1. **State the problem:** A fair coin is flipped. If a head turns up, you win 3. If a tail turns up, you lose 3. We want to find the expected value of the game and determine if the game is fair. 2. **Formula for expected value:** The expected value $E$ of a discrete random variable is given by $$E = \sum (\text{outcome} \times \text{probability of outcome})$$ 3. **Identify outcomes and probabilities:** - Winning 3 with probability $\frac{1}{2}$ (head) - Losing 3 with probability $\frac{1}{2}$ (tail) 4. **Calculate expected value:** $$E = 3 \times \frac{1}{2} + (-3) \times \frac{1}{2}$$ 5. **Simplify:** $$E = \frac{3}{2} - \frac{3}{2}$$ $$E = \cancel{\frac{3}{2}} - \cancel{\frac{3}{2}} = 0$$ 6. **Interpretation:** The expected value is 0, meaning on average, you neither gain nor lose money. 7. **Is the game fair?** A game is fair if the expected value is 0. Since $E=0$, the game is fair. **Final answer:** The expected value of the game is $0$. So, the game is fair.