Question

A machine produces items that are rated as High (A), Medium (B) or Poor (C) in quality. If the machine is properly calibrated, it produces 40% of high quality items and 50% of medium quality items. The remaining are poor quality items. If the machine is NOT properly calibrated, it produces 20% of high quality items and 40% of medium quality items. The remaining are poor quality items. The machine is calibrated 80% of the time.

a. Given the item is of medium quality, what is the probability that the machine is properly calibrated?
b. You tested 3 items and found that two are of medium quality and one is of poor quality. What is the probability that the machine is not well calibrated?

Answer 1

To find the probability that the machine is calibrated given a medium-quality item, one must apply the Bayesian theorem. When testing items for quality, binomial probability formulas are used to determine the probability of a certain outcome given the machine's calibration status.

Step-by-step explanation:

The question requires the use of conditional probabilities and the Bayesian theorem to find the likelihood that a machine is properly calibrated given certain outcomes. For part (a), we have the probability that items are medium quality given the machine is properly calibrated, P(B|A), and the probability the machine is calibrated, P(A). The probability of an item being medium quality, P(B), comes from the sum of the probability of being medium quality and calibrated, P(B|A)P(A), and the probability of being medium quality and not calibrated, P(B|not A)P(not A). The probability that the machine is properly calibrated given that an item is of medium quality is therefore P(A|B) = (P(B|A)P(A)) / P(B).

For part (b), we use a binomial probability formula and the given probabilities to calculate the likelihood that all three items' quality ratings occur given the machine is not well calibrated, P(outcome|not A). This involves calculating the probability of getting two medium-quality and one poor-quality item.