Statistics And Probability Question 5
Question: Assume that each born child is equally likely to be a boy or a girl. If a family has two children, then the conditional probabilities that both are girls given that (i) the youngest is a girl, (ii) at least one is a girl are
Options:
A) $ \frac{1}{2}and\frac{1}{4} $
B) $ \frac{1}{3}and\frac{1}{2} $
C) $ \frac{1}{3}and\frac{1}{4} $
D) $ \frac{1}{2}and\frac{1}{3} $
Show Answer
Answer:
Correct Answer: A
Solution:
- [a] Let b and g represent the boy and the girl child, respectively, if a family has two children, the sample space will be $ S={(b,b),(b,g),(g,b),(g,g)} $ Let A be the event that both children are girls. Therefore, $ A={(g,g)} $ (i) Let B be the event that the youngest child is a girl. Therefore, $ B=[(g,g),(g,g)] $
$ \Rightarrow A\cap B={(g,g)} $
$ \therefore P(B)=\frac{2}{4}=\frac{1}{2} $ and $ P(A\cap B)=\frac{1}{4} $ The conditional probability that both are girls, given that the youngest child is a girl, is given by $ P(A/B) $ $ P(A/B)=\frac{P(A\cap B)}{P(B)}=\frac{1/4}{1/2}=\frac{1}{2} $ (ii) Let C be the event that at least one child is a girl. Therefore, $ C={(b,g),(g,b),(g,g)} $
$ \Rightarrow A\cap C={g,g}\Rightarrow P(C)=\frac{3}{4} $ and $ P(A\cap C)=\frac{1}{4} $ The conditional probability that both are girls, given that at least one child is a girl, is given by $ P(A|C). $ Therefore, $ P(A|C)=\frac{P(A\cap C)}{P(C)}=\frac{1/4}{3/4}=\frac{1}{3} $ .
BETA
