Argh, i hate hate hate google.
The solution is quite simple. My brother solved it by a counting argument, but i think the most obvious thing is to use bayes theorem which is what i did. here goes:
The right answer is 13/27. If we neglect the information about tuesday, the answer is 1/3 - the combinations are BG, GB, BB, and GG, GG is obvisouly out of the question since he has at least one boy, which leave the 3 other combinations. Only one of those has two boyes, hence 1/3.
BUT if we get the information about the day of birth that is wrong. Why?
Imagine the man had said: I have two children, one is the kid who run around just on top of that hill, what is the probability the other is a boy to?
Obviously the answer is 1/2 here, since the chance both kids is the kid on top of the hill is zero. So intuitively, the answer is between 1/2 and 1/3.
However, the exact result is very easy to calculate. What we want to compute is this:
P(2 boys | one is a boy, born on tuesday)
By bayes theorem
P(2 boys | one is a boy, born on tuesday) = P( one is a boy, born on tuesday | 2 boys) p(2 boys) / P( one is a boy, born on tuesday)
Its now easy to compute:
P( one is a boy, born on tuesday | 2 boys) = P( one is born on tuesday) = 1-P( none is born on tuesday ) = 1-(1-1/7)^2
P( one is a boy, born on tuesday ) = 1 - (1-1/14)^2
p(2 boys) = 1/4
Which give the right result.
So here is the roundup. Albert wins the price for a correct answer. ML wins the prize for best numerical estimate, and snowbird win the prize for closest to the actual argument by giving 1/3. Duncan and StAnn wins the price of best googling skills.
John Doe and The Silence wins the "shouldnt have talked so much about my math major prize" - JD, where did you go? Am i still wrong?
and i allmost won a beer...