Question

$A$ and $B$ are independent events. The probability that both $A$ and $B$ occur is $\frac{1}{20}$ and the probability that neither of them occurs is $\frac{3}{5}$. The probability of occurence of $A$ is

• A. $\frac{1}{2}$
• B. $\frac{1}{10}$
• C. $\frac{1}{4}$
• D. $\frac{1}{5}$

Answer 1

Answer: (C), (D)

The correct options are
C $\frac{1}{4}$
D $\frac{1}{5}$

$P(A\cap B)=P\left(A\right)P\left(B\right)=\frac{1}{20}\Rightarrow P\left(B\right)=\frac{1}{20P\left(A\right)}P(\overline{A}\cap \overline{B})=\frac{3}{5}=1-P(A\cup B)\Rightarrow \frac{3}{5}=1-P\left(A\right)-P\left(B\right)+P(A\cap B)\Rightarrow \frac{3}{5}=1-P\left(A\right)-\frac{1}{20P\left(A\right)}+\frac{1}{20}\Rightarrow \frac{3}{5}=\frac{21}{20}-P\left(A\right)-\frac{1}{20P\left(A\right)}\Rightarrow \frac{12-21}{20}=-P\left(A\right)-\frac{1}{20P\left(A\right)}$
Let $P\left(A\right)=x$
$\Rightarrow \frac{-9}{20}=\frac{-20x^2-1}{20x}\Rightarrow 20x^2-9x+1=0\Rightarrow (4x-1)(5x-1)=0\Rightarrow x=\frac{1}{4},\frac{1}{5}i.e.P\left(A\right)=\frac{1}{4},P\left(A\right)=\frac{1}{5}$