5. You go to Atlantic City with $32. You decide to play roulette and always bet

Question

5. 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). 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. a) b) Find p(x) and F(x). Note: F(x)-P(Xx) Find the mean and standard deviation of X. c) 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 S16 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 S31 (winnings of-S31). Let Y-amount 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 the Sample space be defined as xxxx where x = 0 , if there is loss and x=1 if there is a win

Sample Space = {0000, 0001, 0010, 0011, 0100, 0101, 0110, 0111, 1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111}

Value of all 4 losses = -32

Value of 3 loss and 1 win = -16

Value of 2 loss and 2 win = 0

Value of 1 loss and 3 win = 16

Value of all 4 wins = 32

Corresponding value of X = {-32, -16, -16, 0, -16, 0, 0, 16, -16, 0, 0, 16, 0, 16, 16, 32}

As, win or loss on each roll is (1/2), the probability of each outcome of sample space is 1/24 = 1/16

So, probability for each point in sample space = 1/16

(b)

p(x = -32) = p(x=32) = 1/16

p(x = -16) = p(x = 16 ) = 4 * (1/16) = 1/4

p(x = 0) = 6 * (1/16) = 3/8

p(x) is given as,

p = 1/16 for x = -32

p = 1/4 for x = -16

p = 3/8 for x = 0

p = 1/4 for x = 16

p = 1/16 for x = 32

F(X) is given as,

p = 1/16 for x = -32

p = 5/16 for x = -16

p = 11/16 for x = 0

p = 15/16 for x = 16

p = 1 for x = 32

c)

Mean of X, E[X] = (1/16) * (-32) + (1/4) * (-16) + (3/8) * 0 + (1/4) * 16 + (1/16) * 32 = 0

E[X2] = (1/16) * (-322) + (1/4) * (-16)2 + (3/8) * 02 + (1/4) * 162 + (1/16) * 322= 256

Variance of X = E[X2] - E[X]2 = 256 - 02 = 256

So, standard deviation of X = sqrt(256) = 16

Strategy 2:

a)

Let the Sample space be defined as xxxx where x = 0 , if there is loss and x=1 if there is a win

Sample Space = {00000, 00001, 0001, 001, 01, 1}

Winnings on 1st roll = $1

Winnings on 2nd roll = $2 - $1 = $1

Winnings on 3rd roll = $4 - ($2 + $1) = $1

Winnings on 4th roll = $8 - ($4 + $2 + $1) = $1

Winnings on 5th roll = $16 - ($8 + $4 + $2 + $1) = $1

Corresponding value of X = {-31, 1, 1, 1, 1, 1}

So, probability for each point in sample space = {1/2, 1/4, 1/8, 1/16, 1/32}

b)

p(x) is given as,

p = 31/32 for x = 1

p = 1/32 for x = -31

F(X) is given as,

p = 31/32 for x = 1

p = 1/32 for x = -31

c)

Mean of X, E[X] = (31/32) * 0 + (1/32) * (-31) = -0.96875

E[X2] = (31/32) * 02 + (1/32) * (-31)2 = 30.03125

Variance of X = E[X2] - E[X]2 = 30.03125 - (-0.96875)2 = 29.09277

So, standard deviation of X = sqrt(29.09277) = 5.393771

Based on highest expected value criteria, Strategy 1 is best.

Based on lowest standard deviation (variance) criteria, Strategy 2 is best.

Related Questions

Navigate

• Browse (All)
• Browse 5
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.