Question

If $A$ and $B$ are two events such that $P(A\cup B)\ge \frac{3}{4}\text{and}\frac{1}{8}\le P(A\cap B)\le \frac{3}{8}$ 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. $P\left(A\right)+P\left(B\right)\ge \frac{9}{8}$

Answer 1

Answer: (A), (C)

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

$P(A\cup B)\ge \frac{3}{4}\text{and}\frac{1}{8}\le P(A\cap B)\le \frac{3}{8}$

$\therefore P\left(A\right)+P\left(B\right)-P(A\cap B)\ge \frac{3}{4}\Rightarrow P\left(A\right)+P\left(B\right)\ge [\frac{3}{4}+P(A\cap B\left)\right]\ge (\frac{3}{4}+\frac{1}{8})=\frac{7}{8}\cdots \left(i\right)$

Also $P(A\cup B)\le 1\Rightarrow P\left(A\right)+P\left(B\right)\le 1+P(A\cap B)\le (1+\frac{3}{8})=\frac{11}{8}\cdots \left(ii\right)$