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. dollars.

Let's go through the calculation for the expected value again, ensuring clarity and precision:

### Step 1: Define the Outcomes

Kira has 10 cards numbered from 1 to 10. The outcomes depend on whether the drawn card is even or odd:

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

### Step 2: Determine Probabilities

Each card has an equal probability of being drawn, which 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 the Expected Value

The expected value $E$ is calculated using the formula:

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

#### Values 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.

#### Expected Value from Even Cards

Calculate the expected value for winning with even cards:

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

#### Expected Value from Odd Cards

Calculate the expected value for losing with odd cards:

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

### Step 4: Combine the Expected Values

The overall expected value of the game is the sum of the expected values from even and odd cards:

\[
\begin{align*}
E & = \text{Expected value from even cards} + \text{Expected value from odd cards} \\
E & = 3 + (-3.5) \\
E & = 3 - 3.5 \\
E & = -0.5
\end{align*}
\]

### Conclusion

The expected value of playing the game is $-0.5$ dollars. This means that, on average, Kira expects to lose $0.50 each time she plays the game.

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 working with expected values, it’s helpful to separate the contributions of different outcomes and ensure probabilities sum up to 1.

Learn how to calculate the expected value of a game where Kira draws cards numbered 1 through 10, winning money for even cards and losing money for odd cards. This problem involves probability calculations and expected value analysis. Suitable for advanced high school students.

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