Question

If $P\left(A\right)=\frac{1}{8}$ and $P\left(B\right)=\frac{5}{8}$. Which of the following statements is correct?

• A. $P(A\cup B)\le \frac{3}{4}$
• B. $P(A\cap B)\le \frac{5}{8}$
• C. $P(\overline{A}\cap B)\le \frac{5}{8}$
• D. $P(A\cap B)\ge \frac{5}{8}$

Answer 1

The correct option is A $P(A\cup B)\le \frac{3}{4}$

Given $P\left(A\right)=\frac{1}{8},P\left(B\right)=\frac{5}{8}$.

We know that $P(A\cup B)=P\left(A\right)+P\left(B\right)-P(A\cap B)$

$\Rightarrow P(A\cup B)=\frac{1}{8}+\frac{5}{8}-P(A\cap B)$

$=\frac{6}{8}-P(A\cap B)$

$=\frac{3}{4}-P(A\cap B)$

Since $P(A\cup B)$ is $\frac{3}{4}-$ some positive quantity.

We can say that $P(A\cup B)\le \frac{3}{4}$

Hence statement $\left(A\right)$ is correct.