# If P (A) = 0.8, P (B) = 0.5 and P(B|A) = 0.4, find $\begin{array} ((i) P(A \cap B) \quad & (ii) P(A|B) \quad & (iii) P(A ∪ B) \end{array}$

Toolbox:
• $$p(B/A)\;=\;\large \frac{p(A\;\cap\;B)}{p(A)}$$ $$\Rightarrow\;p(A\;\cap\;B)=p(B/A)\;p(A)$$
• $$\;p(A\;\cup\;B)\;=\;p(A)\;+\;p(B)\;-\;p(A\;\cap\;B)\;$$
• $$p(A/B)\;=\;\large \frac{p(A\;\cap\;B)}{p(B)}$$
Given $$p(A)=0.8\;p(B)=0.5\;p(B/A)=0.4$$
$\Rightarrow$ $$p(A\;\cap\;B)=0.4\;\times\;0.8$$ = $$0.32$$
$$p(A/B)=\large \frac{p(A\;\cap\;B)}{p(B)}$$ $\rightarrow$ $$=\large\frac{0.32}{0.5}=\frac{32}{50}$$$$=0.64$$
$$\;p(A\;\cup\;B)\;=\;p(A)\;+\;p(B)\;-\;p(A\;\cap\;B)\;$$ $\rightarrow$ $$p(A\;\cup\;B)=0.8\;+\;0.5\;-\;0.32$$ =$$0.98$$
edited Jun 18, 2013