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The problem is asking about the **expected payback** for a bet on red in a roulette game. Here’s the breakdown of the problem:

- There are **18 red compartments** and **18 black compartments**.
- There are **2 compartments that are neither red nor black** (likely the green 0 and 00 in a typical American roulette wheel).
- If you bet $8 on red:
  - You win $8 (and get your $8 back, meaning a total of $16) if red appears.
  - You lose the $8 if red does not appear.

### Step-by-Step Calculation:

1. **Total number of compartments**:
   \[
   18 \, (\text{red}) + 18 \, (\text{black}) + 2 \, (\text{neither}) = 38
   \]
   
2. **Probability of winning (betting on red)**:
   \[
   P(\text{win on red}) = \frac{18}{38}
   \]
   
3. **Probability of losing (not red)**:
   \[
   P(\text{lose on red}) = 1 - P(\text{win on red}) = \frac{20}{38}
   \]

4. **Expected value calculation**:
   The expected payback is calculated by multiplying the outcomes (win or lose) by their respective probabilities:
   
   \[
   \text{Expected Payback} = ( \text{P(win)} \times \text{winnings}) + (\text{P(lose)} \times \text{loss})
   \]
   
   Plugging in the numbers:
   \[
   \text{Expected Payback} = \left(\frac{18}{38} \times 8\right) + \left(\frac{20}{38} \times (-8)\right)
   \]
   
Let me calculate this value for you.The expected payback for betting $8 on red is approximately **-0.42**. This means that, on average, you would lose about 42 cents for every $8 bet placed on red.

Would you like more details or have any questions?

Here are 5 related questions:
1. How does the expected value change if you bet on black instead of red?
2. What is the expected value if you bet $8 on a single number in roulette?
3. How do the probabilities differ between American and European roulette?
4. What is the expected return if you play multiple rounds of betting on red?
5. How can knowing the expected value help in making gambling decisions?

**Tip**: In games of chance like roulette, the house usually has an advantage, as seen in the negative expected payback.

One option in a roulette game is to bet on the color red or black. (There are 18 red compartments, 18 black compartments, and two compartments that are neither black nor red.) If you bet on a color, you get to keep your bet and win that same amount if the color occurs. If that color does not occur, you will lose the amount of money you wagered on that color to appear. What is the expected payback for this game if you bet $8 on red?

Calculate the expected payback when betting $8 on red in a roulette game. This detailed solution explores key probability concepts and uses the formula for expected value to determine your average outcome. Suitable for high school students and anyone interested in understanding gambling probabilities.

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