Question

Let A and B be two events such that $P(A\cup B)=P(A\cap B)$. Then,
Statement 1: $P(A\cap \overline{B})=P(\overline{A}\cap B)=0$
Statement 2: If $P\left(A\right)+P\left(B\right)=1$.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is C Assertion is correct but Reason is incorrect

We have,
$P(A\cup B)=P(A\cap B)$
$\Rightarrow P\left(A\right)+P\left(B\right)-P(A\cap B)=P(A\cap B)$ ...(i)
$\Rightarrow \left\{P\right(A)-P(A\cap B\left)\right\}+\left\{P\right(B)-P(A\cap B\left)\right\}=0$
$\Rightarrow P(A\cap \overline{B})+P(\overline{A}\cap B)=0$
$\Rightarrow P(A\cap \overline{B})=P(\overline{A}\cap B)=0$
So, statement 1 is true.
From (i), we have
$P\left(A\right)+P\left(B\right)=2P(A\cap B)$
So, $P\left(A\right)+P\left(B\right)=1$ may not be true.