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QUESTION
Large batches of components are delivered to two factories $A$
and $B.$ Each batch is subjected to an acceptance sampling scheme
as follows:
Factory $A$: Accept the batch if a random sample of 10 components
contains less than 2 defectives. Otherwise reject the batch.
Factory $B$:Take a random sample of 5 components. Accept the
batch if this sample contains no defectives. Reject the batch
if this sample contains 2 or more defectives. If the sample
contains 1 defective, take a further sample of 5 and accept the
batch if this sample contains no defectives.
If the fraction defective in the batch is $p,$ find the
probabilities of accepting a batch under each scheme.
Write down an expression for the average number sampled in
factory $B$ and find its maximum value.
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ANSWER
$n=10,$ $p$=proportion defective.\\
$A$: accept if less than 2 defectives.
$P$(accept)$=q^{10}=10q^9p=(1-p)^9(1+9p)$
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Accept} \put(3.8,3.2){$(1-p)^5$}
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$P$(accept)$=(1-p)^5+5p(1-p)^9$
Expected number sampled$=5+5\times 5p(1-p)^4=E\\
\frac{\partial E}{\partial p}=-25p(1-p)^3\times 4 +25(1-p)^4=0$
when $4p=1-p\ \ p=0.2$
$E=5+5\times 0.8^4=7.048$ (check
$\frac{\partial^2 E}{\partial p^2}\leq 0$ when $p=0.2$).
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