Quote:
Originally Posted by omk3
Which makes the overall probability.... 
...13/27?
Oh how I wish I remembered how to calculate these things... |
Yes! We have a winner! 13/27 is indeed the right answer.
I like probability trees myself. In the attached chart, the first branch (top down) is the sex of the first child, the next is whether the child (if a boy) was born on a tuesday. The third branchings are for the sex of the second child, and the last for whether that child (of a boy) was born on a Tuesday.
The probabilities across the bottom are the chance of getting to that outcome, normalised to fractions of 196 (= 2x7x2x7), and are obtained just by multiplying the probabilities of each branch that leads to that outcome.
Now, we know that Dan has at least one boy, so we can ignore the GG outcome. And we also know that at least one of his children is a boy born on a Tuesday. So we can now ignore the outcomes that don't include a Boy born on a Tuesday - that's the OBOB, OBG and GOB outcomes.
We're left with
TBTB - 1/196
TBOB - 6/196
TBG - 7/196
OBTB - 6/196
GTB - 7/196
The total probability of getting one of these outcomes of 27/196. But in only TBTB, TBOB and OBTB do we have two boys - 13/196.
So the probability of Dan having two boys, given that he has at least one boy and that Nick guessed a day correctly is 13 in 27.