1. **State the problem:** You pay 2 dollars to roll a fair six-sided die. You receive back the number of dollars equal to the number rolled. The net gain (payoff) is the amount received minus 2 dollars. 2. **Formula for expected value:** The expected value $E$ of a discrete random variable is given by $$E = \sum (\text{value} \times \text{probability})$$ 3. **Define the payoff values:** If the die shows $k$ dots, the payoff is $k - 2$ dollars. 4. **Calculate probabilities:** Each face has probability $\frac{1}{6}$ since the die is fair. 5. **Calculate expected payoff:** $$E = \sum_{k=1}^6 (k - 2) \times \frac{1}{6} = \frac{1}{6} \sum_{k=1}^6 (k - 2)$$ 6. **Sum the payoffs:** $$\sum_{k=1}^6 (k - 2) = (1-2) + (2-2) + (3-2) + (4-2) + (5-2) + (6-2) = (-1) + 0 + 1 + 2 + 3 + 4 = 9$$ 7. **Calculate expected value:** $$E = \frac{9}{6} = 1.5$$ 8. **Interpretation:** The expected payoff is 1.5 dollars, meaning on average you gain 1.5 dollars per game. 9. **Is the game fair?** Since you pay 2 dollars to play and expect to gain 1.5 dollars, your net expected gain is $1.5 - 2 = -0.5$ dollars, which is a loss. Therefore, the game is not fair; it favors the house. **Final answer:** The expected payoff of the game is $1.5$ dollars. The game is not fair because the expected net gain is negative.