Question

If you draw a card with a value of four or less from a standard deck of cards, I will pay you $59. If not, you pay me $14. (Aces are considered the highest card in the deck.) If you played this game 759 times, how much would you expect to win or lose? Round your answer to two decimal places. Losses must be expressed as negative values.

a. $35.94
b. -$35.94
c. $42.94
d. -$42.94

Answer 1

By calculating the expected value of the game played over 759 times, we find that you would expect to lose $35.94, rounding to two decimal places.

Step-by-step explanation:

Let's calculate the expected value of the card game where you win $59 for drawing a card with a value of four or less, and lose $14 if not. First, we must determine the number of favorable outcomes (cards with a value of four or less) in a standard deck of 52 cards. The cards that meet this condition are the 2s, 3s, and 4s in each of the four suits, which amounts to 3 ranks × 4 suits = 12 cards. Aces are considered high and are not among these cards.

The probability of drawing a card of four or less is therefore P(four or less) = 12/52. The probability of drawing a card with a value of more than four is P(more than four) = 1 - P(four or less) = 1 - (12/52) = 40/52.

Now, we'll use the formula for expected value (EV):

EV = (winning amount × P(four or less)) + (losing amount × P(more than four))

EV = ($59 × (12/52)) + (-$14 × (40/52))

EV = ($59 × 12/52) - ($14 × 40/52)

To find the long-term expectation over 759 games, multiply the EV by 759.

Long-term expectation = 759 × EV

Calculating the values gives us the long-term expectation, rounding to two decimal places:

Long-term expectation = 759 × (($59 × 12/52) - ($14 × 40/52)) = -$35.94

Thus, you would expect to lose $35.94 if you played this game 759 times.