Question

You draw a card from a deck. If you get a red​ card, you win nothing. If you get a​ spade, you win ​$5. For any​ club, you get ​$10 plus an extra ​$50 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

Answer and Explanation:

Given : You draw a card from a deck. If you get a red​ card, you win nothing. If you get a​ spade, you win ​$5. For any​ club, you get ​$10 plus an extra ​$50 for the ace of clubs.

To find :

​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?    

Solution :

In a deck of 52 cards,

Red cards = 26 and Black card = 26

Heart = 13, Diamond=13 and Spade = 13, Club =13

If you get a red​ card, you win nothing.

i.e. At X=0, $P(X)=\frac{26}{52}=0.5$

If you get a spade, you win ​$5.

i.e. At X=5, $P(X)=\frac{13}{52}=0.25$

If you get a club, you get ​$10.

i.e. At X=10, $P(X)=\frac{13}{52}=0.25$

If you get a ace of clubs, you get $50.

i.e. At X=50, $P(X)=\frac{1}{52}=0.2$

a) A probability model for the amount you win at this game.

X     |  0          5        10       50

P(X) | 0.5    0.25    0.25     0.2

b) The expected amount​ you'll win.

$E(X)=\sum XP(X)$

$E(X)=0(0.5)+5(0.25)+10(0.25)+50(0.2)$

$E(X)=0+1.25+2.5+10$

$E(X)=13.75$

c) How much would you be willing to pay to play this​ game.

An amount which is less or equal to the expected value.