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.