Question

If $A$ and $B$ are two independent events such that $P\left(A^{\prime }\right)=0.7,P\left(B^{\prime }\right)=p$ and $P(A\cup B)=0.8,$ then the value of $p$ is

• A. $1$
• B. $0.2$
• C. $0.7$
• D. $0.3$
  

Answer 1

The correct option is D $0.3$

$P\left(A^{\prime }\right)=0.7\Rightarrow P\left(A\right)=1-0.7=0.3$

$P\left(B\right)=1-p$

$P(A\cup B)=0.8$

$P(A\cap B)=P\left(A\right)\times P\left(B\right)=0.3(1-p)$ as they are independent events

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

$0.8=0.3+1-p-0.3(1-p)$

$0.8=0.3+1-p-0.3+0.3p$

$0.8=1-0.7p\Rightarrow 0.7p=0.2$

$p=\frac{2}{7}\approx 0.3$