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Question
Solve the following:
A machine produces parts that are either good (90%), slightly defective (2%), or obviously defective (8%). Produced parts get passed through an automatic inspection machine, which is able to detect any part that is obviously defective and discard it. What is the quality of the parts that make it throught the inspection machine and get shipped?
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Solution
Let event G: The event that machine produces a good part,
Event S: The event that machine produces a slightly defective part,
Event D: The event that machine produces an obviously defective part.
P(G) = `90/100` = 0.90, P(S) = `2/100 = 0.02`, P(D) = `8/100 = 0.08`
Let Dc = G ∪ S. Then
P(G/Dc) = `("P"("G" ∩ "D"^"c"))/("P"("D"^"c"))`
`= ("P"("G"))/("P"("G" ∪ "S"))` ...[∵ G ∩ (G ∪ S) = G]
`= ("P"("G"))/(("P"("G") + "P"("S"))` ...[∵ G and S are disjoint sets]
`= 0.90/(0.90 + 0.02)`
`= 0.90/(0.92)`
`= 90/92 = 45/46`
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