UPSC MainsMANAGEMENT-PAPER-II20136 Marks

Q1.

Question 1

In a post office, three clerks were assigned to process incoming mail. The first clerk, C1, processes 40%, the second clerk, C2, processes 35% and the third clerk, C3, processes 25% of the mail. The first clerk has an error rate of 0.04, the second clerk has an error rate of 0.06 and the third clerk has an error rate of 0.03. A mail selected at random from a day's output is found to have an error. The Postmaster wishes to know the probability that the mail was processed by the first, second or third clerk respectively. Find the probabilities.

How to Approach

This question is a classic application of Bayes' Theorem in a practical scenario. The approach should involve clearly defining the events, stating Bayes' Theorem, and then applying it to calculate the probabilities for each clerk. The answer should be structured logically, showing each step of the calculation. Emphasis should be placed on understanding the conditional probabilities involved. A clear explanation of the reasoning behind each step is crucial for a good score.

Model Answer

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Introduction

In operational management, understanding the reliability and error rates of different processes or personnel is crucial for quality control and efficiency. This scenario presents a typical problem encountered in organizations where tasks are distributed among multiple individuals or departments. The problem requires us to determine the probability that an error in processed mail originated from a specific clerk, given that an error has been detected. This is a classic application of Bayesian inference, a statistical method for updating beliefs based on new evidence. The Postmaster's inquiry necessitates the application of Bayes' Theorem to ascertain the probabilities of each clerk being responsible for the error.

Understanding the Problem

Let's define the events:

C1: The mail was processed by Clerk 1.
C2: The mail was processed by Clerk 2.
C3: The mail was processed by Clerk 3.
E: The mail has an error.

We are given the following probabilities:

P(C1) = 0.40 (Probability that Clerk 1 processed the mail)
P(C2) = 0.35 (Probability that Clerk 2 processed the mail)
P(C3) = 0.25 (Probability that Clerk 3 processed the mail)
P(E|C1) = 0.04 (Probability of an error given the mail was processed by Clerk 1)
P(E|C2) = 0.06 (Probability of an error given the mail was processed by Clerk 2)
P(E|C3) = 0.03 (Probability of an error given the mail was processed by Clerk 3)

We need to find P(C1|E), P(C2|E), and P(C3|E) – the probabilities that the mail was processed by each clerk, given that it has an error.

Bayes' Theorem

Bayes' Theorem states:

P(A|B) = [P(B|A) * P(A)] / P(B)

In our case, we need to calculate P(B), the probability of an error occurring, which can be calculated using the law of total probability:

P(E) = P(E|C1) * P(C1) + P(E|C2) * P(C2) + P(E|C3) * P(C3)

Calculating P(E)

P(E) = (0.04 * 0.40) + (0.06 * 0.35) + (0.03 * 0.25)

P(E) = 0.016 + 0.021 + 0.0075

P(E) = 0.0445

Calculating P(C1|E), P(C2|E), and P(C3|E)

P(C1|E)

P(C1|E) = [P(E|C1) * P(C1)] / P(E)

P(C1|E) = (0.04 * 0.40) / 0.0445

P(C1|E) = 0.016 / 0.0445

P(C1|E) ≈ 0.3596

P(C2|E)

P(C2|E) = [P(E|C2) * P(C2)] / P(E)

P(C2|E) = (0.06 * 0.35) / 0.0445

P(C2|E) = 0.021 / 0.0445

P(C2|E) ≈ 0.4719

P(C3|E)

P(C3|E) = [P(E|C3) * P(C3)] / P(E)

P(C3|E) = (0.03 * 0.25) / 0.0445

P(C3|E) = 0.0075 / 0.0445

P(C3|E) ≈ 0.1685

Summary of Probabilities

Clerk	Probability (Given Error)
C1	0.3596
C2	0.4719
C3	0.1685

Conclusion

The Postmaster can conclude that, given an error in the processed mail, there is approximately a 35.96% probability it was processed by Clerk 1, a 47.19% probability it was processed by Clerk 2, and a 16.85% probability it was processed by Clerk 3. These probabilities highlight that Clerk 2 is the most likely source of errors, despite having a moderate error rate, due to processing the second-highest volume of mail. This information can be used to target training or process improvements to reduce errors and improve overall 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.

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Additional Resources

Key Definitions

Bayes' Theorem

A statistical theorem that describes the probability of an event, based on prior knowledge of conditions that might be related to the event.

Law of Total Probability

The law of total probability states that if a set of mutually exclusive and exhaustive events partition the sample space, then the probability of an event can be calculated by summing the probabilities of the event occurring with each of the partitioning events.

Key Statistics

According to a 2023 report by Statista, the global postal services market was valued at approximately $387.8 billion.

Source: Statista (2023)

In 2022-23, India Post delivered over 4.8 billion letters and parcels.

Source: Department of Posts, Annual Report 2022-23

Examples

Medical Diagnosis

Bayes' Theorem is widely used in medical diagnosis to calculate the probability of a patient having a disease given a positive test result, considering the prevalence of the disease and the accuracy of the test.

Frequently Asked Questions

▶What if the processing percentages were unknown?

If the processing percentages were unknown, we would need to estimate them based on historical data or assume a uniform distribution, which would affect the final probabilities.

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