One of the wagers in the game of roulette is to place a bet that the ball will land on a red number. (Eighteen of the numbers are black, 18 are red, and two are green.) If the ball lands on a red number, the player wins the amount of his bet. If a player bets $7, find the player's expectation. (Round your answer to two decimal places.)

In a standard game of roulette, there are 38 possible outcomes (18 red, 18 black, and 2 green). The player's expectation can be calculated using the formula for expected value:

\[
E = P(\text{win}) \times (\text{amount won}) + P(\text{loss}) \times (\text{amount lost})
\]

Where:
- $ P(\text{win}) $ is the probability of landing on a red number.
- $ P(\text{loss}) $ is the probability of not landing on a red number.
- The amount won is $7 (the amount of the bet).
- The amount lost is the bet itself, $7.

### Step 1: Calculate the probabilities
- $ P(\text{win}) $: The probability of landing on a red number is $\frac{18}{38}$.
- $ P(\text{loss}) $: The probability of not landing on a red number (i.e., landing on black or green) is $\frac{20}{38}$.

### Step 2: Calculate the expectation
Using the formula:

\[
E = \left( \frac{18}{38} \times 7 \right) + \left( \frac{20}{38} \times (-7) \right)
\]

\[
E = \left( \frac{18}{38} \times 7 \right) + \left( \frac{20}{38} \times -7 \right)
\]

\[
E = \left( \frac{126}{38} \right) - \left( \frac{140}{38} \right)
\]

\[
E = \frac{126 - 140}{38} = \frac{-14}{38} \approx -0.37
\]

### Conclusion:
The player's expectation is approximately **-0.37**. This means, on average, the player will lose 37 cents for each $7 bet placed on red.

Would you like further clarification or additional details on this calculation? Here are five related questions to explore:

1. How does the house advantage impact the player's expected value in roulette?
2. What is the expected value if the player places a larger bet?
3. How would the expectation change if there were only 1 green number instead of 2?
4. What is the probability of winning a bet on black numbers instead?
5. How does the expected value compare for different types of roulette wagers (e.g., odd/even or specific numbers)?

**Tip:** In games of chance, the expected value gives you a sense of how much you might win or lose over time, helping you understand the long-term outcome of placing repeated bets.

If a player bets $7 on red in a game of roulette, what is the player's expectation? (18 red, 18 black, 2 green. Round to two decimal places.)

Learn how to calculate the expected value of a $7 bet on red in roulette. Explore probability concepts and expected value formulas to determine the player's average loss over time. Ideal for Grades 10-12 math learners.

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