Question

You draw one card from a standard deck of playing cards. If you pick a heart, you will win $9. If you pick a face card, which is not a heart, you will win $7. If you pick any other card, you lose $5 how do you make this a fair game with a value

Answer 1

Calculating the expected value for the card game with given probabilities of drawing a heart, a non-heart face card, and other cards, it appears the player has an average net gain of $0.86 per game. To make the game fair, the payouts or costs must be adjusted so that the expected value becomes zero.

Step-by-step explanation:

To create a fair game where the expected value is zero, we need to calculate the probabilities of the different outcomes and their corresponding winnings or losses. There are 52 cards in a standard deck of playing cards with 13 hearts, 12 face cards (3 of which are hearts), and 27 other cards (non-face, non-hearts).

• The probability of drawing a heart (and winning $9) is P(heart) = 13/52.
• The probability of drawing a non-heart face card (winning $7) is P(face card, not heart) = (12 - 3)/52 = 9/52.
• The probability of drawing any other card (losing $5) is P(other) = 27/52.

Now, let's calculate the expected value (EV) of the game using these probabilities:

EV = (P(heart) × win for heart) + (P(face card, not heart) × win for face card, not heart) + (P(other) × loss for other)

EV = (13/52 × $9) + (9/52 × $7) + (27/52 × -$5)

EV = ($2.25) + ($1.21) - ($2.60)

EV = $0.86

The positive expected value indicates that the player has an average net gain of $0.86 per game, suggesting the game as it stands is not fair. To make it fair, we would need to adjust the payouts or the costs of the game so that the expected value is zero.