Question

You draw a card from a deck. If you get a red​ card, you win nothing. If you get a​ spade, you win ​$7. For any​ club, you get ​$10 plus an extra ​$30 for the ace of clubs. ​a) Create a probability model for the amount you win at this game. ​b) Find the expected amount​ you'll win. ​c) How much would you be willing to pay to play this​ game?

Answer 1

a) The probability distribution model for the amount we win in different instances is computed here as:

Let X be the winning amount here. Then, we have here:

P(X = 0) = 26/52 = 0.5 as there are 26 red cards
P(X = 7) = 13/52 = 0.25 as there are 13 spade cards
P(X = 10) = 12/52 as there are 12 non Ace club cards
P(X = 40) = 1/52 as there is only 1 ace club card in the deck

This is the required probability model for the amount we win in the game.

b) The expected amount we will win is computed here as:

E(X) = 0*(26/52) + 7*(13/52) + 10*(12/52) + 40*(1/52) = 251/52

Therefore 251/52 is the expected amount we will win here.

c) For this to be a fair game, we would be willing to pay the expected amount for the game that is $251/52