I ask people at random if they have two children and also if one is a boy born on a Tuesday. After a long search I finally find someone who answers yes. What is the probability that this person has two boys? Assume an equal chance of giving birth to either sex and an equal chance to giving birth on any day.
There are 14 possible combinations of day and gender. So for two children there are 196 possible ways for two children.
Picture this as a big 14 by 14 matrix for first child and second child. Then one row represents the first child being a Tuesday boy and one column represents the second child being a Tuesday boy. They intersect when both children are Tuesday boys. There are then 14+14-1=27 ways.
When a person answers yes to Danish their children must be from this limited subset. Within one row or one column there are exactly seven boys possible for the other child.
We can add the row and column count of boys but then subtract one for the overlay cell. Then there are 7+7-1=13 ways the other child is also a boy.
So the probability sought is 13/27.
Solution
