Question

If $A, B$ are two events such that $P\left(A\cup B\right) \ge \frac{3}{4}$ and $\frac{1}{8}\le P\left(A\cap B\right) \le\frac{3}{8}$ then

• A. $P\left(A\right)+P\left(B\right)\le\frac{11}{8}$
• B. $P\left(A\right).P\left(B\right)\le\frac{3}{8}$
• C. $P\left(A\right)+P\left(B\right)\ge\frac{7}{8}$
• D. $None\, of \, these$

Answer 1

Answer: (C)

$P\left(A\right)+P\left(B\right)\ge\frac{7}{8}$

Answer 2

$P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)-P\left(A\cap B\right)$
$\Rightarrow P\left(A\right)+P\left(B\right)=P\left(A\cup B\right)+P\left(A\cap B\right)$
$\frac{3}{4}\le P\left(A\cup B\right)\le1$
$\frac{1}{8}\le P\left(A\cap B\right)\le\frac{3}{8}$
so, $\frac{7}{8}\le P\left(A\cup B\right)+P\left(A\cap B\right)\le\frac{11}{8}$
so, $P\left(A\right)+P\left(B\right)\ge\frac{7}{8} \left(C\right)$
$P\left(A\right)+P\left(B\right)\le\frac{11}{8} \left(A\right)$