Question

Let $A$ and $B$ be two events such that $P\left(\overline{A\cup B}\right)=\frac{1}{6}, P\left(\overline{A \cap B}\right)=\frac{1}{4}$ and $P\left(\bar{A}\right)=\frac{1}{4}$ where $\overline{A}$ 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)

independent but not equally likely

Answer 2

$P(\overline{A \cup B})=\frac{1}{6}$
$\Rightarrow P(A \cup B)=1-\frac{1}{6}=\frac{5}{6}$
$P(\bar{A})=\frac{1}{4}$
$\Rightarrow P(A)=1-\frac{1}{4}=\frac{3}{4}$
$\because P(A \cup B)=P(A)+P(B)-P(A \cap B)$
$\frac{5}{6}=\frac{3}{4}+P(B)-\frac{1}{4}$
$P(B)=\frac{1}{3}$
$\because P(A) \neq P(B)$ so they are not equally likely
Also $P(A) \times P(B)=\frac{3}{4} \times \frac{1}{3}=\frac{1}{4}$
$=P(A \cap B)$
$\because\, P(A \cap B)=P(A) \cdot P(B)$ so A & B are independent