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!

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.