You draw one card from a standard deck of playing cards. If you pick a heart, you will win $10. If you pick a face card, which is not a heart, you win $8. If you pick any other card, you lose 6$. Do you want to play?
E = 10 * 13/52 + 8 * 8/52 - 6 * 31/52
To determine whether we should play the game or not, we need to consider the probabilities and expected values for each outcome.
A standard deck of playing cards contains 52 cards, with 13 hearts (including face cards), and 12 non-heart face cards (3 face cards in each suit other than hearts).
Let's calculate the expected value for each outcome:
1. Picking a heart: The probability of picking a heart is 13/52. If you win $10 for this outcome, the expected value is (13/52) * $10 = $2.50.
2. Picking a non-heart face card: The probability of picking a non-heart face card is 12/52. If you win $8 for this outcome, the expected value is (12/52) * $8 = $1.85.
3. Picking any other card: The probability of picking any other card is (52 - 13 - 12)/52 = 27/52. If you lose $6 for this outcome, the expected value is (27/52) * (-$6) = -$3.00.
To find the overall expected value, we sum up the expected values for each outcome:
Overall expected value = ($2.50) + ($1.85) + (-$3.00) = $1.35
Since the overall expected value is positive ($1.35), it means that statistically, you are expected to make money if you play this game. Therefore, it would be advantageous for you to participate in the game.
To determine whether it is advantageous to play this game, we need to calculate the expected value.
The expected value is calculated by multiplying the value of each outcome by its probability and summing up all the results.
In a standard deck, there are 52 cards. There are 13 hearts (including 1 face card), 12 face cards that are not hearts, and the remaining 27 cards are neither hearts nor face cards.
The probability of drawing a heart is 13/52, or 1/4. So the expected value for picking a heart is (1/4) * $10 = $2.50.
The probability of drawing a face card that is not a heart is 12/52, or 3/13. So the expected value for picking a face card is (3/13) * $8 = $1.85.
The probability of drawing any other card is 27/52, or 9/17. So the expected value for picking any other card is (9/17) * (-$6) = -$3.53.
To calculate the overall expected value of the game, we sum up the expected values for each outcome:
Expected value = (1/4) * $10 + (3/13) * $8 + (9/17) * (-$6) = $2.50 + $1.85 - $3.53 ≈ -$0.18
The negative expected value (-$0.18) suggests that, in the long run, you are expected to lose money if you play this game. Therefore, it is not advantageous to play.