https://file.aiquickdraw.com/mathbot/file/1728873676977j1mqn5dn.jpg

Let's break down the problem step by step to solve it:

### (a) Finding the expected value of playing the game:
- Brian has a deck of 10 cards numbered 1 through 10. 
- He wins an amount equal to the value of the card if an **odd-numbered** card is drawn.
- He loses **$3.40** if an **even-numbered** card is drawn.

The probability of drawing an odd-numbered card is $ \frac{5}{10} = 0.5 $, since there are 5 odd numbers (1, 3, 5, 7, 9) in the deck.

The probability of drawing an even-numbered card is also $ 0.5 $, since there are 5 even numbers (2, 4, 6, 8, 10) in the deck.

#### Step 1: Calculate the expected value for odd-numbered cards:
The values of the odd-numbered cards are 1, 3, 5, 7, and 9. Since each odd-numbered card is equally likely, the expected gain from drawing an odd card is the average of these values:
\[
\text{Average of odd cards} = \frac{1 + 3 + 5 + 7 + 9}{5} = \frac{25}{5} = 5
\]
So, if an odd-numbered card is drawn, Brian can expect to win $5.

#### Step 2: Calculate the expected value for even-numbered cards:
If an even-numbered card is drawn, Brian loses $3.40.

#### Step 3: Find the overall expected value:
The expected value (EV) is the sum of the expected values for both the odd and even-numbered outcomes, weighted by their probabilities:
\[
EV = (0.5 \times 5) + (0.5 \times -3.40)
\]
\[
EV = 2.5 - 1.7 = 0.8
\]

Thus, the expected value of playing the game is **$0.80**.

---

### (b) What can Brian expect in the long run?
Since the expected value per draw is **$0.80**, Brian can expect to gain money in the long run. Specifically, he can expect to **win $0.80 per draw**.

Let me know if you'd like more details, or if you have any questions!

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Here are 5 related questions:
1. What is the variance of this game?
2. How does the expected value change if the loss for even cards increases to $4?
3. What is the probability that Brian loses money after 10 draws?
4. How would changing the deck size affect the expected value?
5. What if Brian lost a different amount for each even-numbered card?

### Tip:
Expected value provides a long-run average but doesn't guarantee a particular outcome for short-term play.

Brian has a deck of 10 cards numbered 1 through 10. He is playing a game where he wins the value of an odd-numbered card or loses $3.40 on even-numbered cards. What is the expected value of this game?

Learn how to calculate the expected value in a card game where Brian wins money for odd-numbered cards and loses $3.40 for even-numbered cards. This problem involves probability and expected value calculations, suitable for Grades 9-12.

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