Questions: Deandre has a deck of 10 cards numbered 1 through 10. He is playing a game of chance. This game is this: Deandre chooses one card from the deck at random. He wins an amount of money equal to the value of the card if an even numbered card is drawn. He loses 3.50 if an odd numbered card is drawn. (a) Find the expected value of playing the game. (b) What can Deandre expect in the long run, after playing the game many times? (He replaces the card in the deck each time.) O Deandre can expect to gain money. He can expect to win dollars per draw. O Deandre can expect to lose money. He can expect to lose dollars per draw. O Deandre can expect to break even (neither gain nor lose money).

Deandre has a deck of 10 cards numbered 1 through 10. He is playing a game of chance. This game is this: Deandre chooses one card from the deck at random. He wins an amount of money equal to the value of the card if an even numbered card is drawn. He loses 3.50 if an odd numbered card is drawn.
(a) Find the expected value of playing the game.
(b) What can Deandre expect in the long run, after playing the game many times? (He replaces the card in the deck each time.)

O Deandre can expect to gain money. He can expect to win dollars per draw.

O Deandre can expect to lose money. He can expect to lose dollars per draw.

O Deandre can expect to break even (neither gain nor lose money).

Transcript text: Deandre has a deck of 10 cards numbered 1 through 10. He is playing a game of chance. This game is this: Deandre chooses one card from the deck at random. He wins an amount of money equal to the value of the card if an even numbered card is drawn. He loses $3.50 if an odd numbered card is drawn. (a) Find the expected value of playing the game. (b) What can Deandre expect in the long run, after playing the game many times? (He replaces the card in the deck each time.) O Deandre can expect to gain money. He can expect to win $ dollars per draw. O Deandre can expect to lose money. He can expect to lose $ dollars per draw. O Deandre can expect to break even (neither gain nor lose money).

Solution

Solution Steps

Step 1: Identify the possible outcomes and their probabilities

The deck has 10 cards numbered 1 through 10. There are 5 even-numbered cards (2, 4, 6, 8, 10) and 5 odd-numbered cards (1, 3, 5, 7, 9). The probability of drawing an even-numbered card is: \[ P(\text{even}) = \frac{5}{10} = 0.5 \] The probability of drawing an odd-numbered card is: \[ P(\text{odd}) = \frac{5}{10} = 0.5 \]

Step 2: Calculate the expected value for each outcome

If Deandre draws an even-numbered card, he wins an amount equal to the value of the card. The average value of the even-numbered cards is: \[ \text{Average even value} = \frac{2 + 4 + 6 + 8 + 10}{5} = 6 \] Thus, the expected value for drawing an even-numbered card is: \[ E(\text{even}) = 6 \times 0.5 = 3 \]

If Deandre draws an odd-numbered card, he loses \$3.50. The expected value for drawing an odd-numbered card is: \[ E(\text{odd}) = -3.50 \times 0.5 = -1.75 \]

Step 3: Calculate the total expected value

The total expected value of playing the game is the sum of the expected values for each outcome: \[ E(\text{total}) = E(\text{even}) + E(\text{odd}) = 3 + (-1.75) = 1.25 \]

Step 4: Interpret the expected value in the long run

Since the expected value is positive (\$1.25), Deandre can expect to gain money in the long run. Specifically, he can expect to win \$1.25 per draw on average.