The panels for the Pulsar 32-inch TV are manufactured in three locations and then shipped to the main
plant for final assembly. Plants A, B, and C supply 50%, 30%, and 20%, respectively of the panels.The
quality-control department of the company has determined that 1% of the panels poduced by Plant A are
defective, and 2% of the panels produced by Plants B and C are defective. If a Pulsar 32-inch TV is
selected at random and the panel is found to be defective, what is the probability that the panel was
manufactured in Plant C?
To solve this problem, we can use Bayes' Theorem. Bayes' Theorem allows us to calculate the probability of an event given some other related information.
Let's define the following probabilities:
- P(A): Probability that the panel is from Plant A.
- P(B): Probability that the panel is from Plant B.
- P(C): Probability that the panel is from Plant C.
- P(D|A): Probability that the panel is defective given that it is from Plant A.
- P(D|B): Probability that the panel is defective given that it is from Plant B.
- P(D|C): Probability that the panel is defective given that it is from Plant C.
According to the problem, we know the following information:
- P(A) = 0.50 (Plant A supplies 50% of the panels)
- P(B) = 0.30 (Plant B supplies 30% of the panels)
- P(C) = 0.20 (Plant C supplies 20% of the panels)
- P(D|A) = 0.01 (1% of Plant A panels are defective)
- P(D|B) = 0.02 (2% of Plant B panels are defective)
- P(D|C) = 0.02 (2% of Plant C panels are defective)
We want to find P(C|D), which is the probability that the panel is from Plant C given that it is defective.
We can use Bayes' Theorem:
P(C|D) = (P(D|C) * P(C)) / (P(D|A) * P(A) + P(D|B) * P(B) + P(D|C) * P(C))
Let's plug in the values:
P(C|D) = (0.02 * 0.20) / (0.01 * 0.50 + 0.02 * 0.30 + 0.02 * 0.20)
P(C|D) = 0.004 / (0.005 + 0.006 + 0.004)
P(C|D) = 0.004 / 0.015
P(C|D) = 0.2667
Therefore, the probability that the panel was manufactured in Plant C given that it is defective is approximately 0.2667 or 26.67%.