20% percent chance of selling 500. 25% chance of selling 1000. 40% chance of selling 1500. 15% chance of selling 2000. A company makes hamburgers for sale at IU home games. Each hamburger costs $4 to make and sells for $11. Any unsold hamburger at the end of the game will be bought by a local food pantry for $3 for immediate distribution to the homeless. Based on past games, the company estimates the following demands with the corresponding probabilities.

We can approach this problem by determining the expected value of the company's profit for each scenario. Here's how:

### Steps:

1. **Revenue from Sales**:
   - Each hamburger is sold for $11. However, we need to account for the unsold hamburgers (which are sold to a food pantry for $3).

2. **Costs**:
   - Each hamburger costs $4 to make.

3. **Unsold Hamburgers**:
   - Unsold hamburgers are sold to the food pantry at $3 each.

4. **Expected Profit Calculation**:
   - We will compute the profit for each possible demand scenario (500, 1000, 1500, 2000 hamburgers sold) and then use their respective probabilities to find the overall expected profit.

### Step-by-Step Profit Calculation:

#### Case 1: Selling 500 hamburgers (20% chance)
- **Revenue from 500 sales**: $500 \times 11 = 5500$
- **Cost of producing 500 hamburgers**: $500 \times 4 = 2000$
- **Profit**: $5500 - 2000 = 3500$

#### Case 2: Selling 1000 hamburgers (25% chance)
- **Revenue from 1000 sales**: $1000 \times 11 = 11000$
- **Cost of producing 1000 hamburgers**: $1000 \times 4 = 4000$
- **Profit**: $11000 - 4000 = 7000$

#### Case 3: Selling 1500 hamburgers (40% chance)
- **Revenue from 1500 sales**: $1500 \times 11 = 16500$
- **Cost of producing 1500 hamburgers**: $1500 \times 4 = 6000$
- **Profit**: $16500 - 6000 = 10500$

#### Case 4: Selling 2000 hamburgers (15% chance)
- **Revenue from 2000 sales**: $2000 \times 11 = 22000$
- **Cost of producing 2000 hamburgers**: $2000 \times 4 = 8000$
- **Profit**: $22000 - 8000 = 14000$

### Expected Profit Calculation:

The expected profit is the weighted average of the profits, considering the probabilities:

\[
\text{Expected Profit} = (0.20 \times 3500) + (0.25 \times 7000) + (0.40 \times 10500) + (0.15 \times 14000)
\]

\[
\text{Expected Profit} = 700 + 1750 + 4200 + 2100 = 8750
\]

### Conclusion:
The expected profit for the company is **$8750**.

Would you like more details on any specific part? 

Here are 5 related questions:
1. How would the profit change if the cost per hamburger increased to $5?
2. What would the expected profit be if the probability distribution changed?
3. How does selling to the food pantry affect the overall profit?
4. Can we calculate the variance in profit to understand the risk involved?
5. How would increasing the selling price per hamburger impact the expected profit?

**Tip:** When calculating expected values, always make sure to weight outcomes by their probabilities to get an accurate prediction.

Learn how to calculate the expected profit for a company selling hamburgers at IU home games. This problem involves probability, expected value, and profit calculations, including revenue from sales and costs of production. The scenario considers multiple sales outcomes and how unsold hamburgers are handled. Suitable for students in Grades 9-12.

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