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 $5, find the player's expectation. (Round your answer to two decimal places.)

$ 

In a game of roulette, there are 18 black numbers, 18 red numbers, and 2 green numbers, making a total of $ 18 + 18 + 2 = 38 $ numbers. A player who bets $5 on black has a chance of winning or losing based on where the ball lands.

Here’s how to calculate the expectation:

1. **Probability of winning** (landing on a black number):  
   \[
   \frac{18}{38}
   \]

2. **Probability of losing** (landing on red or green):  
   \[
   \frac{20}{38}
   \]

3. **Winnings if the player wins**:  
   If the player wins, they receive their bet back plus an additional $5, so they win $5.

4. **Loss if the player loses**:  
   If the player loses, they lose the $5 bet.

Now we can set up the expectation calculation, $ E $:

\[
E = \left(\text{Probability of winning} \times \text{Winnings}\right) + \left(\text{Probability of losing} \times \text{Loss}\right)
\]

Substituting the values:

\[
E = \left(\frac{18}{38} \times 5\right) + \left(\frac{20}{38} \times (-5)\right)
\]

Calculating each part:

\[
E = \left(\frac{18}{38} \times 5\right) - \left(\frac{20}{38} \times 5\right)
\]

\[
E = \frac{18 \cdot 5 - 20 \cdot 5}{38}
\]

\[
E = \frac{90 - 100}{38} = \frac{-10}{38} \approx -0.26
\]

So, the player’s expectation is approximately **-0.26**.

This means that, on average, the player expects to lose $0.26 for every $5 bet they place on black.

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Would you like further explanation, or do you have any questions?

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1. How would the expectation change with different bet amounts?
2. What is the impact of the green numbers on the player’s expectation?
3. How does the expectation differ if the game used a single-zero roulette wheel (European style)?
4. What is the expected value of betting on red instead of black?
5. How would increasing the payout affect the player's expectation?

**Tip:** In expectation calculations, carefully consider the probability of each outcome and multiply it by the gain or loss for that outcome.

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 $5, find the player's expectation. (Round your answer to two decimal places.)

Calculate the expected value for a $5 bet on black in roulette, where there are 18 black numbers, 18 red numbers, and 2 green numbers. Understand the probability of winning and losing, and learn how to compute expectation in gambling scenarios. Suitable for high school students studying probability.

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