Consider the case of a manufacturer who has an automatic machine that produces an important part. Past records indicate that at the beginning of the data the machine is set up correctly 70 percent of the time. Past experience also shows that if the machine is set up correctly it will produce good parts 90 percent of the time. If it is set up incorrectly, it will produce good parts 40 percent of the time. Since the machine will produce 60 percent bad parts, the manufacturer is considering using a testing procedure. If the machine is set up and produces a good part, what is the revised probability that it is set up correctly?
Determine our events:
P[correctly set & part ok] = P(C) * P(C|G)
P[correctly set & part ok] = 70% * 90% = 63%
P[correctly set & part ok] = P(I) * P(I|G)
P[incorrectly set & part ok] = 30% *40% = 12%
P[correctly set | part ok] = P[correctly set & part ok]/(P[correctly set & part ok] + P[incorrectly set & part ok])
P[correctly set | part ok] = 63/(63+12) = 0.84 or 84%
Determine our events:
- C = Correctly Set Machine = 0.7
- C|G = Correctly Set Machine And Good Part = 0.9
- I = Incorrectly Set Machine = 1 - 0.7 = 0.3
- I|G = Incorrectly Set Machine And Good Part = 0.4
- B< = BAD PARTS = 0.60
P[correctly set & part ok] = P(C) * P(C|G)
P[correctly set & part ok] = 70% * 90% = 63%
P[correctly set & part ok] = P(I) * P(I|G)
P[incorrectly set & part ok] = 30% *40% = 12%
P[correctly set | part ok] = P[correctly set & part ok]/(P[correctly set & part ok] + P[incorrectly set & part ok])
P[correctly set | part ok] = 63/(63+12) = 0.84 or 84%