Question

Let $A$ and $B$ be two events such that $P \overline{(A \cup B)}=\frac{1}{6}, P(A \cap B)=\frac{1}{4}$ and $P \overline{(A)}=\frac{1}{4}$, where $\bar{A}$ stands for the complement of the event $A$. Then the events $A$ and $B$ are

• A. Independent but not equally likely
• B. Independent and equally likely
• C. Mutually exclusive and independent
• D. Equally likely but not independent

Answer 1

Answer: (A)

$P(\overline{A \cup B})=\frac{1}{6}$
$\Rightarrow P(A \cup B)=\frac{5}{6} ; P(\bar{A})=\frac{1}{4}$
$\Rightarrow P(A)=\frac{3}{4}$
$P(A \cup B)=P(A)+P(B)-P(A \cap B)$
$\Rightarrow \frac{5}{6}=\frac{3}{4}+P(B)-\frac{1}{4}$
$\Rightarrow P(B)=\frac{5}{6}-\frac{3}{4}+\frac{1}{4}$
$=\frac{10-9+3}{12}=\frac{4}{12}=\frac{1}{3}$
$P(A) P(B)=\frac{3}{4} \times \frac{1}{3}=\frac{1}{4}=P(A \cap B)$
$\Rightarrow A, B$ are independent.
$P(A)=\frac{3}{4}, P(B)=\frac{1}{3}$
$\Rightarrow A, B$ are not equally $1$