To calculate the expected return on a $1 lottery ticket, we need to determine the probability of winning each prize and then compute the expected value based on those probabilities.

Step 1: Determine the probabilities of winning each prize.

• Total number of tickets sold: $4,000,000$

The probabilities of winning each prize are calculated as follows:

• Grand prize: There is 1 grand prize. $P(\text{Grand Prize}) = \frac{1}{4,000,000}$
• Runner-up prizes: There are 2 runner-up prizes. $P(\text{Runner-up Prize}) = \frac{2}{4,000,000}$
• Third-place prizes: There are 6 third-place prizes. $P(\text{Third Place Prize}) = \frac{6}{4,000,000}$
• Consolation prizes: There are 19 consolation prizes. $P(\text{Consolation Prize}) = \frac{19}{4,000,000}$

Step 2: Calculate the expected value for each prize.

The expected value of a lottery ticket is the sum of the products of the prize amounts and their respective probabilities:

$\text{Expected Value} = (1,000,000 \times \frac{1}{4,000,000}) + (100,000 \times \frac{2}{4,000,000}) + (10,000 \times \frac{6}{4,000,000}) + (1,000 \times \frac{19}{4,000,000})$

Step 3: Perform the calculations.

• Grand prize expected value: $1,000,000 \times \frac{1}{4,000,000} = 0.25$
• Runner-up prizes expected value: $100,000 \times \frac{2}{4,000,000} = 0.05$
• Third-place prizes expected value: $10,000 \times \frac{6}{4,000,000} = 0.015$
• Consolation prizes expected value: $1,000 \times \frac{19}{4,000,000} = 0.00475$

Step 4: Add the expected values.

$\text{Total Expected Value} = 0.25 + 0.05 + 0.015 + 0.00475 = 0.31975$

Thus, the expected return on a $1 ticket is approximately $0.32 (rounded to two decimal places).

Final Answer

The expected return on a $1 lottery ticket is $0.32.

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Do you have any questions or want more details?

Here are 5 related questions for further exploration:

1. How does the expected return change if the total number of tickets sold decreases?
2. What is the probability of not winning any prize?
3. How would the expected return change if the grand prize was increased to $2,000,000?
4. How does the variance in the lottery's payouts affect the perceived risk?
5. What would be the expected return if a ticket costs $2 instead?

Tip: When calculating probabilities, always ensure that the sum of all probabilities equals 1 for a complete set of mutually exclusive outcomes.