1. **State the problem:** Calculate the expected value of the winnings from a spinner with 10 equal regions: 3 red, 4 blue, 2 yellow, and 1 green. 2. **Formula for expected value:** $$E(X) = \sum (x_i \cdot P(x_i))$$ where $x_i$ is the value of outcome $i$ and $P(x_i)$ is the probability of outcome $i$. 3. **Identify values and probabilities:** - Red: win 4, probability $\frac{3}{10}$ - Green: win 2, probability $\frac{1}{10}$ - Blue: lose 3, probability $\frac{4}{10}$ - Yellow: lose 1, probability $\frac{2}{10}$ 4. **Calculate expected value:** $$E(X) = 4 \times \frac{3}{10} + 2 \times \frac{1}{10} + (-3) \times \frac{4}{10} + (-1) \times \frac{2}{10}$$ 5. **Simplify each term:** $$= \frac{12}{10} + \frac{2}{10} - \frac{12}{10} - \frac{2}{10}$$ 6. **Combine terms:** $$= \frac{12 + 2 - 12 - 2}{10}$$ $$= \frac{0}{10}$$ 7. **Final answer:** $$E(X) = 0$$ The expected value of the game is 0, meaning on average the player neither wins nor loses money over many spins.