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 $\overline{A}$ stands for complement of event $A$. Then $A$ and $B$ are

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

Answer 1

Answer: (B)

$P(A\cap B)=\frac{1}{4}\ne 0$
Therefore, A and B are not mutually exclusive events
Now,
$\text{ Now, }P(\overline{A\cup B})=\frac{1}{6}$
$1P(A\cup B)=\frac{1}{6}$
$P(A\cup B)=\frac{5}{6}$
$P(A\cup B)P(A)+P(B)-P(A\cap B)$
$\frac{5}{6}=\frac{3}{4}+P(B)-\frac{1}{4}$
$P(B)=\frac{5}{6}+\frac{1}{4}-\frac{3}{4}=\frac{1}{3}$
$P(A\cap B)=\frac{1}{4}$
$P(A)P(B)=\frac{3}{4}\times \frac{1}{3}=\frac{1}{4}$
So, $A$ and $B$ are independent But $P(A)\ne P(B)\Rightarrow A$ and $B$ are not equally likely.