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.

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!