The analysis of shafts for a compressor is summarized by conformance to specifications: roundnessconforms yes no surface finish yes 3455 conforms no 1

The analysis of shafts for a compressor is summarized by conformance to specifications: roundnessconforms yes no surface finish yes 3455 conforms no 128
(a) If we know that a shaft conforms to roundness requirements, what is the probability that it conforms to surface finish requirements?
(b) If we know that a shaft does not conform to roundness requirements, what is the probability that it conforms to surface finish requirements?

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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 $$\displaystyle{P}{\left({S}{\mid}{R}^{{{c}}}\right)}=\frac{{5}}{{{5}+{8}}}={\frac{{{5}}}{{{13}}}}\sim{0.3846}$$.