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QUESTION

 An engineering works receives supplies of a certain component
   from three different factories, 30\% from factory $A,$ 60\% from
   factory $B$ and the remainder from factory $C.$ Past experience has
   shown the percentage defective produced by the factories $A, B$
   and $C$ are 1\%,2\% and 3\%respectively. A random sample of 100
   components all from the same unknown factory are examined and 3
   defectives are found. Find approximately the probability that
   the sample came from factory $A.$

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ANSWER

\begin{tabular}{cccc}
    &A&B&C\\
    \% supplied&30&60&10\\
    \%defective&1 &2 &3\\
    Poisson $\mu$&1&2&3
   \end{tabular}

   Given 100 components, if $x$\% are defective where $x$ is small, the
   number of components which are defective is $P(x)$. Since we
   have three defectives we need to find $P(3).$

   \begin{eqnarray*}
    A:P(3)&=&\frac{e^{-1}1^3}{3!}=0.061\\
    B:P(3)&=&\frac{e^{-2}2^3}{3!}=0.180\\
    C:P(3)&=&\frac{e^{-3}3^3}{3!}=0.224\\
   \end{eqnarray*}

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   $P(3 \textrm{ defectives})=0.3\times 0.61+0.6 \times 0.180+0.1
   \times 0.224=0.1487$\\
   $P(A|3 \textrm{ defectives}\ )=\frac{0.3\times 0.061}{0.1487}=0.123$


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