A computer software firm maintains a telephone hotline service for its customers. The firm finds that 40% of the calls involve questions about the application of the software, 45% involve issues of incompatibility with the hardware, and 15% involve the inability to install the software on the user's machine. These three categories of problems can be resolved with probabilities 0.90, 0.30 and 0.80, respectively. Knowing that an incoming call that does not involves question about the application of the software cannot be resolved, then the probability that it involves issues of incompatibility is

Question 10 Select one:

A.
None of the others

B.
0.8182

C.
0.7778

D.
0.9130

Let's approach the problem using conditional probability.

### Problem Summary:
Given the following dataLet's approach the problem using conditional probability.

### Problem Summary:
Given the following 

- \( P(A) = 0.40 \): Probability that a call involves questions about the application of the software.
- \( P(B) = 0.45 \): Probability that a call involves issues of incompatibility with hardware.
- \( P(C) = 0.15 \): Probability that a call involves the inability to install the software.

The probabilities that these categories can be resolved are:
- \( P(\text{Resolved} \mid A) = 0.90 \)
- \( P(\text{Resolved} \mid B) = 0.30 \)
- \( P(\text{Resolved} \mid C) = 0.80 \)

We also know that if an incoming call does not involve questions about the application of the software (i.e., it’s either about hardware incompatibility or installation issues), it cannot be resolved.

### Objective:
To find the probability that a call involves issues of incompatibility with hardware given that it does **not** involve questions about the application of the software **and** cannot be resolved.

This can be represented as:

\[
P(B \mid \text{Not } A \text{ and Not Resolved})
\]

### Step 1: Determine the Probability of "Not A"
The probability that a call does not involve questions about the application of the software is:

\[
P(\text{Not } A) = 1 - P(A) = 1 - 0.40 = 0.60
\]

### Step 2: Determine the Conditional Probabilities for "Not Resolved" Given \( B \) and \( C \)
- \( P(\text{Not Resolved} \mid B) = 1 - 0.30 = 0.70 \)
- \( P(\text{Not Resolved} \mid C) = 1 - 0.80 = 0.20 \)

### Step 3: Apply the Law of Total Probability for \( P(\text{Not Resolved} \mid \text{Not } A) \)
Since "Not A" includes both categories \( B \) and \( C \):

\[
P(\text{Not Resolved} \mid \text{Not } A) = \frac{P(B) \cdot P(\text{Not Resolved} \mid B) + P(C) \cdot P(\text{Not Resolved} \mid C)}{P(\text{Not } A)}
\]

Substituting the values:

\[
P(\text{Not Resolved} \mid \text{Not } A) = \frac{(0.45 \times 0.70) + (0.15 \times 0.20)}{0.60}
\]

\[
P(\text{Not Resolved} \mid \text{Not } A) = \frac{0.315 + 0.03}{0.60} = \frac{0.345}{0.60} = 0.575
\]

### Step 4: Calculate \( P(B \mid \text{Not } A \text{ and Not Resolved}) \)
Using Bayes’ Theorem:

\[
P(B \mid \text{Not } A \text{ and Not Resolved}) = \frac{P(B) \cdot P(\text{Not Resolved} \mid B)}{P(\text{Not Resolved} \mid \text{Not } A)}
\]

Substituting the values:

\[
P(B \mid \text{Not } A \text{ and Not Resolved}) = \frac{0.45 \times 0.70}{0.575} = \frac{0.315}{0.575} \approx 0.5478
\]

### Step 5: Interpret the Answer Options
Since none of the provided answer choices match \( 0.5478 \), the correct choice would be:

**A. None of the others**

---

Would you like further details or have any other questions? Here are 5 related questions to explore:

1. How does conditional probability apply to real-world scenarios?
2. What is the Law of Total Probability and when is it used?
3. Can you explain the significance of Bayes' Theorem?
4. How would the solution change if different resolution probabilities were given?
5. What if there were additional categories of customer problems?

**Tip:** When dealing with probability problems, always clearly define the conditions and categories involved before applying formulas.

Explore a detailed solution to a probability problem involving conditional probability and Bayes' Theorem. Learn how to calculate the probability of hardware incompatibility given specific conditions and resolutions. Suitable for advanced high school and college-level 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