To find the expected return on a $1 lottery ticket, we calculate the expected value of the winnings for a single ticket. This involves summing up the products of each prize amount and its associated probability.

Step 1: Define the variables

• Total tickets: $N = 13,000,000$
• Ticket cost: $C = 1$ dollar
• Prizes:
  • Grand prize: $1 \times 1,000,000$
  • Runner-up prizes: $4 \times 50,000$
  • Third-place prizes: $14 \times 5,000$
  • Consolation prizes: $28 \times 1,000$

Step 2: Calculate the probability of each prize

The probability of winning a specific prize is: $P(\text{prize}) = \frac{\text{number of prizes}}{N}$

Probabilities for each prize:

• Grand prize: $P(1,000,000) = \frac{1}{13,000,000}$
• Runner-up prize: $P(50,000) = \frac{4}{13,000,000}$
• Third-place prize: $P(5,000) = \frac{14}{13,000,000}$
• Consolation prize: $P(1,000) = \frac{28}{13,000,000}$

Step 3: Calculate the expected value of winnings

The expected value $E$ is given by: $E = \sum (\text{prize amount} \times \text{probability of winning})$

Substituting the values: $E = \left(1,000,000 \times \frac{1}{13,000,000}\right) + \left(50,000 \times \frac{4}{13,000,000}\right) + \left(5,000 \times \frac{14}{13,000,000}\right) + \left(1,000 \times \frac{28}{13,000,000}\right)$

Simplify each term:

• Grand prize: $1,000,000 \times \frac{1}{13,000,000} = \frac{1,000,000}{13,000,000} = 0.07692$
• Runner-up prize: $50,000 \times \frac{4}{13,000,000} = \frac{200,000}{13,000,000} \approx 0.01538$
• Third-place prize: $5,000 \times \frac{14}{13,000,000} = \frac{70,000}{13,000,000} \approx 0.00538$
• Consolation prize: $1,000 \times \frac{28}{13,000,000} = \frac{28,000}{13,000,000} \approx 0.00215$

Sum these values: $E \approx 0.07692 + 0.01538 + 0.00538 + 0.00215 = 0.09983$

Step 4: Subtract the cost of the ticket

The net expected return is: $E_{\text{net}} = E - C = 0.09983 - 1 = -0.90017$

Final Answer:

The expected return on a $1 lottery ticket is approximately $-\$0.90$.

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Would you like a detailed breakdown of any part? Here are 5 related questions:

1. How is the probability of winning calculated for each prize?
2. Can you explain why the expected return is negative?
3. How would the expected return change if fewer tickets were sold?
4. What does the expected value tell us about playing the lottery?
5. How do the probabilities affect the overall fairness of the lottery?

Tip: The expected value reflects the average outcome over a large number of plays. It does not guarantee individual results!