Q.64 An automobile plant contracted to buy shock absorbers from two suppliers X and Y. X supplies
60% and Y supplies 40% of the shock absorbers. All shock absorbers are subjected to a quality test.
The ones that pass the quality test are considered reliable. Of X’s shock absorbers, 96% are reliable.
Of Y’s shock absorbers, 72% are reliable.
The probability that a randomly chosen shock absorber, which is found to be reliable, is made by Y
is
(A) 0.288 (B) 0.334 (C) 0.667 (D) 0.720
Probability That a Reliable Shock Absorber Is Supplied by Y
An automobile plant purchases shock absorbers from two suppliers, X and Y. All shock absorbers
undergo a quality test, and those that pass are considered reliable. Using conditional probability
and Bayes’ theorem, we determine the probability that a reliable shock absorber is supplied by Y.
Problem Overview
- Supplier X supplies 60% of shock absorbers
- Supplier Y supplies 40% of shock absorbers
- Reliability of X’s shock absorbers: 96%
- Reliability of Y’s shock absorbers: 72%
Question: If a randomly chosen shock absorber is found to be reliable,
what is the probability that it was made by supplier Y?
Concept Used: Bayes’ Theorem
Bayes’ theorem helps calculate conditional probabilities:
P(Y | R) = [P(R | Y) × P(Y)] / P(R)
Step-by-Step Solution
Step 1: Define Probabilities
- P(X) = 0.60
- P(Y) = 0.40
- P(R | X) = 0.96
- P(R | Y) = 0.72
Step 2: Calculate Total Probability of Reliability
P(R) = P(R | X) × P(X) + P(R | Y) × P(Y)
P(R) = (0.96 × 0.60) + (0.72 × 0.40)
P(R) = 0.576 + 0.288 = 0.864
Step 3: Apply Bayes’ Theorem
P(Y | R) = (0.72 × 0.40) / 0.864
P(Y | R) = 0.288 / 0.864 = 0.3333 ≈ 0.334
Correct Answer
Option (B) 0.334
Final Conclusion
Given that a shock absorber is reliable, the probability that it was manufactured by supplier Y is:
0.334
