Define events

\(R=\{\text{a shaft meets roundness conformance}\}\)

\(S=\{\text{a shaft meets surface finish conformance}\}\)

(a)We are required to find the probability of S given R. Using the naive definition of conditional probability, it is equal \(P(S|R)=\frac{345}{345+12}=\frac{115}{119} \sim 0.9664\).

(b)We are required to find the probability of S given \(R^{c}\). Using the nauve definition of conditional probability, it is equal to \(P(S|R^{c})=\frac{5}{5+8}=\frac{5}{13} \sim 0.3846\).