Question

Assertion :Let A and B be two events that $P(A\cap B)\ge 3/4$ and $1/8\le P(A\cap B)\ge 3/8$
$P\left(A\right)+P\left(B\right)\ge 7/8$ Reason: $P\left(A\right)+P\left(B\right)\le 11/8$

• 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. Both Assertion and Reason are incorrect

Answer 1

The correct option is D Both Assertion and Reason are incorrect

$P(A\cup B)=P\left(A\right)+P\left(B\right)-P(A\cup B)$

$\therefore 1\ge P\left(A\right)+P\left(B\right)-P(A\cap B)\ge 3/4$

$\Rightarrow P\left(A\right)+P\left(B\right)-1/8\ge 3/4$

[since minimum value of $P(A\cap B)]$ is 1/8]

$\Rightarrow P\left(A\right)+P\left(B\right)\le 1/8+3/4=7/8$

As the maximum value of $P(A\cap B)$ is 3.8, we get

$1\ge P\left(A\right)+P\left(B\right)-3/8$

$\Rightarrow P\left(A\right)+P\left(B\right)\le 1+3/8=11/8$