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Four technicians regularly make repairs when breakdowns occur on an automated production line. Janet, who services \(20 \%\) of the breakdowns, makes an incomplete repair 1 time in 20 ; Tom, who services \(60 \%\) of the breakdowns, makes an incomplete repair 1 time in 10 ; Georgia, who services \(15 \%\) of the breakdowns, makes an incomplete repair 1 time in 10 ; and Peter, who services \(5 \%\) of the breakdowns, makes an incomplete repair 1 time in 20 . For the next problem with the production line diagnosed as being due to an initial repair that was incomplete, what is the probability that this initial repair was made by Janet?
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in Data Science & Statistics by Diamond (38,951 points) | 15 views
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Let \(A\) be the event that the initial repair was incomplete, \(B_{1}\) that the initial repair was made by Janet, \(B_{2}\) that it was made by Tom, \(B_{3}\) that it was made by Georgia, and \(B_{4}\) that it was made by Peter.

Substituting the various probabilities into the formula of Theorem \(3.11\), we get
\[
\begin{aligned}
P\left(B_{1} \mid A\right) &=\frac{(0.20)(0.05)}{(0.20)(0.05)+(0.60)(0.10)+(0.15)(0.10)+(0.05)(0.05)} \\
&=0.114
\end{aligned}
\]

and it is of interest to note that although Janet makes an incomplete repair only 1 out of 20 times, namely, \(5 \%\) of the breakdowns she services, more than \(11 \%\) of the incomplete repairs are her responsibility.
by Diamond (38,951 points)
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