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

Analyzing categorical data
ANSWERED 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? 2021-06-26

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