One of the wagers in the game of roulette is to place a bet that the ball will land on a black number. (Eighteen of the numbers are black, eighteen are red, and two are green.) If the ball lands on a black number, the player wins the amount bet. If a player bets $3, find the player's expectation. (Round your answer to two decimal places.)
$  

To calculate the player's expectation for this bet, we need to analyze the possible outcomes, the probabilities of each outcome, and the amounts won or lost.

### Problem Analysis
1. **Total numbers on the roulette wheel**: 18 black + 18 red + 2 green = 38.
2. **Probability of winning** (landing on a black number): 
   \[
   P(\text{Black}) = \frac{18}{38} = \frac{9}{19}
   \]
3. **Probability of losing** (landing on red or green): 
   \[
   P(\text{Not Black}) = 1 - P(\text{Black}) = 1 - \frac{9}{19} = \frac{10}{19}
   \]

### Winnings and Losses
- **If the player wins**: They win the amount they bet, which is $3.
- **If the player loses**: They lose the amount they bet, which is $3.

### Expected Value Calculation
The expected value (E) can be found by summing the products of each outcome's probability and the respective win/loss amount.

\[
E = (P(\text{Black}) \times \text{Winnings}) + (P(\text{Not Black}) \times \text{Loss})
\]
Plugging in the values:

\[
E = \left(\frac{9}{19} \times 3\right) + \left(\frac{10}{19} \times (-3)\right)
\]

### Calculating Each Term
1. **Winning term**:
   \[
   \frac{9}{19} \times 3 = \frac{27}{19} \approx 1.42
   \]
2. **Losing term**:
   \[
   \frac{10}{19} \times (-3) = \frac{-30}{19} \approx -1.58
   \]

### Summing These Values
\[
E = 1.42 - 1.58 = -0.16
\]

### Conclusion
The player's expectation is approximately **-0.16**. This means that, on average, the player loses 16 cents for every $3 bet on black.

### Final Answer
\[
\boxed{-0.16}
\]

Would you like a deeper explanation on any part of this solution?

---

1. What is the expected value if the player bets a different amount, say $10?
2. How would the expectation change if the game were fair, with an equal number of black, red, and no green slots?
3. How does expectation relate to probability and outcomes in games of chance?
4. Why do casinos set up games with negative expected values for players?
5. How would the expectation change if there were no green slots (only red and black)?

**Tip:** Understanding expected value helps you assess the long-term outcome of repetitive bets or games of chance.

One of the wagers in the game of roulette is to place a bet that the ball will land on a black number. (Eighteen of the numbers are black, eighteen are red, and two are green.) If the ball lands on a black number, the player wins the amount bet. If a player bets $3, find the player's expectation. (Round your answer to two decimal places.)

Explore the calculation of expected value in roulette when betting on a black number. This problem covers probability and expected value with detailed steps, showing how a $3 bet on black results in an expected value of -0.16, meaning an average loss of 16 cents per $3 wager. Suitable for Grades 10-12.

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