In American roulette, the wheel has the 38 numbers, , and 36 , marked on equally spaced slots. If a player bets on a number and wins, then the player keeps
and receives an additional . Otherwise, the player is awarded nothing, and the casino takes the player’s . Find the expected value to the player for one play of the
game. If is the gain to a player in a game of chance, then is usually negative. This value gives the average amount per game the player can expect to lose.
The expected value is .
(Round to the nearest cent as needed.)

Ask by Nguyen Mathis. in the United States

Jan 21,2025

Question

In American roulette, the wheel has the 38 numbers, , and 36 , marked on equally spaced slots. If a player bets on a number and wins, then the player keeps
and receives an additional . Otherwise, the player is awarded nothing, and the casino takes the player’s . Find the expected value to the player for one play of the
game. If is the gain to a player in a game of chance, then is usually negative. This value gives the average amount per game the player can expect to lose.
The expected value is .
(Round to the nearest cent as needed.)

Ask by Nguyen Mathis. in the United States

Jan 21,2025

UpStudy AI · Accepted Answer

To calculate the expected value $$ E(x) $$ for the player in one play of American roulette, let's break down the possible outcomes and their probabilities.

### Understanding the Game:
- **Total Numbers on the Wheel:** 38 (including $$ 00, 0, 1, 2, \ldots, 36 $$).
- **Bet:** The player bets \$4 on a single number.
- **Winning Payout:** If the player wins, they keep their \$4 and receive an additional \$140.
- **Losing Outcome:** If the player loses, they forfeit the \$4 bet.

### Calculating Possible Gains:
1. **Winning Scenario:**
   - **Probability:** $$ \frac{1}{38} $$ (since there are 38 possible outcomes).
   - **Gain:** \$140 (net profit, since the original \$4 is returned).

2. **Losing Scenario:**
   - **Probability:** $$ \frac{37}{38} $$ (the remaining outcomes).
   - **Gain:** $$-\$4$$ (the player loses their bet).

### Expected Value Formula:
$$
E(x) = \left( \frac{1}{38} \times 140 \right) + \left( \frac{37}{38} \times (-4) \right)
$$

### Performing the Calculations:
1. **Winning Contribution:**
   $$
   \frac{1}{38} \times 140 = \frac{140}{38} \approx 3.6842
   $$

2. **Losing Contribution:**
   $$
   \frac{37}{38} \times (-4) = \frac{-148}{38} \approx -3.8947
   $$

3. **Total Expected Value:**
   $$
   E(x) = 3.6842 - 3.8947 \approx -0.2105
   $$

### Rounding to the Nearest Cent:
$$
E(x) \approx -\$0.21
$$

**Conclusion:** On average, the player can expect to lose **\$0.21** per game.

**Answer:** $$-\$0.21$$
The expected value $$ E(x) $$ is $$-\$0.21$$.

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