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# 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