1. **State the problem:** We have a spinner with 10 equal regions: 3 red, 4 blue, 2 yellow, and 1 green. 2. **Define the winnings:** - Red: win 3 - Green: win 1 - Blue: lose 1 (which is -1) - Yellow: lose 3 (which is -3) 3. **Calculate probabilities:** - Probability(red) = \frac{3}{10} - Probability(blue) = \frac{4}{10} - Probability(yellow) = \frac{2}{10} - Probability(green) = \frac{1}{10} 4. **Expected value formula:** $$E(X) = \sum (\text{value} \times \text{probability})$$ 5. **Calculate expected value:** $$E(X) = 3 \times \frac{3}{10} + (-1) \times \frac{4}{10} + (-3) \times \frac{2}{10} + 1 \times \frac{1}{10}$$ 6. **Simplify step-by-step:** $$E(X) = \frac{9}{10} - \frac{4}{10} - \frac{6}{10} + \frac{1}{10}$$ 7. **Combine terms:** $$E(X) = \frac{9 - 4 - 6 + 1}{10} = \frac{0}{10}$$ 8. **Final answer:** $$E(X) = 0$$ The expected value of the game is 0, meaning on average the player neither wins nor loses money per spin.