Question

If events $A$ and $B$ are independent and $P\left(A\right)=0.4,P(A\cup B)=0.6$, then $P\left(B\right)=$

• A. $\frac{1}{3}$
• B. $\frac{1}{4}$
• C. $\frac{2}{3}$
• D. None of these

Answer 1

Answer: (A)

$\frac{1}{3}$

Given $A$ and $B$ are independent events, $P\left(A\right)=0.4,P(A\cup B)=0.6$. We have to find $P\left(B\right)$.

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

We know that, $P(A\cup B)=P\left(A\right)+P\left(B\right)-P(A\cap B)$

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

Substitute the values we get

$0.6=0.4+P\left(B\right)-0.4\times P\left(B\right)$

$\Rightarrow 0.6-0.4=(1-0.4)P\left(B\right)$

$\Rightarrow 0.6P\left(B\right)=0.2$

$\Rightarrow P\left(B\right)=\frac{0.2}{0.6}=\frac{2}{6}=\frac{1}{3}$

That is $P\left(B\right)=\frac{1}{3}$