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

$R$: Roundness conforms to specifications
$F$: Surface finish conforms to specifications
The given information can be summarized as:
$P(R\text{ conforms })=1$
$P(F\text{ conforms }|R\text{ conforms })=\frac{345}{345+128}$
$P(F\text{ conforms }|R\text{ does not conform })=\frac{128}{128+345}$
(a) We need to find the probability that surface finish conforms to specifications given that roundness conforms to specifications. Mathematically, we want to calculate $P(F\text{ conforms }|R\text{ conforms })$.
Using the provided values, we have:
$P(F\text{ conforms }|R\text{ conforms })=\frac{345}{345+128}$
(b) We need to find the probability that surface finish conforms to specifications given that roundness does not conform to specifications. Mathematically, we want to calculate $P(F\text{ conforms }|R\text{ does not conform })$.
Using the provided values, we have:
$P(F\text{ conforms }|R\text{ does not conform })=\frac{128}{128+345}$
Therefore, the solutions are:
(a) The probability that a shaft conforms to surface finish requirements given that it conforms to roundness requirements is $\frac{345}{345+128}$.
(b) The probability that a shaft conforms to surface finish requirements given that it does not conform to roundness requirements is $\frac{128}{128+345}$.

Step 1:
(a) Let's denote the event that a shaft conforms to roundness requirements as $R$ and the event that it conforms to surface finish requirements as $S$. We want to find the conditional probability $P(S|R)$.
According to the given information, we have the following data:
Roundness: $R=\text{yes}$
Surface finish: $S=\text{yes}$ with a count of 345 conforming shafts and $S=\text{no}$ with a count of 128 non-conforming shafts.
Using the conditional probability formula, we can calculate $P(S|R)$ as follows:
$P(S|R)=\frac{P(S\cap R)}{P(R)}$
The joint probability $P(S\cap R)$ represents the probability of a shaft conforming to both roundness and surface finish requirements. From the given information, we don't have the exact count for this joint event, so we can't determine its probability directly. However, we can make an assumption of independence between roundness and surface finish requirements for the sake of calculation. This assumption implies that the probability of a shaft conforming to surface finish requirements is the same, regardless of whether it conforms to roundness requirements.
Under this assumption, we can write $P(S\cap R)$ as the product of the individual probabilities: $P(S\cap R)=P(S)·P(R)$.
The probability $P(R)$ represents the proportion of shafts conforming to roundness requirements, which is given as 1 since we know that a shaft conforms to roundness requirements.
Substituting these values into the conditional probability formula, we get:
$P(S|R)=\frac{P(S)·P(R)}{P(R)}=P(S)=\frac{345}{345+128}$
Therefore, the probability that a shaft conforms to surface finish requirements given that it conforms to roundness requirements is $\frac{345}{345+128}$.
Step 2:
(b) In this case, we want to find the conditional probability $P(S|\neg R)$, where $\neg R$ denotes the event that a shaft does not conform to roundness requirements.
Using the same conditional probability formula as before, we have:
$P(S|\neg R)=\frac{P(S\cap \neg R)}{P(\neg R)}$
The joint probability $P(S\cap \neg R)$ represents the probability of a shaft conforming to surface finish requirements but not conforming to roundness requirements. Unfortunately, the given information does not provide us with the count or probability for this specific joint event, so we cannot determine it directly.
However, we can assume independence between roundness and surface finish requirements for the purpose of calculation, just as in the previous case. Under this assumption, we can write $P(S\cap \neg R)$ as the product of the individual probabilities: $P(S\cap \neg R)=P(S)·P(\neg R)$.
The probability $P(\neg R)$ represents the proportion of shafts not conforming to roundness requirements, which is given as $\frac{128}{345+128}$.
Substituting these values into the conditional probability formula, we get:
$P(S|\neg R)=\frac{P(S)·P(\neg R)}{P(\neg R)}=P(S)=\frac{345}{345+128}$
Therefore, the probability that a shaft conforms to surface finish requirements given that it does not conform to roundness requirements is also $\frac{345}{345+128}$.

Answer:
(a) $P(S|R)=\frac{345}{473}$
(b) $P(S|\neg R)=\frac{345}{473}$
Explanation:
To solve the given problem, let's define the following events:
- R: The shaft conforms to roundness requirements.
- S: The shaft conforms to surface finish requirements.
The information given can be summarized in a table as follows:
(a) We need to find the probability that a shaft conforms to surface finish requirements given that it conforms to roundness requirements, i.e., we need to find $P(S|R)$.
From the given table, we know that there are 345 shafts that conform to both roundness and surface finish requirements. We also know that the total number of shafts that conform to roundness requirements is 345 + 128 = 473.
Therefore, we can calculate $P(S|R)$ as:
$P(S|R)=\frac{\text{Number of shafts conforming to both roundness and surface finish requirements}}{\text{Number of shafts conforming to roundness requirements}}=\frac{345}{473}$
(b) We need to find the probability that a shaft conforms to surface finish requirements given that it does not conform to roundness requirements, i.e., we need to find $P(S|\neg R)$.
From the given table, we know that there are 128 shafts that do not conform to roundness requirements. We also know that the total number of shafts that do not conform to roundness requirements is 128 + 345 = 473.
Therefore, we can calculate $P(S|\neg R)$ as:
$P(S|\neg R)=\frac{\text{Number of shafts conforming to both roundness and surface finish requirements}}{\text{Number of shafts not conforming to roundness requirements}}=\frac{345}{473}$