1. **State the problem:** We have a spinner with 10 equal regions: 3 red, 4 blue, 2 yellow, and 1 green. The winnings are: - Red: 3 dollars - Green: 1 dollar - Blue: -1 dollar (lose 1) - Yellow: -3 dollars (lose 3) We want to find: a) The expected value of the winnings per spin. b) The interpretation of this expected value over 10 games. 2. **Formula for expected value:** $$E(X) = \sum (\text{value} \times \text{probability})$$ 3. **Calculate probabilities:** - Probability(red) = $\frac{3}{10}$ - Probability(blue) = $\frac{4}{10}$ - Probability(yellow) = $\frac{2}{10}$ - Probability(green) = $\frac{1}{10}$ 4. **Calculate expected value:** $$E(X) = 3 \times \frac{3}{10} + (-1) \times \frac{4}{10} + (-3) \times \frac{2}{10} + 1 \times \frac{1}{10}$$ 5. **Simplify step-by-step:** $$E(X) = \frac{9}{10} - \frac{4}{10} - \frac{6}{10} + \frac{1}{10}$$ 6. **Combine terms:** $$E(X) = \frac{9 - 4 - 6 + 1}{10} = \frac{0}{10} = 0$$ 7. **Interpretation:** The expected value per spin is 0, meaning on average, the player neither wins nor loses money per spin. 8. **Over 10 games:** Since the expected value per game is 0, over 10 games the expected total winnings is: $$10 \times 0 = 0$$ This means the player can expect to break even over 10 games. **Answer for b:** D. Over 10 games, the player can expect to break even.