Final exan Problem 5. [10 points] There are 3 children of different ages. What is the probability that at least two are boys, given that at least one of the two youngest children is a boy? Assume that each child is equally likely to be a boy or a girl and that their genders are mutually independent. A correct answer alone is sufficient. However, to be eligible for partial credit, you must include a clearly-labeled tree diagram Solution. Let M be the event that there are at least two boys and let y be the event that at least one of the two youngest children is a boy. In the tree diagram below, all edge probabilities are 1 /2 B B B<GB 11111 22222222 youngest oldest M Y Prob Pr(My Pr(M∩Y)Final Exam 7 Problem 5. [10 points] There are 3 children of different ages. What is the probability that at least two are boys, given that at least one of the two youngest children is a boy? Assume that each child is equally likely to be a boy or a girl and that their genders are mutually independent. A correct answer alone is sufficient. However, to be eligible for partial credit, you must include a clearly­labeled tree diagram. Solution. Let M be the event that there are at least two boys, and let Y be the event that at least one of the two youngest children is a boy. In the tree diagram below, all edge probabilities are 1/2. �� � HH B H B G × × × × 1/2 1/2





@@ @ HHH B G B G × × × 1/2 1/2 A A A A A A�� � HHH G B B G × × × 1/2 1/2 @ @ @ HHH G B G 1/2 1/2 youngest oldest M Y Prob Pr (M | Y ) = Pr (M ∩ Y ) Pr (Y ) 1/2 = 3/4 = 2/3