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, 18 are red, and two are green.) If the ball lands on a black number, the player wins the amount of his bet. If a player bets $2, find the player's expectation. (Round your answer to two decimal places.)

In this roulette game, the player bets that the ball will land on a black number. To calculate the player's expectation (expected value of the bet), we need to evaluate the possible outcomes and their associated probabilities.

### Step 1: Define the outcomes
- **Number of black numbers**: 18
- **Number of red numbers**: 18
- **Number of green numbers**: 2 (which correspond to the numbers 0 and 00)

The total number of outcomes on a roulette wheel is 18 (black) + 18 (red) + 2 (green) = 38.

### Step 2: Calculate probabilities
- The probability of landing on a black number:
  \[
  P(\text{black}) = \frac{18}{38}
  \]
  
- The probability of not landing on a black number (either red or green):
  \[
  P(\text{not black}) = 1 - P(\text{black}) = \frac{20}{38}
  \]

### Step 3: Define winnings and losses
- If the player wins (ball lands on black), they win the amount of their bet, which is $2.
- If the player loses (ball does not land on black), they lose their bet, which is $2.

### Step 4: Calculate the expectation
The expected value $ E $ of a bet is calculated as:
\[
E = (\text{win amount} \times P(\text{win})) + (\text{loss amount} \times P(\text{loss}))
\]

Substitute the values:
\[
E = (2 \times \frac{18}{38}) + (-2 \times \frac{20}{38})
\]

Simplify:
\[
E = \frac{36}{38} - \frac{40}{38} = \frac{-4}{38}
\]

Thus, the expected value is:
\[
E = -\frac{2}{19} \approx -0.11
\]

### Conclusion:
The player's expected value of the bet is **-$0.11**. This means that, on average, the player will lose 11 cents for every $2 bet placed on black.

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Would you like more details on any part of this solution or further clarification? Here are 5 related questions to explore:

1. What is the expectation if a player bets $5 instead of $2?
2. How does the expectation change if the roulette wheel has only one green number?
3. What is the expected value if the player bets on a red number instead of black?
4. How is the expectation affected by changing the payout ratio?
5. How can probability theory explain the house advantage in roulette?

**Tip:** In gambling, a negative expected value means that the game is unfavorable in the long run, favoring the house over the player.

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, 18 are red, and two are green.) If the ball lands on a black number, the player wins the amount of his bet. If a player bets $2, find the player's expectation. (Round your answer to two decimal places.)

Find the player's expected value when betting $2 on a black number in roulette. The solution involves calculating probabilities for different outcomes and applying the expected value formula. This problem covers key probability concepts and is suitable for students in Grades 10-12.

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