You go to Atlantic City with $32. You decide to play roulette and always bet on
ID: 3180999 • Letter: Y
Question
You go to Atlantic City with $32. You decide to play roulette and always bet on red. Note that the true probability that a spin results in red is 18/38, but assume that it is ½ in answering this question.
Strategy 1: Bet $8 on red on four consecutive rolls. Let X= amount you win following strategy 1. (For example, if you win on all four rolls you win $32).
a) Write down the sample space based on the outcomes of the four rolls, the probability for each point in the sample space and the value of X for each point in the sample space.
b) Find p(x) and F(x). Note: F(x) =P(X x)
c) Find the mean and standard deviation of X.
Strategy 2: Bet $1 on the first roll. If you win go home. If you lose then bet $2 on the second roll. If you win the second roll go home. If you lose then bet $4 on the third roll. If you win on the third roll then go home. If you lose then bet $8 on the fourth roll. If you win go home. If you lose bet $16 on the fifth roll. You go home no matter what the outcome is on the fifth roll, because if you lose on all rolls your total loss is $31 (winnings of -$31). Let Y=amount you win following strategy 2. Repeat parts a), b) and c) for Strategy 2.
Think of two criteria for comparing Strategy 1 and Strategy 2 and discuss which strategy is best based on each of these criteria.
Explanation / Answer
Strategy 1:
Let W ="Win" and L = "Lose
Sample Space Probability X
WWWW 1/16 $32
LWWW 1/16 $16
WLWW 1/16 $16
WWLW 1/16 $16
WWWL 1/16 $16
LLWW 1/16 $0
LWLW 1/16 $0
LWWL 1/16 $0
WLLW 1/16 $0
WLWL 1/16 $0
WWLL 1/16 $0
LLLW 1/16 -$16
LLWL 1/16 -$16
LWLL 1/16 -$16
WLLL 1/16 -$16
LLLL 1/16 -$32
b)
X p(X) F(X)
-32 1/16 1/16
-16 1/4 5/16
0 3/8 11/16
16 1/4 15/16
32 1/16 1
c)
Strategy 2
a)
Sample space Probability X
W 1/2 1
LW 1/4 1
LLW 1/8 1
LLLW 1/16 1
LLLLW 1/32 1
LLLLL 1/32 -31
b)
X p(X) F(x)
-31 1/32 1/32
1 31/32 31/32
c)
Criteria 1: Maximum value of X
In Strategy 1, the maximuma a customer could win was $32 in the four consecutive rolls, but in Strategy 2, one could win only a maximum of $1. Hence Strategy 1 is better in this criteria.
Criteria 2: Risk
Standard deviation can be used to measure the risk of the action.
The standard deviation of Strategy 2 is smaller than that of Strategy 1, so Strategy 2 is better than Strategy 1 according to this criteria
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