*(c) If each ticket costs $1, how many tickets should be purchased to get an*

*expected return of $1000?*

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?

---

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.

Learn how to determine the number of tickets needed to achieve an expected return of $1000. This problem involves calculating expected value using probability concepts, with each ticket costing $1. Suitable for advanced high school students.

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