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 \(\displaystyle{P}{\left({S}{\mid}{R}\right)}=\frac{{345}}{{{345}+{12}}}={\frac{{{115}}}{{{119}}}}\sim{0.9664}\).

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