It looks like you’ve uploaded a problem related to the expected value of a game where Eric draws cards. Let me break it down.

From the image, it appears that:

• Eric draws a card from a deck numbered 1 to 10.
• If Eric draws an even-numbered card, he wins the dollar amount equal to the card's value.
• If Eric draws an odd-numbered card, he loses $1.

Part (a): Find the expected value of playing the game.

To solve this, we calculate the expected value (E), which is a weighted average of all possible outcomes based on their probabilities.

1. Winning outcomes (even cards):

   • The even-numbered cards are 2, 4, 6, 8, and 10.
   • The probability of drawing an even-numbered card is 5 out of 10, or $\frac{5}{10} = 0.5$.
   • The average amount won (the expected winnings) from even-numbered cards is: $\frac{2 + 4 + 6 + 8 + 10}{5} = 6$ So, for even cards, Eric wins $6 on average.

2. Losing outcomes (odd cards):

   • The odd-numbered cards are 1, 3, 5, 7, and 9.
   • The probability of drawing an odd-numbered card is also $\frac{5}{10} = 0.5$.
   • For each odd-numbered card, Eric loses $1, which is constant.

Now, let's calculate the expected value (E):

$E = (\text{probability of even} \times \text{average win}) + (\text{probability of odd} \times \text{average loss})$ $E = (0.5 \times 6) + (0.5 \times (-1)) = 3 - 0.5 = 2.5$ Thus, the expected value of playing the game is $2.50 per draw.

Part (b): What can Eric expect in the long run?

Since the expected value is positive ($2.50), over the long run, Eric can expect to gain money on average. Hence, his net expected change will be positive.

Would you like further details on any part of the solution or have additional questions? Here are some questions to explore further:

1. How would the expected value change if the payouts were different for the cards?
2. What does it mean for a game to be fair in terms of expected value?
3. How would you calculate the variance of this game?
4. How do you apply the concept of expected value in other types of probability games?
5. How does the number of cards (or choices) in the game affect the expected value?

Tip: Always make sure to break the problem down into manageable parts: calculate probabilities, find the average payouts for each outcome, and then compute the expected value using those components.