GMAT Focus EditionQuantitative ReasoningHard
A manufacturing plant has two machines, A and B, that produce identical parts. Machine A produces 60% of the total parts, and Machine B produces the remaining 40%. The defect rate for Machine A is 3%, and for Machine B is 5%. If a randomly selected part is found to be defective, what is the probability that it was produced by Machine A?
- A0.45
- B0.30
- C0.36
- D0.50
Show answer & explanationAnswer & explanation
Correct answer: C. 0.36
This is a Bayes' Theorem problem. First, calculate the total probability of a part being defective: P(Defective) = P(Defective|A)P(A) + P(Defective|B)P(B) = (0.03 * 0.60) + (0.05 * 0.40) = 0.018 + 0.020 = 0.038. Then, P(A|Defective) = P(Defective|A)P(A) / P(Defective) = 0.018 / 0.038 = 18/38 = 9/19 ≈ 0.4737.
Why the other options are wrong
- A. Incorrect application of Bayes' Theorem.
- B. This is P(Defective|A) * P(A) = 0.018, not the conditional probability P(A|Defective).
- D. Incorrect application of Bayes' Theorem.
Bayes' Theorem
Bayes' Theorem describes the probability of an event, based on prior knowledge of conditions that might be related to the event. It calculates conditional probability.
- P(A|B) = [P(B|A) * P(A)] / P(B)
- P(B) is often calculated using the law of total probability.
- Used to update beliefs based on new evidence.
Memory trick: Prior Beliefs Updated by Evidence.