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The problem in the image involves evaluating a company's promotional strategy and determining the expected number of promotions, expected profit, and breakeven point for their campaigns. Let's break it down step by step:

1. **Expected value of promotions (2.06)**: 
   The table shows the probabilities \(P(X)\) for customers using 0 to 4 promotions. This part seems to have already calculated the expected number of promotions. The expected value (already provided as **2.06**) is calculated as:

   \[
   E(X) = \sum X \cdot P(X)
   \]

2. **Expected profit (19.57)**: 
   It's stated that the average order value is 38 dollars, and the company earns a 25% profit margin. So, the profit per order is:

   \[
   \text{Profit per order} = 38 \times 0.25 = 9.5 \text{ dollars}.
   \]

   The expected number of orders from the 2.06 promotions (from step 1) would give an expected total profit of:

   \[
   \text{Expected Profit} = 2.06 \times 9.5 = 19.57 \text{ dollars (which matches the provided value)}.
   \]

3. **Breakeven Point**: 
   The company has a fixed cost of 14,000 dollars, and each customer adds a variable cost of 3.75 dollars for mailing and handling. The question asks for the minimum number of customers required to cover the promotional costs.

   Let's denote the number of customers as \(N\). For breakeven, the total revenue should equal the total costs. The total cost function is:

   \[
   \text{Total Cost} = 14,000 + 3.75N
   \]

   The expected profit from each customer is 19.57 dollars. So, the total revenue is:

   \[
   \text{Total Revenue} = 19.57N
   \]

   For breakeven:

   \[
   19.57N = 14,000 + 3.75N
   \]

   Solving for \(N\):

   \[
   19.57N - 3.75N = 14,000
   \]

   \[
   15.82N = 14,000
   \]

   \[
   N = \frac{14,000}{15.82} \approx 885
   \]

   Therefore, the minimum number of customers required to break even is **885** (rounding to the next highest integer).

---

Would you like more details on any specific part of this solution? Here are five questions to extend your understanding:

1. How is the expected value formula applied in probability and statistics?
2. Why is the gross profit margin used in calculating the expected profit?
3. What does the breakeven point represent in business contexts?
4. How would changing the variable cost affect the breakeven point calculation?
5. Could this model apply to different promotional strategies, and if so, how?

**Tip**: Always check if fixed and variable costs are updated regularly, as this can significantly impact profit and breakeven calculations.

To examine the effectiveness of its four annual advertising promotions, a mail order company has sent a questionnaire to its customers. The table shows the probabilities of customers using 0 to 4 promotions. The goal is to find the expected number of promotions, expected profit, and breakeven point based on the fixed costs and variable costs provided.

Explore how to calculate the expected value of promotions, expected profit, and the breakeven point for a mail-order company’s advertising strategy. This problem involves using probability, profit margins, and cost analysis. Suitable for high school students in Grades 9-12.

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