In a courier company, three office assistants are assigned to process incoming mail. The first assistant A processes 40%, the second assistant B processes 35% and the third assistant C processes 25% of the mail. The first Assistant has an error rate of 0.04, the second has an error rate of 0.06 and the third has an error rate of 0.03. A mail selected at random is found to have an error. The Manager of the company wishes to know the probability that the mail was processed by first, second or the third assistant respectively.

This problem is a classic application of **Bayes' Theorem**, which is used to find the probability of an event based on prior knowledge of conditions related to the event.

### Step-by-Step Breakdown:

We are given the following:

- The total probability that a mail is processed by Assistant A is \( P(A) = 0.40 \), by Assistant B is \( P(B) = 0.35 \), and by Assistant C is \( P(C) = 0.25 \).
- The probability that an error occurs given it was processed by Assistant A is \( P(E|A) = 0.04 \), by Assistant B is \( P(E|B) = 0.06 \), and by Assistant C is \( P(E|C) = 0.03 \).

We need to find the probability that the mail was processed by each assistant given that an error was found, i.e., \( P(A|E), P(B|E), P(C|E) \).

### 1. Total Probability of Error \( P(E) \)

The total probability of error \( P(E) \) is calculated using the **Law of Total Probability**:

\[
P(E) = P(E|A)P(A) + P(E|B)P(B) + P(E|C)P(C)
\]

Substituting the given values:

\[
P(E) = (0.04 \times 0.40) + (0.06 \times 0.35) + (0.03 \times 0.25)
\]

\[
P(E) = 0.016 + 0.021 + 0.0075 = 0.0445
\]

### 2. Applying Bayes' Theorem

We can now apply **Bayes' Theorem** to find the probabilities \( P(A|E), P(B|E), P(C|E) \).

Bayes' Theorem is given by:

\[
P(A|E) = \frac{P(E|A)P(A)}{P(E)}, \quad P(B|E) = \frac{P(E|B)P(B)}{P(E)}, \quad P(C|E) = \frac{P(E|C)P(C)}{P(E)}
\]

#### For Assistant A:

\[
P(A|E) = \frac{P(E|A)P(A)}{P(E)} = \frac{0.04 \times 0.40}{0.0445} = \frac{0.016}{0.0445} \approx 0.3596
\]

#### For Assistant B:

\[
P(B|E) = \frac{P(E|B)P(B)}{P(E)} = \frac{0.06 \times 0.35}{0.0445} = \frac{0.021}{0.0445} \approx 0.4722
\]

#### For Assistant C:

\[
P(C|E) = \frac{P(E|C)P(C)}{P(E)} = \frac{0.03 \times 0.25}{0.0445} = \frac{0.0075}{0.0445} \approx 0.1685
\]

### 3. Final Probabilities:

- \( P(A|E) \approx 0.3596 \)
- \( P(B|E) \approx 0.4722 \)
- \( P(C|E) \approx 0.1685 \)

So, if a mail with an error is selected at random, the probability that it was processed by:
- Assistant A is about **35.96%**,
- Assistant B is about **47.22%**,
- Assistant C is about **16.85%**.

---

Would you like further details or have any questions? 

Here are 5 related questions to explore this further:

1. How would the probabilities change if the error rates for the assistants were adjusted?
2. What happens to the final probabilities if the proportion of mail processed by each assistant is changed?
3. How would you calculate the expected number of errors given the number of mails processed daily?
4. Can Bayes’ Theorem be applied to non-error events in this context? How?
5. How do the probabilities change if there is a fourth assistant with a different error rate?

**Tip:** Bayes' Theorem is extremely useful when you have conditional probabilities and need to reverse the conditions to find the likelihood of causes based on observed effects.

In a courier company, three office assistants are assigned to process incoming mail. The first assistant A processes 40%, the second assistant B processes 35%, and the third assistant C processes 25% of the mail. The first assistant has an error rate of 0.04, the second has an error rate of 0.06, and the third has an error rate of 0.03. A mail selected at random is found to have an error. The manager of the company wishes to know the probability that the mail was processed by the first, second, or third assistant, respectively.

Understand how to calculate the probability that mail processed by office assistants A, B, or C contains an error using Bayes' Theorem. This problem covers Bayesian and conditional probability, suitable for Grades 11-12 students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation