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.

$A$ and $B$ are not independent

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.