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. **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 = $1 - 2 = -1$ - If the die shows 2, net gain = $2 - 2 = 0$ - If the die shows 3, net gain = $3 - 2 = 1$ - If the die shows 4, net gain = $4 - 2 = 2$ - If the die shows 5, net gain = $5 - 2 = 3$ - If the die shows 6, net gain = $6 - 2 = 4$ 4. **Calculate expected value:** Since the die is fair, each outcome has probability $\frac{1}{6}$. $$E = \frac{1}{6}(-1) + \frac{1}{6}(0) + \frac{1}{6}(1) + \frac{1}{6}(2) + \frac{1}{6}(3) + \frac{1}{6}(4)$$ 5. **Simplify:** $$E = \frac{-1 + 0 + 1 + 2 + 3 + 4}{6} = \frac{9}{6} = 1.5$$ 6. **Interpretation:** The expected net gain is $1.5$. Since you pay 2 dollars to play, on average you gain 1.5 dollars net, so the game is favorable to the player and not fair (fair would mean expected net gain of 0).