To solve this problem, we will first calculate the expected value of a single game by considering the probability and outcome of each possible event. Then, we'll interpret this value in the context of playing ten games.

Part 1: Expected Value Calculation

The expected value $E(X)$ can be calculated using the formula:

$E(X) = \sum_{i} P(x_i) \times x_i$

where:

• $P(x_i)$ is the probability of the outcome $x_i$,
• $x_i$ is the outcome (monetary value) for that event.

Step 1: List the outcomes and their probabilities:

1. Red:

   • Probability: $\frac{3}{10}$
   • Outcome: +$1

2. Blue:

   • Probability: $\frac{4}{10}$
   • Outcome: -$1

3. Yellow:

   • Probability: $\frac{2}{10}$
   • Outcome: -$2

4. Green:

   • Probability: $\frac{1}{10}$
   • Outcome: +$5

Step 2: Calculate the expected value $E(X)$:

$E(X) = \left(\frac{3}{10} \times 1\right) + \left(\frac{4}{10} \times -1\right) + \left(\frac{2}{10} \times -2\right) + \left(\frac{1}{10} \times 5\right)$

$E(X) = \left(\frac{3}{10} \times 1\right) + \left(\frac{4}{10} \times -1\right) + \left(\frac{2}{10} \times -2\right) + \left(\frac{1}{10} \times 5\right)$

$E(X) = \frac{3}{10} - \frac{4}{10} - \frac{4}{10} + \frac{5}{10}$

$E(X) = \frac{3 - 4 - 4 + 5}{10} = \frac{0}{10} = 0$

Final Answer for Part 1:

The expected value $E(X)$ is $0.00 (after rounding to the nearest cent).

Part 2: Interpretation Over 10 Games

Since the expected value for one game is $0.00, over 10 games, the player can expect to break even on average. Therefore, the correct interpretation is:

B. Over 10 games, the player can expect to break even.

Questions:

1. What would the expected value be if the spinner had more red regions?
2. How would the expected value change if the payout for landing on green increased?
3. What is the expected value if all regions had equal payouts but different probabilities?
4. How does the variance in outcomes affect the reliability of the expected value?
5. How would the expected value change if there were more regions and outcomes?

Tip: When calculating expected values, consider both the probabilities and the outcomes—they determine whether the game is fair, favorable, or unfavorable.