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This document presents questions about expected values and probability distributions in a scenario where two companies, \( A \) and \( B \), use two computer systems \( W \) and \( M \) to collaborate. Employees are assigned to use these systems with various probabilities, and we are tasked with finding the expected values based on the distributions provided. Here is an outline of the approach to solve the questions:

### Analysis of the Document Content:
1. **Problem Context**:
   - Two companies \( A \) and \( B \) are using two systems \( W \) and \( M \).
   - Probabilities for each employee using \( W \) or \( M \) are defined differently across questions.

2. **Questions**:
   - Each problem (1, 2, and 3) has two parts:
      - Calculate the expected profit for employees using system \( W \) and those using system \( M \) for Company \( A \).

3. **Provided Values and Probabilities**:
   - Probabilities are given as fractions or decimals, with specific values changing for each problem. 
   - Expected values should likely be computed by multiplying the probability by a constant factor, representing the profit for each type of employee.

### Solution Steps:
To solve each part, we need to:
- Multiply the probability of using each system by the expected benefit associated with it.
- Sum these values to get the overall expected profit for each scenario.

Would you like me to calculate specific values for each part, or would you like an explanation on setting up probability-based expected value calculations? 

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#### Questions to Consider
1. How do probability distributions affect expected value calculations?
2. What are common applications of expected value in business decisions?
3. How would changes in probability impact overall company profit expectations?
4. Could we use simulations to verify the expected outcomes calculated?
5. What other factors could influence the distribution of system use among employees?

#### Tip:
Expected values are simply weighted averages, where each probability contributes proportionally to the overall outcome. This is useful in decision-making, as it provides a balanced forecast based on likelihoods.

Two companies, A and B, are using computer systems W and M for collaborative work. Employees are assigned to these systems with different probabilities, and we need to calculate the expected profit for company A's employees based on the probability distributions given. Each question provides specific probabilities for employees using systems W and M in different scenarios, and we are asked to determine expected values.

Explore how to calculate expected profit for employees of two companies, A and B, using computer systems W and M with different probability distributions. This problem covers key probability concepts and demonstrates how to use weighted averages in decision-making. Suitable for undergraduate-level probability and statistics.

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