Kira has a deck of 10 cards numbered 1 through 10 . She is playing a game of chance.
This game is this: Kira chooses one card from the deck at random. She wins an amount of money equal to the value of the card if an even numbered card is drawn. She loses $7 if an odd numbered card is drawn. (a)Find the expected value of playing the game.

To find the expected value of the game Kira is playing, we need to consider all possible outcomes and their associated probabilities. Here's how we can calculate the expected value step-by-step:

### Step 1: List Possible Outcomes
Kira has a deck of 10 cards, numbered 1 through 10. The outcomes and their probabilities depend on whether a card is even or odd:

- **Even cards**: 2, 4, 6, 8, 10
- **Odd cards**: 1, 3, 5, 7, 9

### Step 2: Determine Probabilities
Since each card is equally likely to be drawn, the probability of drawing any specific card is $\frac{1}{10}$.

- Probability of drawing an even card: $\frac{5}{10} = \frac{1}{2}$
- Probability of drawing an odd card: $\frac{5}{10} = \frac{1}{2}$

### Step 3: Calculate Expected Value
The expected value $E$ is given by the formula:

\[
E = \sum (\text{probability of outcome} \times \text{value of outcome})
\]

#### Value of Outcomes:
- **Even card**: Kira wins the value of the card. Therefore, the possible winnings are 2, 4, 6, 8, and 10.
- **Odd card**: Kira loses $7.

We compute the expected value by calculating the contributions from both even and odd outcomes separately:

#### Contribution from Even Cards:
The contribution from even cards is the sum of the products of each card's probability and its value:

\[
\begin{align*}
\text{Expected value from even cards} & = \frac{1}{10}(2) + \frac{1}{10}(4) + \frac{1}{10}(6) + \frac{1}{10}(8) + \frac{1}{10}(10) \\
& = \frac{2 + 4 + 6 + 8 + 10}{10} \\
& = \frac{30}{10} \\
& = 3
\end{align*}
\]

#### Contribution from Odd Cards:
The contribution from odd cards is the product of the probability of drawing an odd card and the loss:

\[
\begin{align*}
\text{Expected value from odd cards} & = \frac{1}{2}(-7) \\
& = -\frac{7}{2} \\
& = -3.5
\end{align*}
\]

### Step 4: Combine Contributions
Now, we combine the contributions from the even and odd outcomes to find the overall expected value:

\[
E = \text{Expected value from even cards} + \text{Expected value from odd cards}
\]

\[
E = 3 + (-3.5) = 3 - 3.5 = -0.5
\]

### Conclusion
The expected value of playing the game is $-0.5$. This means that, on average, Kira can expect to lose $0.50 per game played.

---

If you have any further questions or need more clarification, feel free to ask!

Here are some related questions that might deepen your understanding:

1. What is the expected value if the game is modified such that Kira wins double the card value for even cards?
2. How would the expected value change if there are only 8 cards, with numbers from 1 to 8?
3. What would be the expected value if Kira only loses $5 for drawing an odd card?
4. How does changing the deck to include cards from 1 to 12 affect the expected value?
5. If Kira plays this game 100 times, what is her expected total loss or gain?
6. What is the expected value if Kira wins $10 for every even card instead of the card value?
7. How would the expected value change if the probability of drawing an even card is increased to $\frac{2}{3}$?
8. How can the game be adjusted to make the expected value zero?

**Tip:** When calculating expected values in probability games, ensure you account for all possible outcomes and their respective probabilities.

Learn how to calculate the expected value of a game where Kira draws cards numbered 1 through 10. If Kira draws an even card, she wins the card's value; if she draws an odd card, she loses $7. Understand the probabilities involved and how to compute the expected value step-by-step.

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