To calculate the expected value $E(X)$, we sum the products of the probabilities and the values for each possible outcome:

1. Calculate the probabilities for each prize:

   • Total tickets sold: 5,000
   • Probability of winning a $500 prize: $\frac{1}{5000}$
   • Probability of winning a $200 prize: $\frac{3}{5000}$
   • Probability of winning a $50 prize: $\frac{5}{5000}$
   • Probability of winning a $5 prize: $\frac{20}{5000}$
   • Probability of winning no prize: $\frac{4971}{5000}$

2. Calculate the contribution to $E(X)$ from each outcome:

   • For $500: \frac{1}{5000} \times 500 = 0.1
   • For $200: \frac{3}{5000} \times 200 = 0.12
   • For $50: \frac{5}{5000} \times 50 = 0.05
   • For $5: \frac{20}{5000} \times 5 = 0.02
   • For $0 (no prize): \frac{4971}{5000} \times 0 = 0

3. Sum these values: $E(X) = 0.1 + 0.12 + 0.05 + 0.02 = 0.29$

Thus, the expected value $E(X)$ is $0.29. This means that, on average, each ticket buyer can expect to win 29 cents per ticket.

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Do you want more details or have questions about this calculation?

Here are some follow-up questions for better understanding:

1. How does the probability of winning each prize affect the expected value?
2. What happens to $E(X)$ if more tickets are sold for the same prizes?
3. How is the expected value helpful in evaluating whether to buy a ticket?
4. What is the variance and standard deviation of this raffle's winnings?
5. Can we design a new prize structure to increase $E(X)$?

Tip: Expected value is a long-term average, not a guarantee for individual outcomes!