Question

The probability that at least one of the events $A$ and $B$ occurs is $\frac{3}{5}$. If $A$ and $B$ occur simultaneously with probability $\frac{1}{5}$ then $P\left(A^{\prime }\right)+P\left(B^{\prime }\right)$ is

• A. $\frac{2}{5}$
• B. $\frac{4}{5}$
• C. $\frac{6}{5}$
• D. $\frac{7}{5}$

Answer 1

Answer: (C)

$\frac{6}{5}$

$P(A\cup B)=\frac{3}{5}andP(A\cap B)=\frac{1}{5}P(A\cup B)=P\left(A\right)+P\left(B\right)-P(A\cap B)\Rightarrow P\left(A\right)+P\left(B\right)=P(A\cup B)+P(A\cap B)=\frac{3}{5}+\frac{1}{5}=\frac{4}{5}\Rightarrow 1-P\left(\overline{A}\right)+1-P\left(\overline{B}\right)=\frac{4}{5}\Rightarrow P\left(\overline{A}\right)+P\left(\overline{B}\right)=2-\frac{4}{5}=\frac{6}{5}\Rightarrow P\left(A^c\right)+P\left(B^c\right)=\frac{6}{5}$