Question

Let $A$ and $B$ be independent events with $P\left(A\right)=0.3$ and $P\left(B\right)=0.4$. Find
(i) $P(A\cap B)$
(ii) $P(A\cup B)$
(iii) $P\left(A|B\right)$
(iv) $P\left(B|A\right)$

Answer 1

It is given that $P\left(A\right)=0.3$ and $P\left(B\right)=0.4$
(i) If $A$ and $B$ are independent events, then
$P(A\cap B)=P\left(A\right)\cdot P\left(B\right)=0.3\times 0.4=0.12$
(ii) It is known that, $P(A\cup B)=P\left(A\right)+P\left(B\right)-P(A\cap B)$
$\Rightarrow P(A\cup B)=0.3+0.4-0.12=0.58$
(iii) It is known that, $P\left(A|B\right)=\frac{P(A\cap B)}{P\left(B\right)}$
$\Rightarrow P\left(A|B\right)=\frac{0.12}{0.4}=0.3$
(iv) It is known that, $P\left(B|A\right)=\frac{P(A\cap B)}{P\left(A\right)}$
$\Rightarrow P\left(B|A\right)=\frac{0.12}{0.3}=0.4$