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This problem involves calculating the expected value of a random game based on a deck of cards, where Jessica wins or loses money depending on whether the card drawn is odd or even.

Here's a breakdown of the problem:

- Jessica has a deck of 10 cards numbered from 1 to 10.
- If she draws an **odd-numbered** card, she wins an amount equal to the value of the card.
- If she draws an **even-numbered** card, she loses $5.

### Step-by-step solution:

1. **List of odd and even cards:**
   - Odd-numbered cards: 1, 3, 5, 7, 9.
   - Even-numbered cards: 2, 4, 6, 8, 10.

2. **Calculating expected value:**
   - The probability of drawing an odd card is $ \frac{5}{10} = 0.5 $.
   - The probability of drawing an even card is also $ 0.5 $.

   For the odd cards:
   - If she draws card 1, she wins $1.
   - If she draws card 3, she wins $3.
   - If she draws card 5, she wins $5.
   - If she draws card 7, she wins $7.
   - If she draws card 9, she wins $9.

   The expected winnings from an odd-numbered card are:
   \[
   \text{Average winnings from odd cards} = \frac{1 + 3 + 5 + 7 + 9}{5} = \frac{25}{5} = 5
   \]
   Therefore, if she draws an odd card, she expects to win $5.

   For the even cards:
   - She always loses $5 if she draws an even-numbered card.

3. **Expected value calculation:**
   The overall expected value (EV) of the game can be calculated as follows:
   \[
   \text{EV} = (0.5 \times 5) + (0.5 \times -5) = 2.5 - 2.5 = 0
   \]
   So, the expected value of the game is **0 dollars**.

4. **Interpretation:**
   Since the expected value is 0, Jessica can expect to **break even** in the long run.

### Final answers:
- (a) The expected value of playing the game is **0 dollars**.
- (b) Jessica can expect to **break even (neither gain nor lose money)** after playing the game many times.

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Would you like more details or have any other questions about this problem?

Here are 5 related questions for further exploration:

1. What is the concept of expected value in probability theory?
2. How does the probability of winning or losing affect the expected value of a game?
3. What would happen to the expected value if Jessica lost more than $5 for even cards?
4. How can expected value be applied to real-world decision-making?
5. How does the variance of a game impact risk assessment?

**Tip:** When calculating expected value, always consider both the probabilities and the outcomes to get a complete picture!

Jessica has a deck of 10 cards numbered 1 through 10. She wins an amount of money equal to the value of the card if an odd-numbered card is drawn. She loses $5 if an even-numbered card is drawn. What is the expected value of playing the game?

Solve the card game probability problem where Jessica wins if an odd card is drawn and loses $5 for an even card. This solution involves expected value calculation, probability concepts, and detailed step-by-step explanations. Suitable for high school students in Grades 10-12.

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