Let ${E_1} =$ Event that item is manufactured on $A$,
${E_2} =$ Event that an item is manufactured on $B$,
${E_3} =$ Event that an item is manufactured on $C$,
Let $E$ be the event that an item is defective.
$\therefore$ $\quad P\left( {{E_1}} \right) = \frac{{50}}{{100}} = \frac{1}{2},P\left( {{E_2}} \right) = \frac{{30}}{{100}} = \frac{3}{{10}}$ and $P\left( {{E_3}} \right) = \frac{{20}}{{100}} = \frac{1}{5}$
$P\left( {\frac{E}{{{E_1}}}} \right) = \frac{2}{{100}} = \frac{1}{{50}},P\left( {\frac{E}{{{E_2}}}} \right) = \frac{2}{{100}} = \frac{1}{{50}}$ and $P\left( {\frac{E}{{{E_3}}}} \right) = \frac{3}{{100}}$
$\therefore$ $P\left( {\frac{{{E_1}}}{E}} \right) = \frac{{P\left( {{E_1}} \right) \cdot P\left( {\frac{E}{{{E_1}}}} \right)}}{{P\left( {{E_1}} \right) \cdot P\left( {\frac{E}{{{E_1}}}} \right) + P\left( {{E_2}} \right) \cdot P\left( {\frac{E}{{{E_2}}}} \right) + P\left( {{E_3}} \right) \cdot P\left( {\frac{E}{{{E_3}}}} \right)}}$
$= \frac{{\frac{1}{2} \cdot \frac{1}{{50}}}}{{\frac{1}{2} \cdot \frac{1}{{50}} + \frac{3}{{10}} \cdot \frac{1}{{50}} + \frac{1}{5} \cdot \frac{3}{{100}}}}$
$= \frac{{\frac{1}{{100}}}}{{\frac{1}{{100}} + \frac{3}{{500}} + \frac{3}{{500}}}} = \frac{{\frac{1}{{100}}}}{{\frac{{5 + 3 + 3}}{{500}}}} = \frac{5}{{11}}$