1. **State the problem:** You pay 5 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 5 dollars. 2. **Formula for expected value:** The expected value $E$ of the net gain is given by $$E = \sum (\text{net gain} \times \text{probability})$$ 3. **Calculate net gains for each outcome:** - If the die shows 1, net gain = $1 - 5 = -4$ - If the die shows 2, net gain = $2 - 5 = -3$ - If the die shows 3, net gain = $3 - 5 = -2$ - If the die shows 4, net gain = $4 - 5 = -1$ - If the die shows 5, net gain = $5 - 5 = 0$ - If the die shows 6, net gain = $6 - 5 = 1$ 4. **Calculate expected value:** Since the die is fair, each outcome has probability $\frac{1}{6}$. $$E = \frac{1}{6}(-4) + \frac{1}{6}(-3) + \frac{1}{6}(-2) + \frac{1}{6}(-1) + \frac{1}{6}(0) + \frac{1}{6}(1)$$ 5. **Simplify the sum:** $$E = \frac{-4 - 3 - 2 - 1 + 0 + 1}{6} = \frac{-9}{6}$$ 6. **Simplify fraction:** $$E = \frac{\cancel{-9}}{\cancel{6}} = -1.5$$ 7. **Interpretation:** The expected value of the game is $-1.5$, meaning on average you lose 1.5 dollars per game. 8. **Is the game fair?** A fair game has an expected value of 0. Since $E = -1.5$, the game is not fair; it favors the house. **Final answer:** The expected payoff of the game is $-1.5$. The game is not fair.