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The problem in the image is asking to calculate the expected winnings for a player placing a bet on red in American roulette. Here's a summary of the important details:

1. **Roulette Details**: The wheel contains numbers 1 through 36, alternating between black and red. There are also two green spaces, 0 and 00, making a total of 38 possible outcomes.
2. **Bet Details**: The player places a $0.25 bet on red. If the ball lands on red, the player wins $0.25 plus gets their bet back. If the ball lands on any other color (black or green), the player loses the $0.25.
3. **Expected Outcome**: The player places the same bet on red 34 times, and we need to calculate the expected winnings and round to the nearest cent.

### Step-by-Step Solution:

1. **Probability of Winning (Landing on Red)**: 
   Since there are 18 red numbers out of 38 total numbers on the wheel:
   \[
   P(\text{Red}) = \frac{18}{38}
   \]
   
2. **Probability of Losing (Not Landing on Red)**: 
   The probability of landing on black or green is:
   \[
   P(\text{Not Red}) = \frac{20}{38}
   \]
   
3. **Expected Winnings Per Bet**: 
   The expected value for a single $0.25 bet can be calculated as:
   \[
   \text{Expected Value} = P(\text{Red}) \times (\text{Winnings on Red}) + P(\text{Not Red}) \times (\text{Loss on Not Red})
   \]
   The winnings on red are $0.25 (net win) and the loss on not red is -$0.25 (complete loss of the bet).

   \[
   \text{Expected Value} = \left(\frac{18}{38} \times 0.25\right) + \left(\frac{20}{38} \times (-0.25)\right)
   \]

4. **Expected Value Over 34 Bets**: 
   Multiply the expected value per bet by the number of times the player places the bet (34):
   \[
   \text{Total Expected Winnings} = \text{Expected Value per bet} \times 34
   \]

Let's calculate this.The player's expected winnings after placing the $0.25 bet on red 34 times is **-$0.45**. This means that, on average, the player is expected to lose 45 cents over the course of 34 bets.

Would you like any further details or clarification?

Here are 5 related questions to expand on this concept:
1. What is the expected loss for 100 bets on red in American roulette?
2. How does the house edge in American roulette compare to European roulette?
3. How would the expected winnings change if the bet amount was $1 instead of $0.25?
4. What is the effect of the two green spaces (0 and 00) on the player's expected outcome?
5. How does the probability of landing on black compare to landing on red?

**Tip**: The house edge in American roulette comes from the inclusion of the 0 and 00 spaces, which make the probability of winning on a red or black bet less than 50%.

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 $0.25 on red to play the game. If the ball lands on red, the player gets $0.25 for winning and receives the money back. If the ball does not land on red, then the player simply loses the $0.25 placed on the bet. If the player places the same bet on red 34 times, what is the player’s expected winnings? Round your answer to the nearest cent.

Learn how to calculate the expected winnings in American roulette for a player who bets $0.25 on red over 34 spins. Explore key concepts in probability and expected value, including detailed calculations using the probabilities of winning and losing on red.

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