$\begin{array}{1 1}(A)\;\large\frac{5}{11}\\(B)\;\large\frac{6}{11}\\(C)\;\large\frac{3}{11}\\(D)\;\large\frac{9}{11}\end{array} $

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- \(E_1=item\;is\;manufactured\;by\;machin\;A\)
- \(E_2==item\;is\;manufactured\;by\;machin\;B\)
- \(E_3=item\;is\;manufactured\;by\;machin\;c\)
- A=item is defective
- \(P(A/E_1)=P(item\;found\;defective/item\;is\;manufactured\;by\;machin\;A\))
- P(E\(_2\)/A)=\(\Large\frac{{P(E_2)}{P(A/E_2)}}{{P(E_1)}{P(A/E_1)}+{P(E_2)}{P(A/E_2)}}\)
- \(since\; 50%\;of\; item\; are\; manufactured\; by\; A\)

\(P(E-1)\)=\(\Large\frac{50}{100}\)=\(\Large\frac{1}{2}\)

= \(30%\;of\; item\; are\; manufactured\; by\; B\)

\(P(E_2)\)=\(\Large\frac{30}{100}\)=\(\Large\frac{3}{10}\)

= \(20%\;of\; item\; are\; manufactured\; by\; c\)

\(P(E_1)\)=\(P(one\; in\; thousand \;having\; T\;B\))

=\(\Large\frac{1}{1000}\)

\(p(E_3)=1-\Large\frac{20}{100}=\frac{1}{5}\)

\(since\; 2% \;produced\; by\; A\; machin\; is\; defective\)

P(A/E\(_1)\)=\(\Large\frac{2}{100}\)

\(since\; 2%\; produced\; by\; B \;machin\; is\; defective\)

\(P(A/E_2)\)=\(\Large\frac{2}{100}\)

\(since\; 3%\; produced\; by\; C\; machin\; is\; defective\)

\(P(A/E_3)\)=\(\Large\frac{23}{100}\)

\(P(E_1\)A)\)=

\(\huge\frac{\frac{1}{2}\times\frac{2}{100}}{\frac{1}{2}\times\frac{2}{100}+\frac{3}{10}\times\frac{2}{100}+\frac{1}{5}\times\frac{3}{100}}\)

=\(\Large\frac{10}{22}\)

=\(\Large\frac{5}{11}\)

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