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.