the underspecified problem. When you hear, "A family has two children, at least one of whom is a boy," one tends to think that you are getting the answer to, "Do you have at least one boy?" and not "What is the sex of one of your children?" Because Bart is such an unusual name, it's more natural to think that you are receiving the answer to the question, "What is the name of one of your boy(s)?" In short, these questions are notoriously difficult not so much because the probability is hard, but because it's hard to pin down the hidden assumptions. David
participants (2)
-
Andy Latto -
David Wolfe