Let's work through the question in parts.

Part (a): Expected Value of the Game

Lamar draws a card from a deck of 10 cards numbered 1 through 10.

• If he draws an even-numbered card (2, 4, 6, 8, or 10), he wins an amount equal to the card's value.
• If he draws an odd-numbered card (1, 3, 5, 7, or 9), he loses $3.50.

Step 1: Calculate the Probability of Drawing an Even or Odd Card

• Total cards = 10
• Even cards = 5 (2, 4, 6, 8, 10) → Probability of drawing an even card = $\frac{5}{10} = 0.5$
• Odd cards = 5 (1, 3, 5, 7, 9) → Probability of drawing an odd card = $\frac{5}{10} = 0.5$

Step 2: Calculate the Expected Value of the Game

The expected value (EV) is the sum of the expected values for each outcome.

1. Expected value from even-numbered cards:

   • Possible winnings: 2, 4, 6, 8, or 10 dollars.
   • Average winning amount = $\frac{2 + 4 + 6 + 8 + 10}{5} = 6$.
   • Expected winnings from even cards = $0.5 \times 6 = 3$.

2. Expected value from odd-numbered cards:

   • Lamar loses $3.50 for each odd card.
   • Expected loss from odd cards = $0.5 \times -3.50 = -1.75$.

3. Total Expected Value: $\text{EV} = 3 + (-1.75) = 1.25$ So, the expected value of playing the game is $1.25.

Part (b): Long-Run Expectation

Based on the expected value, Lamar can expect to gain money in the long run, with an average win of $1.25 per draw.

Would you like a breakdown of these calculations or have any questions?

Related Questions

1. How would the expected value change if Lamar lost $5.00 for drawing an odd-numbered card?
2. What is the probability of Lamar winning more than $8 in a single draw?
3. How would the result differ if the deck had 12 cards instead of 10, with the same rules applied?
4. If Lamar could choose not to play when he draws an odd-numbered card, how would that affect his expected value?
5. How would the expected value change if Lamar won only half the card's value for even numbers?

Tip

When calculating expected value, consider both probability and the amount of gain or loss for each outcome. This helps in analyzing the long-run effects of repeated games.