Question

Assertion : Two events A and B are such that $P\left(A\right)=\frac{1}{3}$ and $P\left(B\right)=\frac{2}{3}$ then $P(A\cup B)\ge \frac{2}{3}$
Reason : If $P\left(B\right)>P\left(A\right)$, then $P(A\cup B)\ge P\left(B\right)$

• A. Both assertion and reason are true and reason is the correct explanation of assertion.
• B. Both assertion and reason are 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 A Both assertion and reason are true and reason is the correct explanation of assertion.

We know that, P(B) = P(A), then,
$P\left(B\right)\le P(A\cup B)\le 1$ and $0\le P(A\cap B)\le P\left(A\right)$
Now $P\left(B\right)=\frac{2}{3},P\left(A\right)=\frac{1}{3}$
$\therefore P\left(B\right)>P\left(A\right)\Rightarrow P(A\cup B)\ge P\left(B\right)$
$\Rightarrow P(A\cup B)\ge \frac{2}{3}$
So, assertion and reason both are true and reason is the correct explanation of assertion.