Question

1. Nobel is a serial gambler and regularly plays several rounds of a gamble in which he wins $10,000 if the head comes on top twice when a fair coin is flipped twice and loses $5,000 for any other outcome from the two flips of the coin. In other words, the outcome of the gamble is determined by flipping a coin twice on each round of the gamble and Nobel wins only if head comes on top on both flips of the coin and loses if any other outcome occurs. Nobel is considering to play 10 rounds of such a gamble next week hoping that he will win back the $20,000 he lost in a similar gamble last week. If we let X represent the number of wins for Nobel out of the next 10 rounds of the gamble, X can be assumed to have a binomial probability distribution. Please answer the following questions based on the information given above.

a. Please calculate the probability of success for Nobel on each round of the gamble. Show how you arrived at your answer.

b. What is the probability that Nobel will win none of the 10 rounds of the gamble?  

c. What is the probability that Nobel will lose more than 5 of the 10 rounds of the gamble?   

d. What is the probability that Nobel will win at least 8 out of the 10 rounds of the gamble?   

e. What is the probability that Nobel will lose less than 4 of the 10 rounds of the gamble?   

f. What is the probability that Nobel will lose no more than 7 out of the 10 rounds of the gamble? g. How much money is Nobel expected to win in the 10 rounds of the gamble? How much money is he expected to lose? Given your results, do you think Nobel is playing a smart gamble? Please show how you arrived at your results and explain your final answer.

h. Calculate and interpret the standard deviation for the number of wins for Nobel in the next 10 rounds of the gamble. Show your work.

2. Vehicles arrive at a toll bridge at an average rate of 180 an hour. Only one toll booth is currently open and can process arrivals (collect tolls) at a mean rate of 22 seconds per vehicle.

A. How many vehicles should be expected at the toll bridge in a 10-minute period? Please show how you arrived at your answer.

B. Define X to be the number of vehicles arriving at the toll bridge in a 10-minute period and assume that X has a Poisson Probability distribution. Please calculate the probability that 25 vehicles will arrive at the toll bridge in a 10-minute period.

C. Please calculate the probability of at least 20 vehicles arriving at the bridge within 10-minute period.                                                                  

D. Please calculate the probability of less than 28 vehicles arriving at the bridge within 10-minute period.

E. Please calculate the probability that no vehicle will arrive at the bridge within a 10-minute period.

F. Please calculate the probability of more than 35 vehicles arriving at the bridge within a 10-minute period. Show your work.   

G. Please calculate the probability of at most 22 vehicles arriving at the bridge within a 10-minute period. Show your work.

G. Calculate and interpret the standard deviation of X.   

I. For what purpose can the kind of information you have calculated above be used? If the state department of transportation wants to reduce the average wait time for the drivers to less than 25 seconds at the toll bridge, do you think they should open another toll booth at the bridge? Please explain.             

Answer 1

Question 1

a. Please calculate the probability of success for Nobel on each round of the gamble. Show how you arrived at your answer.

We know that the probability of getting Head if we flip a coin is 0.5.

If we flip coin twice, the outcomes are (HH, HT, TH, TT)

Probability of success = P(HH) = P(H)*P(H) = 0.5*0.5 = 0.25

b. What is the probability that Nobel will win none of the 10 rounds of the gamble?

Here, we have to find P(X=0)

We are given n=10, p=0.25

P(X=x) = nCx*p^x*(1 – p)^(n – x)

P(X=0) = 10C0*0.25^0*(1 – 0.25)^(10 – 0)

P(X=0) = 1*1*0.75^10

P(X=0) = 0.056313515

Required probability = 0.056313515

c. What is the probability that Nobel will lose more than 5 of the 10 rounds of the gamble?  

Here, we have to find P(X>5)

Probability of losing = 1 – 0.25 = 0.75

So, we have n=10, p=0.75

P(X>5) = 1 – P(X≤5)

P(X≤5) = 0.078126907 (by using binomial table or excel)

P(X>5) = 1 – 0.078126907

P(X>5) = 0.921873093

Required probability = 0.921873093

d. What is the probability that Nobel will win at least 8 out of the 10 rounds of the gamble?  

We have to find P(X≥8)

We are given n=10, p = 0.25

P(X≥8) = 1 – P(X≤7) = 1 - 0.999584198 (by using binomial table or excel)

P(X≥8) = 0.000415802

Required probability = 0.000415802

e. What is the probability that Nobel will lose less than 4 of the 10 rounds of the gamble?  

n=10, p=0.75

P(X<4) = P(X≤3) = 0.003505707

(by using binomial table or excel)

f. What is the probability that Nobel will lose no more than 7 out of the 10 rounds of the gamble?

n=10, p=0.75

P(X≤7) = 0.474407196

(by using binomial table or excel)

g. How much money is Nobel expected to win in the 10 rounds of the gamble? How much money is he expected to lose? Given your results, do you think Nobel is playing a smart gamble? Please show how you arrived at your results and explain your final answer.

Expected number of wins = n*p = 10*0.25 = 2.5

He expected to lose 75% of money. We do not think that Nobel is playing a smart gamble because if he plays a game for long time, then the probability of winning is 0.25 which is very less as compared to the probability of losing.

h. Calculate and interpret the standard deviation for the number of wins for Nobel in the next 10 rounds of the gamble. Show your work.

We are given n=10, p=0.25, q = 1 – p = 1 – 0.25 = 0.75

Standard deviation = sqrt(npq) = sqrt(10*0.25*0.75) = 1.369306394

Number of wins will be deviate on an average 1.37 number of wins.