Chapter 17 Conditional Probability732
are two children in every household and that girls and boys are equally likely to be
children and equally likely to open the door.
A sample space for this experiment has outcomes that are triples whose first
element is eitherBorGfor the sex of the elder child, whose second element is
eitherBorGfor the sex of the younger child, and whose third element isEorY
indicating whether theelder child oryounger child opened the door. For example,
.B;G;Y/is the outcome that the elder child is a boy, the younger child is a girl, and
the girl opened the door.
(a)LetObe the event that a girl opened the door, and letTbe the event that the
household has two girls. List the outcomes inOandT.
(b)What is the probability PrT jO, that both children are girls, given that a
girl opened the door?
(c)What mistake is made in the following argument? (Note: merely stating the
correct probability is not an explanation of the mistake.)
If a girl opens the door, then we know that there is at least one girl in the
household. The probability that there is at least one girl is1 PrŒboth children are boysçD 1 .1=21=2/D3=4: (17.8)So,PrT jthere is at least one girl in the household(17.9)
D
PrŒT\there is at least one girl in the householdç
PrŒthere is at least one girl in the householdç(17.10)
D
PrŒTç
PrŒthere is at least one girl in the householdç(17.11)
D.1=4/=.3=4/D1=3: (17.12)
Therefore, given that a girl opened the door, the probability that there
are two girls in the household is 1/3.Problem 17.21.
A guard is going to release exactly two of the three prisoners, Sauron, Voldemort,
and Bunny Foo Foo, and he’s equally likely to release any set of two prisoners.
(a)What is the probability that Voldemort will be released?