Events $A$ and $B$ are such that $P(A) = \frac{1}{2}, P(B) = \frac{7}{12}$ and $P(\overline{A} \cup \overline{B}) = \frac{1}{4}$. Find whether the events $A$ and $B$ are independent or not.
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → $A$ and $B$ are not independent ##
Given $P(A) = \frac{1}{2}, P(B) = \frac{7}{12}$ and $P(\overline{A} \cup \overline{B}) = \frac{1}{4}$
For $A$ and $B$ are independent:
$P(A \cap B) = P(A) \cdot P(B) \quad \dots(i)$
Now, $P(\overline{A} \cup \overline{B}) = P(\overline{A \cap B})$
$\Rightarrow P(\overline{A} \cup \overline{B}) = 1 - P(A \cap B)$
$\Rightarrow P(A \cap B) = 1 - P(\overline{A} \cup \overline{B})$
$\Rightarrow P(A \cap B) = 1 - \frac{1}{4} = \frac{3}{4} \quad \dots(ii)$
Now, $P(A) \cdot P(B) = \frac{1}{2} \times \frac{7}{12} = \frac{7}{24} \quad \dots(iii)$
Since, from eqs. $(ii)$ & $(iii)$:
$P(A \cap B) \neq P(A) \cdot P(B)$
Therefore, events $A$ and $B$ are not independent.