Questions: Kaitlin has a deck of 10 cards numbered 1 through 10. She is playing a game of chance. This game is this: Kaitlin chooses one card from the deck at random. She wins an amount of money equal to the value of the card if an odd numbered card is drawn. She loses 4.20 if an even numbered card is drawn. (a) Find the expected value of playing the game. dollars (b) What can Kaitlin expect in the long run, after playing the game many times? (She replaces the card in the deck each time.) Kaitlin can expect to gain money. She can expect to win dollars per draw. Kaitlin can expect to lose money. She can expect to lose dollars per draw. Kaitlin can expect to break even (neither gain nor lose money).

Kaitlin has a deck of 10 cards numbered 1 through 10. She is playing a game of chance. This game is this: Kaitlin chooses one card from the deck at random. She wins an amount of money equal to the value of the card if an odd numbered card is drawn. She loses 4.20 if an even numbered card is drawn.

(a) Find the expected value of playing the game. dollars

(b) What can Kaitlin expect in the long run, after playing the game many times? (She replaces the card in the deck each time.) Kaitlin can expect to gain money. She can expect to win dollars per draw. Kaitlin can expect to lose money. She can expect to lose dollars per draw. Kaitlin can expect to break even (neither gain nor lose money).

Transcript text: Kaitlin has a deck of 10 cards numbered 1 through 10 . She is playing a game of chance. This game is this: Kaitlin chooses one card from the deck at random. She wins an amount of money equal to the value of the card if an odd numbered card is drawn. She loses $\$ 4.20$ if an even numbered card is drawn. (a) Find the expected value of playing the game. $\square$ dollars (b) What can Kaitlin expect in the long run, after playing the game many times? (She replaces the card in the deck each time.) Kaitlin can expect to gain money. She can expect to win $\square$ dollars per draw. Kaitlin can expect to lose money. She can expect to lose $\square$ dollars per draw. Kaitlin can expect to break even (neither gain nor lose money).

Solution

Solution Steps

To find the expected value of playing the game, we need to calculate the probability-weighted average of all possible outcomes. There are 5 odd-numbered cards (1, 3, 5, 7, 9) and 5 even-numbered cards (2, 4, 6, 8, 10). For odd cards, Kaitlin wins the value of the card, and for even cards, she loses $4.20. The expected value is the sum of the products of each outcome's value and its probability.

Step 1: Determine the Probability of Drawing Each Type of Card

Kaitlin has a deck of 10 cards, with 5 odd-numbered cards and 5 even-numbered cards. The probability of drawing an odd card is:

\[ P(\text{odd}) = \frac{5}{10} = 0.5 \]

Similarly, the probability of drawing an even card is:

\[ P(\text{even}) = \frac{5}{10} = 0.5 \]

Step 2: Calculate the Expected Value for Odd-Numbered Cards

The expected value for drawing an odd-numbered card is the average of the odd card values, multiplied by the probability of drawing an odd card. The odd-numbered cards are 1, 3, 5, 7, and 9. The average value is:

\[ \text{Average of odd cards} = \frac{1 + 3 + 5 + 7 + 9}{5} = 5 \]

Thus, the expected value for odd cards is:

\[ E(\text{odd}) = 5 \times 0.5 = 2.5 \]

Step 3: Calculate the Expected Value for Even-Numbered Cards

For even-numbered cards, Kaitlin loses $4.20. The expected value for drawing an even card is:

\[ E(\text{even}) = -4.20 \times 0.5 = -2.10 \]

Step 4: Calculate the Total Expected Value of the Game

The total expected value of the game is the sum of the expected values for odd and even cards:

\[ E(\text{total}) = E(\text{odd}) + E(\text{even}) = 2.5 + (-2.10) = 0.40 \]