Question

Consider the following card game with a well-shuffled deck of cards. If you draw a red card, you don't win. If you get a space you win $1. If you draw a club, you win $3, if you draw the ace of clubs you win $30.  

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

b. Find the expected value for winning a single game.  

c. Find the standard deviation of the winnings.  

d. Now suppose you are running this game. How much should you charge a person to play per game if you want an average profit of $1.50 per game?

Answer 1

a)below is probability model for the amount you win

P(X=0)=P(red card)=26/52=1/2 =0.5

P(X=1)=P(spade)=13/52 =0.25

P(X=3)=P(club which is not ace)=12/52 =0.2308

P(X=30)=P(ace of clubs)=1/52 =0.0192

b)

x	P(x)	xP(x)	x2P(x)
0	0.5000	0.000	0.000
1	0.2500	0.250	0.250
3	0.2308	0.692	2.077
30	0.0192	0.577	17.308
	total	1.519	19.635
	E(x) =μ=	ΣxP(x) =	1.5192
	E(x2) =	Σx2P(x) =	19.6346
	Var(x)=σ2 =	E(x2)-(E(x))2=	17.327
	std deviation=	σ= √σ2 =	4.1625

expected value =1.52

c) standard deviation =4.16

d) you should charge a person to play per game =1.5+1.52 =3.02