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 gender and an equal chance to giving birth on any day.

This is the classic conditional probability puzzle. The sample space is all 2-child families where at least one child is a boy born on Tuesday. There are 27 equally likely possibilities (7x7 - 1 + 7 for the double-counted BB-Tue/BB-Tue case). Of these, 13 have two boys (BB-Tue/BB-Tue + 12 others where one is BB-Tue and the other is a boy on a different day). Thus P(2 boys | at least 1 boy born Tuesday) = 13/27.