To solve this problem, we'll use the concept of expected value.

Expected Value Formula

The expected value $E(X)$ of a random variable $X$ is given by:

$E(X) = \sum (x \times p(x))$

where:

• $x$ represents the outcome.
• $p(x)$ represents the probability of that outcome.

Problem Breakdown

Given:

• The cost per ticket is $1.
• The expected return is $1000.

Let:

• $n$ be the number of tickets to purchase.
• $P$ be the probability of winning per ticket.
• $R$ be the prize amount if a ticket wins.

We want the expected return (total expected value of all tickets) to equal $1000:

$n \times (R \times P) = 1000$

Since the cost of each ticket is $1:

$R \times P = 1$

Therefore:

$n \times 1 = 1000$

$n = 1000$

So, 1000 tickets should be purchased to achieve an expected return of $1000.

Would you like any further details or explanations?

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Here are some related questions:

1. How does changing the prize amount affect the number of tickets needed for a specific expected return?
2. What is the concept of expected value, and how is it applied in other scenarios?
3. How does the probability of winning a ticket affect the expected return?
4. If the ticket cost were $2, how would that change the number of tickets needed?
5. What happens to the expected return if more tickets are purchased?
6. How do you calculate the expected value in a scenario with multiple prize levels?
7. How does variance impact decision-making in scenarios involving expected return?
8. What are some real-world applications of expected value calculations in finance or business?

Tip: Expected value is a fundamental concept in probability theory that helps in making informed decisions in uncertain situations.