If $P(A) = 0.8,P(B) = 0.5$ and $P(B|A) = 0.4$ , find

(i) $P(A \cap B)$

(ii) $P(A|B)$

(iii) $P(A \cup B)$

.: Given, $P(B|A) = 0.4 \Rightarrow \cfrac{{P(A \cap B)}}{{P(A)}} = 0.4$

$\Rightarrow P(A \cap B) = P(A) \times 0.4 = 0.8 \times 0.4 = 0.32$

(ii) $P(A|B) = \cfrac{{P(A \cap B)}}{{P(B)}} = \cfrac{{0.32}}{{0.5}} = \cfrac{{16}}{{25}} = 0.64$

(iii) $P(A \cup B) = P(A) + P(B) - P(A \cap B)$

$= 0.8 + 0.5 - 0.32 = 1.3 - 0.32 = 0.98$