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