If $P(A) = \cfrac{6}{{11}},\;P(B) = \cfrac{5}{{11}}\;{\rm{and}}\;P(A \cup B) = \cfrac{7}{{11}}$ , find

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

(ii) $P(A|B)$

(iii) $P(B|A)$

.: (i) Given, $P(A \cup B) = \cfrac{7}{{11}}$

$\Rightarrow$ $P(A) + P(B) - P(A \cap B) = \cfrac{7}{{11}}$

$\Rightarrow$ $\Rightarrow \cfrac{6}{{11}} + \cfrac{5}{{11}} - P(A \cap B) = \cfrac{7}{{11}}$

$\Rightarrow P(A \cap B) = \cfrac{6}{{11}} + \cfrac{5}{{11}} - \cfrac{7}{{11}} = \cfrac{4}{{11}}$ $P(A \cap B) = \cfrac{6}{{11}} + \cfrac{5}{{11}} - \cfrac{7}{{11}} = \cfrac{4}{{11}}$

(ii) $P(A/B) = \cfrac{{P(A \cap B)}}{{P(B)}} = \cfrac{{4/11}}{{5/11}} = \cfrac{4}{5}$

(iii) $P(B/A) = \cfrac{{P(A \cap B)}}{{P(A)}} = \cfrac{{4/11}}{{6/11}} = \cfrac{4}{6} = \cfrac{2}{3}$