# An item is manufactured by three machines A,B and C.Out of the total number of items manufactured during a specified period,50% are manufactured on A,30% on B and 20% on C.2% of the items produced on A and 2% of items produced on B are defective,and 3% of these produced on C are defective.All the items are stored at one godown.One item is drawn at random and is found to be defective.What is the probability that it was manufactured on machine A?

\$\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}  ## 1 Answer Toolbox: • \(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)\)=
$$\huge\frac{\frac{1}{2}\times\frac{2}{100}}{\frac{1}{2}\times\frac{2}{100}+\frac{3}{10}\times\frac{2}{100}+\frac{1}{5}\times\frac{3}{100}}$$
=$$\Large\frac{10}{22}$$
=$$\Large\frac{5}{11}$$

edited Jun 4, 2013