Question

Let $A$ and $B$ be two events such that $P ( B \mid A )=\frac{2}{5}, P ( A \mid B )=\frac{1}{7}$ and $P ( A \cap B )=\frac{1}{9}$. Consider (S1) $P \left( A ^{\prime} \cup B \right)=\frac{5}{6}$, (S2) $P \left( A ^{\prime} \cap B ^{\prime}\right)=\frac{1}{18}$. Then

• A. Both (S1) and (S2) are true
• B. Both (S1) and (S2) are false
• C. Only (S1) is true
• D. Only (S2) is true

Answer 1

Answer: (A)

$P ( A \mid B )=\frac{1}{7} \Rightarrow \frac{ P ( A \cap B )}{ P ( B )}=\frac{1}{7}$
$ \Rightarrow P ( B )=\frac{7}{9}$
$P ( B \mid A )=\frac{2}{5} \Rightarrow \frac{ P ( A \cap B )}{ P ( A )}=\frac{2}{5}$
$ \Rightarrow P ( A )=\frac{5}{18} $
$\text { Now, } P \left( A ^{\prime} \cup B \right)=1- P ( A \cup B )+ P ( B )$
$ =1- P ( A )+ P ( A \cap B )=\frac{5}{6}$
$ P \left( A ^{\prime} \cap B ^{\prime}\right)=1- P ( A \cup B ) $
$ =1- P ( A )- P ( B )+ P ( A \cap B )=\frac{1}{18} $
$ \Rightarrow $ Both (S1) and (S2) are true