Question

A high school spends approximately $9,000 hosting a football game (e.g. the hourly pay for staff, the electricity, water usage, etc.). The school usually collects $15,000 in ticket sales and vending if it doesn't rain. If it rains during the game, the school only collects about $5,000. During any day during football season, there is about a 12% chance of rain.

What is the expected payoff (net income) for the school for a football game?

Question 10 options:

$4,800


$8,520


$14,000


$2,700

Answer 1

To calculate the expected payoff (net income) for the school for a football game, we first need to determine the outcomes based on whether it rains or does not rain, and then incorporate their probabilities into the calculation.

1. **If it doesn't rain:**
- Revenue from ticket sales and vending = $15,000
- Expenses = $9,000
- Net income = Revenue - Expenses = $15,000 - $9,000 = $6,000

2. **If it rains:**
- Revenue from ticket sales and vending = $5,000
- Expenses = $9,000
- Net income = Revenue - Expenses = $5,000 - $9,000 = -$4,000

3. **Probabilities:**
- Probability of no rain (successful game) = 1 - 0.12 = 0.88
- Probability of rain (unsuccessful game) = 0.12

4. **Expected payoff (net income):**
- Expected income if no rain = Probability of no rain × Net income if no rain
- Expected income if rain = Probability of rain × Net income if it rains

Calculating these:

- Expected income if no rain:
\[
0.88 \times 6,000 = 5,280
\]

- Expected income if it rains:
\[
0.12 \times (-4,000) = -480
\]

Now, we add these two expected incomes together to find the total expected payoff:

\[
\text{Total Expected Payoff} = 5,280 - 480 = 4,800
\]

Thus, the expected payoff (net income) for the school for a football game is **$4,800**.

The correct option is **$4,800**.