1. **State the problem:** Calculate the expected value of the spinner game where the wheel has 10 equal regions: 3 red, 4 blue, 2 yellow, and 1 green. 2. **Formula for expected value:** $$E(X) = \sum (\text{probability of outcome} \times \text{value of outcome})$$ 3. **Calculate probabilities:** - Probability(red) = $\frac{3}{10}$ - Probability(blue) = $\frac{4}{10}$ - Probability(yellow) = $\frac{2}{10}$ - Probability(green) = $\frac{1}{10}$ 4. **Values for each color:** - Red: +4 - Green: +2 - Blue: -3 - Yellow: -1 5. **Calculate expected value:** $$E(X) = \frac{3}{10} \times 4 + \frac{4}{10} \times (-3) + \frac{2}{10} \times (-1) + \frac{1}{10} \times 2$$ 6. **Simplify step-by-step:** $$E(X) = \frac{3}{10} \times 4 + \frac{4}{10} \times (-3) + \frac{2}{10} \times (-1) + \frac{1}{10} \times 2$$ $$= \frac{12}{10} - \frac{12}{10} - \frac{2}{10} + \frac{2}{10}$$ 7. **Cancel terms:** $$= \cancel{\frac{12}{10}} - \cancel{\frac{12}{10}} - \frac{2}{10} + \frac{2}{10}$$ $$= 0$$ 8. **Interpretation:** The expected value is $0$, meaning on average, the player neither wins nor loses money per game. 9. **Answer to part b:** Since the expected value per game is $0$, over 10 games the player can expect to break even. **Final answers:** - a. Expected value = $0.00$ - b. Answer: C. Over 10 games, the player can expect to break even.