1. **State the problem:** We have a game where a player pays 5 to draw a coin from a chest containing 1 gold, 4 silver, 10 copper, and 25 brass coins. Each coin has a different winning amount. We want to find the expected value of the game and determine if the game is fair. 2. **List the outcomes and probabilities:** - Total coins = 1 + 4 + 10 + 25 = 40 - Probability of gold = $\frac{1}{40}$ - Probability of silver = $\frac{4}{40} = \frac{1}{10}$ - Probability of copper = $\frac{10}{40} = \frac{1}{4}$ - Probability of brass = $\frac{25}{40} = \frac{5}{8}$ 3. **List the outcomes (winnings) for each coin:** - Gold: 40 - Silver: 15 - Copper: 5 - Brass: 2 4. **Calculate $x \cdot P(x)$ for each coin:** - Gold: $40 \times \frac{1}{40} = 1$ - Silver: $15 \times \frac{1}{10} = 1.5$ - Copper: $5 \times \frac{1}{4} = 1.25$ - Brass: $2 \times \frac{5}{8} = 1.25$ 5. **Calculate the expected value $E(x)$:** $$ E(x) = 1 + 1.5 + 1.25 + 1.25 = 5 $$ 6. **Interpretation:** The expected winnings per game is 5, but the player pays 5 to play. 7. **Is the game fair?** A game is fair if the expected winnings equal the cost to play. Here, expected winnings = 5 and cost = 5, so the game is fair. **Final answer:** The expected value $E(x)$ is 5, so the game is fair because the expected winnings equal the cost to play.