Let's analyze the problem to calculate the expected value of this game.

Given Information:

• Deck of 10 cards numbered 1 through 10.
• Game Rules:
  • If Mary draws an odd-numbered card, she wins the value of that card (in dollars).
  • If she draws an even-numbered card, she loses $6.

Part (a): Calculating the Expected Value

1. Probability of drawing an odd or even card:

   • There are 5 odd-numbered cards (1, 3, 5, 7, 9) and 5 even-numbered cards (2, 4, 6, 8, 10).
   • Therefore, the probability of drawing an odd-numbered card $P(\text{odd}) = \frac{5}{10} = 0.5$.
   • Similarly, the probability of drawing an even-numbered card $P(\text{even}) = 0.5$.

2. Expected gain for an odd-numbered card:

   • For each odd-numbered card drawn, Mary wins an amount equal to the card’s value. So, we calculate the expected value of drawing an odd card: $\text{Average of odd values} = \frac{1 + 3 + 5 + 7 + 9}{5} = \frac{25}{5} = 5$
   • Thus, the expected gain for drawing an odd card is $5.

3. Expected loss for an even-numbered card:

   • If Mary draws an even-numbered card, she loses $6.

4. Total Expected Value of the Game:

   • The expected value (EV) can be calculated by combining the gains and losses based on their probabilities: $\text{EV} = (0.5 \times 5) + (0.5 \times -6) = 2.5 - 3 = -0.5$
   • So, the expected value of playing the game is -0.5 dollars.

Answer to Part (a):

Mary can expect to lose 0.5 dollars per game on average.

Part (b): Long-term Expectation

Since the expected value of each game is -0.5 dollars, over the long run, Mary can expect to lose money at an average rate of 0.5 dollars per game.

• Correct Option: Mary can expect to lose money. She can expect to lose 0.5 dollars per draw.

Would you like further clarification on any of these calculations, or do you have additional questions?

Related Questions

1. How would the expected value change if she lost $4 instead of $6 for even-numbered cards?
2. What if the deck had 20 cards, with an equal number of odd and even numbers?
3. How would the expected value change if odd cards paid double their face value?
4. What is the probability of Mary winning after 10 draws?
5. How could Mary modify the game to have an expected value of zero?

Tip:

In probability calculations, breaking down complex expectations into smaller parts (like separating odd and even cases) often simplifies the problem significantly.