Question

Assertion (A) : If $A\& B$ are two events such that $P\left(A\right) = \frac{2}{5} P\left(B\right) = \frac{3}{4}$ then $\frac{3}{20} \le P\left(A \cap B\right) \le \frac{2}{5}$
Reason (R) : $P\left(A \cup B\right) \le$ max $\left\{P\left(A\right) P\left(B\right)\right\}\& P\left(A \cap B\right) \ge$ min $\left\{P\left(A\right), P\left(B\right)\right\}$

• A. Both (A) & (R) are individually true & (R) is correct explanation of (A).
• B. Both (A) & (R) are individually true but (R) is not the correct (proper) explanation of (A).
• C. (A) is true but (R) is false.
• D. (A) is false but (R) is true.

Answer 1

Answer: (C)

$P\left(A \cup B\right) \ge$ max $\left\{P\left(A\right) P\left(B\right)\right\}=$ max $\left\{\frac{2}{5}, \frac{3}{4}\right\}$
$\Rightarrow P\left(A \cup B\right) \ge \frac{3}{4}$
Since $P\left(A \cap B\right) =P\left(A\right) +P\left(B\right) -P\left(A \cup B\right) \ge P\left(A\right) +P\left(B\right) -1 = \frac{2}{5} +\frac{3}{4} -1 = \frac{3}{20}$ ....(i)
and $P\left(A \cap B\right) \le$ min $\left\{P\left(A\right), P\left(B\right)\right\} =$ min $\left\{\frac{2}{5}, \frac{3}{4}\right\}$
$\Rightarrow P\left(A \cap B\right) \le\frac{2}{5}$ ...(ii)
$\therefore $ From (i) &(ii), we have $\frac{3}{20} \le P\left(A \cap B\right) \le \frac{2}{5}$
$\Rightarrow $ Assertion (A) is true but Reason is false.