Question

Mike and dave play the following game. mike picks a card from a deck of cards. if he selects an ace, dave give him $5. if not, he gives dave$2. determine mike’s expectation?

a. $1
b. $2
c. $3
d. $4

Answer 1

Let's calculate Mike's expectation. The probability of Mike selecting an ace is \( \frac{4}{52} \) (there are four aces in a standard deck of 52 cards), and the probability of not selecting an ace is \( \frac{48}{52} \).

Mike's expected value (\( E \)) can be calculated as follows:

\[ E = (\text{Probability of Ace} \times \text{Amount for Ace}) + (\text{Probability of Not Ace} \times \text{Amount for Not Ace}) \]

\[ E = \left(\frac{4}{52} \times \$5\right) + \left(\frac{48}{52} \times -\$2\right) \]

Now, calculate the value to determine Mike's expectation.

Answer 2

To determine Mike's expectation, we calculate the probability of him selecting an ace and not selecting an ace. Then we calculate the expected value in each case and sum them up to find the overall expectation. Mike's expectation is approximately -$1.46.

Step-by-step explanation:

To determine Mike's expectation, we need to calculate the probability of him selecting an ace and the probability of him not selecting an ace. Let's assume there are 52 cards in a deck, 4 of which are aces. The probability of selecting an ace is 4/52 = 1/13. The probability of not selecting an ace is 1 - 1/13 = 12/13.

Now we can calculate the expectation. If Mike selects an ace, Dave gives him $5. So, the expected value in this case is (1/13) * $5 = $5/13. If Mike does not select an ace, Dave gives him $2. So, the expected value in this case is (12/13) * (-$2) = -$24/13.

To find Mike's overall expectation, we sum up the two expected values: ($5/13) + (-$24/13) = -$19/13.

Therefore, Mike's expectation is -$19/13, which is approximately -$1.46. So, the correct answer is d. -$1.46.