Year: 1999
Paper: 2
Question Number: 12
Course: LFM Stats And Pure
Section: Conditional Probability
No solution available for this problem.
Difficulty Rating: 1600.0
Difficulty Comparisons: 0
Banger Rating: 1484.0
Banger Comparisons: 1
It is known that there are
three manufacturers $A, B, C,$ who can produce
micro chip MB666. The probability that a randomly selected MB666
is produced by $A$ is $2p$, and the corresponding probabilities for
$B$ and $C$ are $p$ and $1 - 3p$, respectively, where
${{0} \le p \le {1 \over 3}}.$ It is also known that $70\%$ of
MB666 micro chips from $A$ are sound and that the corresponding
percentages for $B$ and $C$ are $80\%$ and $90\%$, respectively.
Find in terms of $p$, the conditional probability, $\P(A {\vert} S)$,
that if a randomly selected
MB666 chip is found to be sound then it came from $A$, and also the
conditional probability, $\P(C {\vert} S)$, that
if it is sound then it came from $C$.
A quality inspector took a random sample
of one MB666 micro chip and found it to be sound. She then traced
its place of manufacture to be $A$, and
so estimated $p$ by calculating the value of $p$
that corresponds to the greatest value
of $\P(A {\vert} S)$. A second quality
inspector also a took random sample of one MB666 chip and
found it to be sound. Later he traced its place of manufacture
to be $C$ and so estimated $p$ by applying the procedure of his
colleague to $\P(C {\vert} S)$.
Determine the values of the two estimates and comment
briefly on the results obtained.