Question

A and B are two events such that $P\left(A\right)=0.5,$ $P\left(A\cup \overline{B}\right)=0.7$. If A and B are independent events $P\left(B\right)=$

• A. 1.5
• B. 2.5
• C. $\frac{3}{5}$
• D. $\frac{4}{5}$

Answer 1

The correct option is C $\frac{3}{5}$

Given that $A$ and $B$ are independent events, then

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

Given:$P\left(A\right)=0.5,P(A\cup \overline{B})=0.7$

$\Rightarrow 0.7=0.5+P\left(\overline{B}\right)-P\left(A\right)P\left(\overline{B}\right)$ since $A$ and $B$ are independent events

$\Rightarrow 0.7-0.5=P\left(\overline{B}\right)(1-P\left(A\right))$

$\Rightarrow 0.2=P\left(\overline{B}\right)(1-0.5)$

$\Rightarrow 0.2=0.5\times P\left(\overline{B}\right)$

$\Rightarrow P\left(\overline{B}\right)=\frac{0.2}{0.5}\times \frac{10}{10}=\frac{2}{5}$

$\Rightarrow P\left(B\right)=1-P\left(\overline{B}\right)=1-\frac{2}{5}=\frac{5-2}{5}=\frac{3}{5}$