1. **State the problem:** You pay 2 dollars to play a game where a fair six-sided die is rolled. You receive a payout equal to the number of dots on the die face. The net gain is the payout minus the 2 dollars paid. 2. **Define the random variable and payoff:** Let $X$ be the number rolled on the die, so $X \in \{1,2,3,4,5,6\}$ with equal probability $\frac{1}{6}$ each. The net gain (payoff) is $G = X - 2$. 3. **Expected value formula:** The expected value $E(G)$ is given by $$ E(G) = E(X - 2) = E(X) - 2 $$ 4. **Calculate $E(X)$:** Since the die is fair, $$ E(X) = \sum_{k=1}^6 k \cdot \frac{1}{6} = \frac{1+2+3+4+5+6}{6} = \frac{21}{6} = 3.5 $$ 5. **Calculate $E(G)$:** $$ E(G) = 3.5 - 2 = 1.5 $$ 6. **Interpretation:** The expected payoff is 1.5 dollars per game. 7. **Is the game fair?** A game is fair if the expected net gain is zero. Here, $E(G) = 1.5 > 0$, so the player expects to gain 1.5 dollars on average. **Therefore, the game is favorable to the player and not fair to the house.**