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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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