To calculate the expected value (or expectation) for a person buying one ticket, we need to consider the probabilities and outcomes of the different prize amounts.

Information given:

• 1000 tickets are sold, each for $4.
• The prizes:
  • One $1000 prize
  • One $500 prize
  • Two $100 prizes
• The total number of tickets is 1000.

Step 1: Calculate the probability of each outcome

• Probability of winning the $1000 prize:
  $\frac{1}{1000}$

• Probability of winning the $500 prize:
  $\frac{1}{1000}$

• Probability of winning one of the $100 prizes:
  $\frac{2}{1000} = \frac{1}{500}$

• Probability of not winning any prize:
  There are 1000 tickets, and 4 total prizes (1 + 1 + 2). So, the probability of no prize is:
  $\frac{1000 - 4}{1000} = \frac{996}{1000}$

Step 2: Determine the expected value for each outcome

The expected value is calculated by multiplying each prize by its corresponding probability and adding these products together.

$\text{Expected value} = \left( \text{Prize amount} \times \text{Probability} \right)$

• Expected value for $1000 prize:
  $1000 \times \frac{1}{1000} = 1$

• Expected value for $500 prize:
  $500 \times \frac{1}{1000} = 0.5$

• Expected value for $100 prize:
  $100 \times \frac{2}{1000} = 0.2$

• Expected value for no prize (loss of $4 for ticket):
  $-4 \times \frac{996}{1000} = -3.984$

Step 3: Add up the expected values

Now, we add these values together to find the total expected value for the person buying one ticket:

$\text{Total expected value} = 1 + 0.5 + 0.2 - 3.984 = -2.284$

Step 4: Round the result to the nearest cent

The expected value is approximately $-2.28$ dollars.

Final Answer:

The expectation for a person buying one ticket is -2.28 dollars.

Would you like further details or have any questions about this calculation? Here are some related questions that might interest you:

1. How does the number of tickets affect the expected value in a lottery?
2. What would the expected value be if the ticket cost was different (say $5)?
3. How can the concept of expected value be applied in other scenarios, such as insurance or investments?
4. What happens to the expected value if there are more prizes of smaller amounts?
5. How do probabilities influence the decision to play in a lottery?

Tip: The expected value can often help you understand the "average" outcome over time, but it doesn't guarantee your result on a single trial.