Question

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

• A. mutually exclusive and independent
• B. equally likely but not independent
• C. independent but not equally likely
• D. independent and equally likely

Answer 1

Answer: (D)

independent and equally likely

Given $P\left(\overline{A\cup B}\right)=\frac{1}{6},P(A\cap B)=\frac{1}{4}$ and $P\left(\overline{A}\right)=\frac{1}{4}$

$P\left(A\cup B\right)=\frac{5}{6},P\left(A\right)=\frac{3}{4}$

$P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)-P(A\cap B)=\frac{5}{6}$

$P\left(B\right)=\frac{5}{6}-\frac{3}{4}+\frac{1}{4}=\frac{1}{3}$

$P(A\cap B)=P\left(A\right).P\left(B\right)$
$\frac{1}{4}=\frac{3}{4}\times \frac{1}{3}$

So, $A$ and $B$ are independent but not equally likely