Question

Assertion :If A & B are two events such that $P\left(A\right)=\frac{2}{5}$, $P\left(B\right)=\frac{3}{4}$ then $\frac{1}{20}\le P(A\cap B)\le \frac{2}{5}$. Reason: $P(A\cup B)\le max\{P\left(A\right),P\left(B\right)\}$ & $P(A\cap B)\ge min\{P\left(A\right),P\left(B\right)\}$.

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

Answer 1

The correct option is C Assertion is true but Reason is false

We have $P(A\cup B)\ge max\{P\left(A\right),P\left(B\right)\}=max\{\frac{2}{5},\frac{3}{4}\}$
$\Rightarrow P(A\cup B)\ge \frac{3}{4}$
$\Rightarrow P(A\cap B)=P\left(A\right)+P\left(B\right)-P(A\cup B)$
$\ge P\left(A\right)+P\left(B\right)-1$
$=\frac{2}{5}+\frac{3}{4}-1=\frac{3}{20}$

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