Question

A person has $32. The person decides to play roulette. She always bets on Red. (To simplify the problem assume that P(Red) = .5)

a) Let X be the random variable for the amount that she makes if she bets $8 on each roll and does this for four rolls. What is the probability function for X? What is E(X)?

b) Let Y be the amount that she makes if she follows the following strategy.

She bets $1 on roll 1 and stops if she wins;

Otherwise she bets $2 on roll 2 and stops if she wins;

Otherwise she bets $4 on roll 3 and stops if she wins;

Otherwise she bets $8 on roll 4 and stops if she wins;

Otherwise she bets $16 on roll 5 and stops no matter whether she wins or loses.

What is the probability function of Y? What is E(Y)?

Answer 1

(a) X can take values -32, -8, 0, 8, 32

P(X = -32) = P(X = 32) = = 1/16

P(X = -8) = P(X = 8) = = 4/16

P(X = 0) = = 6/16

X	-32	-8	0	8	32
P(X = x)	1/16	1/4	3/8	1/4	1/16

E(X) = = 0

(b)

Y = 1 if she wins in roll 1

Y = 1 if she loses in roll 1 and wins in roll 2

Y = 1 if she loses in roll 1 and roll 2 and wins in roll 3

Y = 1 if she loses in roll 1 and roll 2 and roll 3 and wins in roll 4

Y = 1 if she loses in roll 1, roll 2, roll 3, roll 4 and wins in roll 5

Y = -31 if she loses in all the five rolls

Y = -15 if she loses in roll 4

P(Y = 1) = = 31/32

P(Y = -15) = 1/32

Thus, E(Y) = 1*31/32 + (-15)*1/32 = $0.5