1. **State the problem:** We have 5000 tickets sold at 1 each for a charity raffle. Prizes are: 1 prize of 800, 3 prizes of 300, 5 prizes of 10, and 20 prizes of 5. We want to find the expected value $E(X)$ of the amount won if you buy 1 ticket. 2. **Formula for expected value:** $$E(X) = \sum (\text{value of prize} \times \text{probability of winning that prize})$$ 3. **Calculate probabilities:** - Probability of winning 800: $\frac{1}{5000}$ - Probability of winning 300: $\frac{3}{5000}$ - Probability of winning 10: $\frac{5}{5000}$ - Probability of winning 5: $\frac{20}{5000}$ - Probability of winning nothing: $1 - \frac{1+3+5+20}{5000} = 1 - \frac{29}{5000} = \frac{4971}{5000}$ 4. **Calculate expected value:** $$E(X) = 800 \times \frac{1}{5000} + 300 \times \frac{3}{5000} + 10 \times \frac{5}{5000} + 5 \times \frac{20}{5000} + 0 \times \frac{4971}{5000}$$ 5. **Simplify each term:** $$= \frac{800}{5000} + \frac{900}{5000} + \frac{50}{5000} + \frac{100}{5000} + 0$$ 6. **Sum the numerators:** $$= \frac{800 + 900 + 50 + 100}{5000} = \frac{1850}{5000}$$ 7. **Simplify the fraction:** $$= \frac{1850}{5000} = \frac{\cancel{1850}}{\cancel{5000}} = \frac{37}{100} = 0.37$$ 8. **Interpretation:** The expected value of the raffle ticket is 0.37 dollars, meaning on average you expect to win 37 cents per ticket. **Final answer:** $$E(X) = 0.37$$