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**Problem**:
In American roulette, the wheel contains the numbers 1 through 36, alternating between black and red. There are two green spaces numbered 0 and 00. A player places a bet of $9.75 on red to play the game. If the ball lands on red, the player gets $9.75 for winning and receives the money back. If the ball does not land on red, then the player simply loses the $9.75 placed on the bet. If the player places the same bet on red 35 times, what is the player’s expected winnings? Round your answer to the nearest cent.

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Let's break down the solution to this expected value problem.

1. In American roulette, there are **38 total spaces** (36 numbers plus 0 and 00).
2. The probability of the ball landing on **red** is:
   \[
   P(\text{red}) = \frac{18}{38}
   \]
3. The probability of the ball **not landing on red** (either black or green) is:
   \[
   P(\text{not red}) = \frac{20}{38}
   \]
4. If the player wins, they receive $9.75 back in addition to their original bet. If they lose, they lose $9.75.

### Expected value for one spin:
- **Winning**: The player gains $9.75 with a probability of $\frac{18}{38}$.
- **Losing**: The player loses $9.75 with a probability of $\frac{20}{38}$.

The expected value $E$ for one bet is given by:
\[
E = (9.75 \times \frac{18}{38}) + (-9.75 \times \frac{20}{38})
\]

Now, let's calculate this and then determine the total expected winnings after 35 spins.

The player's expected winnings after placing the same $9.75 bet on red for 35 spins is approximately **- $17.96**. This means that, on average, the player can expect to lose about $17.96 over the course of 35 spins.

Would you like more details or have any questions?

Here are 5 related questions to consider:
1. How would the expected value change if the player increased the number of bets?
2. What is the expected value if the player bets on black instead of red?
3. How would the house edge be calculated in this American roulette game?
4. What would happen to the expected value if there were no green spaces (0 and 00)?
5. How does the expected value change if the bet amount is doubled?

**Tip:** In gambling games like roulette, the house edge ensures that, on average, the casino always wins over time, as indicated by a negative expected value.

In American roulette, the wheel contains the numbers 1 through 36, alternating between black and red. There are two green spaces numbered 0 and 00. A player places a bet of $9.75 on red to play the game. If the ball lands on red, the player gets $9.75 for winning and receives the money back. If the ball does not land on red, then the player simply loses the $9.75 placed on the bet. If the player places the same bet on red 35 times, what is the player’s expected winnings? Round your answer to the nearest cent.

Learn how to calculate the expected value of a $9.75 bet on red in American roulette. This example walks through the probability of winning and losing, using the expected value formula over 35 spins, and highlights key concepts in probability theory. Suitable for high school students in Grades 10-12.

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