Consider the following card game with a well-shuffled deck of cards. If you draw a red card, you win nothing. If you get a spade, you win $5. For any club, you win $10 plus an extra $20 for the ace of clubs.

(a)
Create a probability model for the amount you win at this game. Also, find the expected winnings for a single game and the standard deviation of the winnings.(b)
What is the maximum amount you would be willing to pay to play this game? Explain.

(a)x               0          5        10       30

   P(X=x)   26/52  13/52  12/52   1/52

E(x) = $4.135

σ = $5.43

(b) The maximum amount I would be willing to play this game is $5.

Step-by-step explanation:

Let X be the amount you win at this game.

The values of X can be 0, 5, 10, 30.

X=0 for red cards and there are 13 diamonds and 13 hearts in a deck of 52 cards. So, the probability of X=0 is:

P(X=0) = 26/52

X=5 for spades and there are 13 spades out of 52 so,

P(X=5) = 13/52

X=10 for any club except ace so there are 12 clubs excluding the ace.

P(X=10) = 12/52

You win $10 for any club and an extra $20 for the ace of clubs which makes it $30 for the ace of clubs.

P(X=30) = 1/52

So, the probability model can be made as:

x               0          5        10       30

P(X=x)   26/52  13/52  12/52   1/52

Expected winnings E(x) can be calculated as:

E(x) = ∑x*P(x)

     = (0)*(26/52) + (5)*(13/52) + (10)*(12/52) + (30)*(1/52)

     = 0 + 1.25 + 2.3077 + 0.5769

E(x) = $4.135

Standard deviation (σ) can be calculated as:

σ² = ∑x²P(x) - (E(x))²

  = (0)²*(26/52) + (5)²*(13/52) + (10)²*(12/52) + (30)²*(1/52) - (4.135)²

  = 0 + 6.25 + 23.077 + 17.3077 - 17.098

σ² = 29.537

σ = √29.537

σ = $5.43

(b) The maximum amount I would be willing to play this game is $5 because the expected winning amount is also $5 ($4.135≅$5 out of the prize money that can be won). Also, the probability of winning $0 is 1/2 and the probability of winning a prize is also 1/2 so, $5 is the maximum amount which I will be willing to pay since there is a 50% chance of winning and losing.