This is a classic problem in conditional probability. We are asked to find the probability that a part is good given that it has passed the inspection machine. Let's solve this step-by-step.

Step 1: Defining Events

• Let $G$ be the event that the part is good.
• Let $S$ be the event that the part is slightly defective.
• Let $O$ be the event that the part is obviously defective.
• Let $P$ be the event that the part passes the inspection machine.

We are given:

• $P(G) = 0.90$ (90% of the parts are good)
• $P(S) = 0.02$ (2% of the parts are slightly defective)
• $P(O) = 0.08$ (8% of the parts are obviously defective)

The inspection machine detects and discards all obviously defective parts, so a part will pass the inspection only if it is either good or slightly defective.

Step 2: Finding $P(P)$ (Probability that a part passes inspection)

A part will pass the inspection if it is not obviously defective, which means it is either good or slightly defective. Therefore:

$P(P) = P(G) + P(S) = 0.90 + 0.02 = 0.92$

Step 3: Applying Conditional Probability Formula

We are interested in the probability that the part is good given that it has passed the inspection, which is represented as $P(G|P)$. Using the conditional probability formula:

$P(G|P) = \frac{P(G \cap P)}{P(P)}$

Since all good parts pass the inspection, $P(G \cap P) = P(G) = 0.90$. So:

$P(G|P) = \frac{0.90}{0.92} \approx 0.978$

Final Answer:

The probability that a part is good given that it passed the inspection is 0.978.

Thus, the correct answer is 0.978.

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Would you like more details or have any questions?

Here are 5 related questions:

1. What would the probability be if slightly defective parts were also discarded?
2. How would the result change if the machine had a 5% chance of failing to detect obviously defective parts?
3. How do we calculate the probability if there were more categories for defective parts?
4. How would the solution change if good parts had a small chance of being rejected?
5. Can you extend this problem to more than one inspection stage?

Tip: Always double-check conditional probabilities by reviewing how the conditions influence the entire sample space.