Model Answer
0 min readIntroduction
In modern manufacturing operations, quality control and production efficiency are paramount. Statistical analysis plays a crucial role in assessing product quality and identifying the source of defects. This question presents a scenario involving scooter production across three plants with varying production volumes and quality rates. Applying probability theory allows us to determine the overall probability of a scooter being of standard quality and, conversely, the probability that a standard-quality scooter originated from a specific plant. This is a fundamental application of statistical quality control in operations management.
Part (i): Probability of a Scooter Being of Standard Quality
Let 'S' denote the event that a scooter is of standard quality. We are given the following information:
- P(A) = Probability that a scooter comes from plant A = 500/1000 = 0.5
- P(B) = Probability that a scooter comes from plant B = 300/1000 = 0.3
- P(C) = Probability that a scooter comes from plant C = 200/1000 = 0.2
- P(S|A) = Probability that a scooter is of standard quality given it comes from plant A = 0.9
- P(S|B) = Probability that a scooter is of standard quality given it comes from plant B = 0.92
- P(S|C) = Probability that a scooter is of standard quality given it comes from plant C = 0.95
We can use the law of total probability to find P(S):
P(S) = P(S|A)P(A) + P(S|B)P(B) + P(S|C)P(C)
P(S) = (0.9 * 0.5) + (0.92 * 0.3) + (0.95 * 0.2)
P(S) = 0.45 + 0.276 + 0.19
P(S) = 0.916
Therefore, the probability that a scooter selected at random is of standard quality is 0.916 or 91.6%.
Part (ii): Probability that a Scooter Comes from Plant 'B' Given it is of Standard Quality
We need to find P(B|S), the probability that a scooter comes from plant B given that it is of standard quality. We can use Bayes' Theorem:
P(B|S) = [P(S|B) * P(B)] / P(S)
We already know:
- P(S|B) = 0.92
- P(B) = 0.3
- P(S) = 0.916 (calculated in part i)
Therefore:
P(B|S) = (0.92 * 0.3) / 0.916
P(B|S) = 0.276 / 0.916
P(B|S) ≈ 0.3011
Therefore, if a scooter selected at random is of standard quality, the probability that it comes from plant 'B' is approximately 0.3011 or 30.11%.
Conclusion
In conclusion, we have successfully applied probability principles to analyze the quality control scenario presented. The probability of a randomly selected scooter being of standard quality is 91.6%, and the probability that a standard-quality scooter originated from plant 'B' is approximately 30.11%. These calculations are vital for operational decision-making, allowing management to identify areas for improvement in production processes and quality control measures at each plant. Continuous monitoring and statistical analysis are essential for maintaining high-quality standards and optimizing production efficiency.
Answer Length
This is a comprehensive model answer for learning purposes and may exceed the word limit. In the exam, always adhere to the prescribed word count.