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