ProbabilityDistributions Problem Set

Solutions are provided at the end of the document

P-1 The salespeople at Owl Realty sell up to 9 houses per month. The probability distribution of a salesperson selling n houses in a month is as follows:
Sales
Probability

a. What is the probability of selling 5 houses in a month?
b. What are the expected value and the standard deviation for the number houses sold by the salespeople per month?

P-2 The following data has been recorded regarding the rental history of a high pressure cleaner at a home improvement store.
Demand
Frequency

Find the expected demand for the high pressure cleaner.

P-3Forty percent of all high school graduate work during the summer to earn money for college tuition for the upcoming fall term. Assuming a binomial distribution, if 6 graduates are selected at random, what is the probability that:
a) 3 have a summer job
b) None have a summer job
c) All have a summer job
d) At least 2 have a summer job

P-4Suppose the mean number of hurricanes a year is 2.35 and the number can be modeled by a Poisson distribution with this mean.
a. What is the probability of no hurricanes next year?
b. What's the probability that during the next 2 years, there's exactly 1 hurricane?

P-5A local paving company obtained a contract with the county to maintain roads serving a large urban center. The roads recently paved by thiscompany revealed an average of two defects per mile after beingused for one year.
If the county retains this paving company, what is the probability of one defect in any five mile of road after carrying traffic for one year? Assume a Poisson distribution and use the formula.

P-6You are the only bank teller on duty and you want to take a break for 10 minutes but you don't want to miss any customers. Suppose the arrival of customers can be models by a Poisson distribution with mean of 2 customers per hour.
a. What's the probability that no one will arrive in the next 10 minutes?
b. What's the probability that 2 or more people arrive in the next 10 minutes?
c. You have just served 2 customers, who came in one after the other. Is this a better time to take a break?

P-7Assume the speed of vehicles along a stretch of I-10 has an approximately normal distribution with a mean of 71 mph and a standard deviation of 8 mph.
a. The current speed limit is 65 mph. What is the proportion of vehicles less than or equal to the speed limit?
b. What proportion of the vehicles would be going less than 50 mph?
c. A new speed limit will be initiated based on the current mean of 71 mph such that approximately 20% of vehicles will be over the speed limit. What is the new speed limit based on this criterion?

P-8An instructor grades on a curve (normal distribution) and your grade for each test is determined by the following where S = your score.
A-grade: S ≥ μ + 2σ
B-grade: μ + σ ≤ S < μ + 2σ
C-grade: μ - σ ≤ S < μ + σ
D-grade: μ - 2σ ≤ S < μ - σ
F-grade: S < μ - 2σ

If on a particular test, the average on the test was μ = 66, the standard deviation was σ = 15.
a. What is the range of scores for each grade?
b. If you got an 82%, what grade did you get on that test?
c. Using the grading scheme in the above problem, what is the range of scores for each letter-grade on a test if the average was μ = 75 and the standard deviation was σ = 6?
d. If you got an 82%, what grade did you get on that test?

P-9 In a sample of 1000 cases, the mean of a certain test is 14 and the standard deviation is 2.5. Assume the distribution to be normal.
a. How many students score between 12 and 15?
b. How many score above 18?
c. How many score below 18?
d. What is probability that the test score is between 15 and 18?
e. The top 20% of the students will score how many points above the mean?

P-10An investment service is currently recommending the purchase of shares of Dollar Department Store selling at $18 per share. The price is approximately normally distributed with a mean of 20 and a standard deviation of 2.

What is the probability that in a year the shares will be selling for:
More than $20
Less than $18
Between $21 and $24
More than $21
More than $19
10% of the time shares will be selling at a price higher than what price?

P-11On a test at Microsoft, which has a mean of 500 and a standard deviation of 100, only the top 2% of the test takers will get an interview with Bill Gates. What score or better will they have to earn on this test to get the interview?

P-12The time required to assemble a part of a machine follows an exponential distribution. The average time to assemble the part is 10 minutes.
a. What is the probability that the part can be assembled in 7 minutes or less?
b. What is the probability that part can be assembled in 3 to 7 minutes?

P-13
The time it takes to complete an examination follows an exponential distribution with a mean of 40 minutes.
a. What is the probability of completing the examination in 30 minutes or less?
b. What is the probability of completing the examination in 30 to 35 minutes?