Question

(10 pts) In a game, a player flips a coin three times. The player wins $3 for every head that turns up. The player must pay S5 to play the game. Let the random variable W represent the total winnings after playing the game. (a) Construct the pmf of W (b) Find the expectation and variance of W. (c) Would you play the game? Why or why not?

Answer 1

outcomes of flipping three times

HHH
HHT
HTH
HTT
THT
TTH
TTT
THH
P(0 head) = 1/8 =0.125

so, W will be = 0*3 - 5 = -5

P(1 head) = 3/8 =0.375

so, Winning will be = 1*3-5 = -2

P(2 head) =3/8 =0.375

so, winning will be = 2*3 - 5 = 1

P(3 head) = 1/8 = 0.375

so, winning will be = 3*3 - 5 = 4

------------------------

a)

pmf of W is

W	P(W)
-5	0.125
-2	0.375
1	0.375
4	0.125

b)

W	P(W)	W*P(W)	W² * P(W)
-5	0.125	-0.625	3.125
-2	0.375	-0.75	1.5
1	0.375	0.375	0.375
4	0.125	0.5	2

	P(W)	W*P(W)	W² * P(W)
total sum =	1	-0.5	7

expectation = E[W] = ΣW*P(W) =            -0.5
          
E [ W² ] = ΣW² * P(W) =            7
          
variance = E[ W² ] - (E[ W ])² =            6.75
          
c)

we would not like to play the game , because expecation is negative

it means in long run ,there would be a loss of $0.5 in playing the game