Question 1019571: In the process of producing engine valves, the valves are subjected to a first grind. Valves whose thicknesses are within the specification are ready for installation. Those valves whose thicknesses are above the specification are reground, while those whose thicknesses are below the specification are scrapped. Assume that after the first grind, 65% of the valves meet the specification, 17% are reground, and 18% are scrapped. Furthermore, assume that of those valves that are reground, 90% meet the specification, and 10% are scrapped. Given that a valve is not reground, what is the probability that it is scrapped?
Answer by mathmate(429)   (Show Source):
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Question:
In the process of producing engine valves, the valves are subjected to a first grind. Valves whose thicknesses are within the specification are ready for installation. Those valves whose thicknesses are above the specification are reground, while those whose thicknesses are below the specification are scrapped. Assume that after the first grind, 65% of the valves meet the specification, 17% are reground, and 18% are scrapped. Furthermore, assume that of those valves that are reground, 90% meet the specification, and 10% are scrapped. Given that a valve is not reground, what is the probability that it is scrapped?
Solution:
First grind:
M=meet specifications, P(M)=0.65
H=above specifications, regrind, P(H)=0.17
L=below specifications, reject, P(L)=0.18
Second grind:
P(M)=0.9
P(L)=0.1
Need to find
P(L|not reground)
=P(L∩not reground)/P(not reground) [by definition of conditional probability]
=P(L)/(P(E)+P(L)) [P(H) is excluded, because this represents the reground batch]
=0.18/(0.65+0.18)
=0.217 (approximately)
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