Question

Let $A$and $B$ be two events such that $P\left(A\right)=0.3$ and $P(A\cup B)=0.8$. If $A$and $B$ are independent events, then $P\left(B\right)$ is

• A. $\frac{5}{6}$
• B. $\frac{5}{7}$
• C. $\frac{3}{5}$
• D. $\frac{2}{5}$

Answer 1

The correct option is B

$\frac{5}{7}$

Determine the value of $P\left(B\right)$

Let $A$and $B$ be two events.

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

since, $A$and $B$ are independent events, $P\left(A\cap B\right)=P\left(A\right).P\left(B\right)$

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

Substitute the given values

$$\begin{gathered} 0.8=0.3+P\left(B\right)-0.3\times P\left(B\right) \\ 0.8-0.3=\left(1-0.3\right)P\left(B\right) \\ 0.5=0.7P\left(B\right) \\ P\left(B\right)=\frac{0.5}{0.7} \\ P\left(B\right)=\frac{5}{7} \end{gathered}$$

Hence, option (B) is the correct answer