Suppose you were given two unfair coins with PH 23 and a fa

Suppose you were given two unfair coins with P(H) = 2/3 and a fair die with face of each die is marked with numbers 0,1,2.3,4 and 5. If you toss two coins and roll the die at the same time, What is the sample space (list all possible outcomes)? Suppose the outcomes of the coin. \"H\" is marked as \"1\" and \"0\" for \"T\". Let X be a random variable representing the sum of three outcomes, what are the possible values for X? What is the probability of each outcome of X? What is the expected value of X? What is the standard deviation of X? Suppose you are playing a game and X (referring to 3 b ) is your outcome, each time you will pay $5 to enter the game. If you get the highest possible value of X you will win $20, if you get a prime number you will win $5 and anything else you will lose the game. What is the expected net value that you will win/lose?

Solution

a) Sample space = {HH0, HH1, HH2, HH3,H H4, HH5, HH6, HT0,HT1,H T2, HT3, HT4, HT5, HT6,T H0, TH1, TH2, TH3, TH4,T H5, TH6,T T0,TT1, TT2, TT3, TT4, TT5,T T6}

x = SUM OF three outcomes

minimum of x = 0 and max of x = 7

Mean = 3.5

E(X^2) = 15.6667

Var(X) = 15.6667-12.25

= 3.4167

x	0	1	2	3	4	5	6	7		Total
Fvble	TT0	HTO TH0 TT1	HH0 TT2 HT1 TH1	HH1 HT2 TH2 TT3	HH2, HT3, TH3, TT4	HH3 HT4 TH4 TT5	HH4 HT5 TH5	HH5
p	1/24	1/8	1/6	1/6	1/6	1/6	1/8	1/24		1
PX	0	1/8	1/3	1/2	2/3	5/6	3/4	7/24		3 1/2
PX^2	0	1/8	2/3	1 1/2	2 2/3	4 1/6	4 1/2	2 1/24		15 2/3