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To solve this problem, we need to create a decision tree and calculate the expected profits for each choice (Short, Medium, Long) using the probabilities given for each prevailing style (Short, Medium, Long).

### Steps:
1. **Determine the probabilities for each style:**
   - Probability of **Short** prevailing: \( P(\text{Short}) = 0.20 \)
   - Probability of **Medium** prevailing: \( P(\text{Medium}) = 0.35 \)
   - Probability of **Long** prevailing: \( P(\text{Long}) = 0.45 \)

2. **Set up the decision tree:**
   - There are three possible choices for the designer: **Short**, **Medium**, and **Long**.
   - For each choice, we calculate the expected profit based on the probabilities of the prevailing styles and the corresponding profits from the table.

### Expected Profit Calculation:
For each designer’s choice, we compute the expected profit by multiplying the profit values by the corresponding probabilities and summing them.

#### 1. Designer's Choice: **Short**
\[
\text{Expected Profit} = (8 \times 0.20) + (3 \times 0.35) + (1 \times 0.45)
= 1.6 + 1.05 + 0.45 = 3.1
\]

#### 2. Designer's Choice: **Medium**
\[
\text{Expected Profit} = (1 \times 0.20) + (9 \times 0.35) + (2 \times 0.45)
= 0.2 + 3.15 + 0.9 = 4.25
\]

#### 3. Designer's Choice: **Long**
\[
\text{Expected Profit} = (4 \times 0.20) + (3 \times 0.35) + (10 \times 0.45)
= 0.8 + 1.05 + 4.5 = 6.35
\]

### Conclusion:
- **Short**: Expected profit = 3.1
- **Medium**: Expected profit = 4.25
- **Long**: Expected profit = 6.35

The designer should choose **Long** to maximize expected profits, as it results in the highest expected profit of 6.35.

Would you like more details or further clarification? Here are five related questions:

1. How do we incorporate probabilities into decision-making in general?
2. What is the role of expected value in decision analysis?
3. Could a risk-averse designer still prefer a lower expected profit with less variability?
4. How would the decision change if one probability was adjusted?
5. Can we graphically represent this decision-making process with a tree diagram?

**Tip:** Always verify if the probabilities add up to 1 before making any calculations!

A designer is trying to predict which style will prevail in the upcoming fashion season and must choose between three design options: short, medium, and long. Given probabilities for each prevailing style and profits based on the choice of design, what decision should the designer make to maximize expected profits?

Learn how to use probability and expected value to make informed decisions. This example focuses on a fashion designer predicting prevailing styles and maximizing profits by calculating the expected values of each design choice.

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