Question

Assertion :Odd in favour of an event A are $2:1$ & odd in favour of $A\cup B$ are $3:1$ then $\frac{1}{12}\le P\left(B\right)\le \frac{3}{4}$. Reason: If $A\cap B\subset A$ then $P(A\cap B)\le P\left(A\right)$

• 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 A Both Assertion & Reason are individually true & Reason is correct explanation of Assertion

Given odd in favour of $A$ is $2:1$
$\therefore P\left(A\right)=\frac{2}{3}$
Odd in favour of $A\cup Bis3:1$
$\therefore P(A\cup B)=\frac{3}{4}$
Now $\therefore P(A\cup B)=P\left(A\right)+P\left(B\right)-P(A\cap B)$
$\Rightarrow \frac{3}{4}=\frac{2}{3}+P\left(B\right)-P(A\cap B)$
$\Rightarrow P(A\cap B)=P\left(B\right)+\frac{2}{3}-\frac{3}{4}$
$\Rightarrow P(A\cap B)=P\left(B\right)-\frac{1}{12}$ ....( i )
$\Rightarrow P\left(B\right)\ge \frac{1}{12}$ ......( * )
Again $P(A\cap B)\le P\left(A\right)$
$\Rightarrow P\left(B\right)-\frac{1}{12}\le \frac{2}{3}$ Using (i)
$\Rightarrow P\left(B\right)\le \frac{2}{3}+\frac{1}{12}=\frac{3}{4}$ ...........( * * )
By (*) & (* *) we have
$\frac{1}{12}\le P\left(B\right)\le \frac{3}{4}$