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# [Phys-L] two boy problems

• From: Carl Mungan <mungan@usna.edu>
• Date: Tue, 28 Oct 2014 09:05:30 -0400

We've discussed problems similar to the following one before. (Look up
"two aces" in the archives.) If you think there are ambiguities in how
I have worded things below, feel free to reword them. The issue I am
interested in is not the semantics, but how additional information
affects probabilities p.

Consider all two-child families in the world such that X. What
fraction of those families have two boys?

Substitute X with one of the following statements:
case A) "at least one child is a boy"
case B) "at least one child is a boy born on a Tuesday"
case C) "the older child is a boy"
case D) "the older child is a boy born on a Tuesday"

(Assume equal probability that any baby is a boy as a girl, and equal
probability that a baby is born on any day of the week.)

Hypothesis: Providing additional positive information that is relevant
will increase the fraction. (By "positive" I mean you're not giving me
information about girls in families rather than boys. By "relevant" I
mean you're not for example telling me "It rained on March 21, 2008 in
Knoxville.")

What I find is that that p(B) > p(A) and p(C) > p(A) which both
support the hypothesis. However, I also find p(C) = p(D) which
contradicts it. So apparently (and surprisingly to me) the hypothesis
is false.

Feel free to think about what the fractions (probabilities) are before
you peek at this solution:

http://usna.edu/Users/physics/mungan/_files/documents/Scholarship/ProblemOfTwoBoys.pdf

Please feel free to comment and challenge anything I've said here. I'd
like to clarify better ways to think about these kinds of probability
problems. -Carl

--
Carl E Mungan, Assoc Prof of Physics 410-293-6680 (O) -3729 (F)
Naval Academy Stop 9c, 572C Holloway Rd, Annapolis MD 21402-1363
mailto:mungan@usna.edu http://usna.edu/Users/physics/mungan/