Question

Assertion :If the probability of an event $A=0.2$ & the probability of the event B is 0.3, then the probability of neither A nor B occurs depends upon the fact that A & B are mutually exclusive or not. Reason: Two events A & B are mutually exclusive if they do not have common elements between them.

• 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 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 B Both Assertion & Reason are individually true but Reason is not the correct explanation of Assertion

If events are mutually exclusive, then $A\cap B=\varphi$
$P(A\cap B)=0$
Again $P(\overline{A}\cap \overline{B})=P\left(\overline{A\cup B}\right)=1-P(A\cup B)$
$\therefore P(\overline{A}\cap \overline{B})=1-\{P\left(A\right)+P\left(B\right)\}(\because P(A\cap B)=0)$
$=1-(0.2+0.3)$ $=0.5$ .....( i )
and if events are independent, then $P(A\cap B)=P\left(A\right)\cdot P\left(B\right)$
$\therefore P(\overline{A}\cap \overline{B})=P\left(\overline{A}\right)P\left(\overline{B}\right)$
$=\{1-P\left(A\right)\}\{1-P\left(B\right)\}$
$=(1-0.2)(1-0.3)$
$=0.8\times 0.7$ $=0.56$......( ii )
So the probability of neither A nor B occurs depends upon the fact that A
& B are mutually exclusive or not. Hence Assertion (A) & Reason
(R) both are correct but Reason (R) is not the proper explanation of Assertion (A).