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QUESTION

\begin{description}

\item[(a)]
Components produced by a certain manufacturing process have a 4\%
failure rate, the distribution of failures being random. To detect
the failures a screening test is devised. This test picks out 95\%
of the failed components, but unfortunately also picks out 2\% of
the components which are perfectly produced.

\begin{description}

\item[(i)]
Find the percentage of components that are picked out by the
screening test.

\item[(ii)]
Determine the probability that a component which is picked out by
the screening test is a faulty one.

\end{description}

\end{description}



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ANSWER

\begin{description}

\item[(a)]
From the given information,


$p$(failure) $=p(F)=0.04,$

$p$(perfect) $=p$(not fail) $=p(NF)=1-0.04=0.96$

$p$(positive test$|F)=0.95,$

$p$(positive test$|NF)=0.02$


\begin{description}

\item[(i)]

$p$(positive test)

$=p$(positive test$|F)p(F)+p$(positive test$|NF)p(NF)$

$=(0.95)(0.04)+(0.02)(0.96)=0.0572$

i.e. the test picks out 5.72\% of components.

\item[(ii)]
We want
\begin{eqnarray*}
p(F|\textrm{pos. test})&=&\frac{p(F\textrm{ and pos.
test})}{p(\textrm{pos. test})}\\ &=&\frac{p(\textrm{pos.
test}|F)p(F)}{p(\textrm{pos. test})}\\
&=&\frac{(0.95)(0.04)}{(0.0572)}\\ &=&0.664 \end{eqnarray*}

\end{description}

\end{description}




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