1. **State the problem:** We want to find the expected winnings for a ticket buyer in a raffle with two prizes and 11000 tickets sold. 2. **Identify the prizes and probabilities:** - Grand prize: European cruise valued at 5500 - Second prize: Las Vegas getaway valued at 800 - Number of tickets sold: 11000 - Cost per ticket: 4 3. **Calculate the probability of winning each prize:** - Probability of winning grand prize = \frac{1}{11000} - Probability of winning second prize = \frac{1}{11000} - Probability of winning no prize = 1 - \frac{1}{11000} - \frac{1}{11000} = 1 - \frac{2}{11000} = \frac{10998}{11000} 4. **Calculate expected winnings before subtracting ticket cost:** $$ E = 5500 \times \frac{1}{11000} + 800 \times \frac{1}{11000} + 0 \times \frac{10998}{11000} = \frac{5500 + 800}{11000} = \frac{6300}{11000} $$ 5. **Simplify the fraction:** $$ E = \frac{6300}{11000} = \frac{\cancel{6300}}{\cancel{11000}} = \frac{63}{110} \approx 0.5727 $$ 6. **Subtract the cost of the ticket to find net expected winnings:** $$ \text{Net expected winnings} = 0.5727 - 4 = -3.4273 $$ 7. **Interpretation:** On average, a ticket buyer expects to lose about 3.427 dollars per ticket. **Final answer:** The expected winnings for a ticket buyer is approximately **-3.427** dollars.