Imagine a gambling game played with an ordinary deck of 52 cards. You pay $2.50
ID: 3224997 • Letter: I
Question
Imagine a gambling game played with an ordinary deck of 52 cards. You pay $2.50 to play and you play by shuffling the deck and then turning over the top card.
If the top card is a diamond, you win $4.50
If the top card is a picture (face) card, you win $4.50, but J, Q, or K of diamonds wins $7.00
If the top card is an ace, you win $5.00, but the ace of diamonds wins $10.00
If the top card is any other card, you win nothing
a.Let f be your payoff, that is your win minus the $2.50. Then f has a value for every card turned up, but only has 5 different values. Determine the probability distribution for these values.
b. Calculate the expected value for f.
c. Why is the average payoff per play negative
Explanation / Answer
Total cards = 52
Amount to play = $2.50
Total diamond cards = 13
total face cards = 12
total ace cards = 4
> For 9 diamond cards(13 diamond cards - 1 ace card - 3 face cards) - money win = $4.50
payoff = 4.5 - 2.5 = $2.00
> 3 face cards of diamond - win = $7.00
payoff = 7 - 2.5 = $4.50
> rest 9 face cards - win = $4.50
payoff = 4.5 - 2.5 = $2.00
> ace of diamonds - win = $10.00
payoff = 10 - 2.5 = $7.50
> rest 3 ace cards - win = $5.00
payoff = 5 - 2.5 = $2.50
a) Probability distribution for these values
Value Probability
2 $ 9/52
4.5$ 3/52
2$ 9/52
7.5$ 1/52
2.5$ 3/52
b)
Expected value
(2*9 + 4.5*3 + 2*9 + 7.5*1 + 2.5*3 + -2.5*27) / 52 = -12
c)
The expected payoff per play is negative because we are not winning anything for te 27 hands and our playoffs for those hands is negative which overall is greater than the winning in the rest of the hands