Question

Let A and B be two events such that $P(A\cup B)\ge 3/4$ and $1/8\le P(A\cap B)\le 3/8$.
Statement 1: $P\left(A\right)+P\left(B\right)\ge 7/8$
statement 2: $P\left(A\right)+P\left(B\right)\le 11/8$

• A. Both the statements are TRUE and STATEMENT 2 is the correct explanation of STATEMENT 1.
• B. Both the statements are TRUE and STATEMENT 2 is not the correct explanation of STATEMENT 1.
• C. STATEMENT 1 is TRUE and STATEMENT 2 is FALSE.
• D. STATEMENT 1 is FALSE and STATEMENT 2 is TRUE.

Answer 1

The correct option is B Both the statements are TRUE and STATEMENT 2 is not the correct explanation of STATEMENT 1.

$P(A\cup B)=P\left(A\right)+P\left(B\right)-P(A\cap B)\because \frac{3}{4}\le P(A\cup B)\le 1\Rightarrow \frac{3}{4}\le P\left(A\right)+P\left(B\right)-P(A\cap B)\le 1\Rightarrow \frac{3}{4}+P(A\cap B)\le P\left(A\right)+P\left(B\right)\le 1+P(A\cap B)\Rightarrow \frac{3}{4}+\frac{1}{8}\le P\left(A\right)+P\left(B\right)\le 1+\frac{3}{8}\Rightarrow \frac{7}{8}\le P\left(A\right)+P\left(B\right)\le \frac{11}{8}$
both the statements are correct but statement $1$ is not the correct explanation of statement $2$