Question

Exercises for Probability

Exercise 2-17

Evertight, a leading manufacturer of quality nails, produces 1-, 2-, 3-, 4-, and 5-inch nails for various uses. In the production process, if there is an over- run or the nails are slightly defective, they are placed in a common bin. Yesterday, 651 of the 1-inch nails, 243 of the 2-inch nails, 41 of the 3-inch nails, 451 of the 4-inch nails, and 333 of the 5-inch nails were placed in the bin.

1. What is the probability of reaching into the bin and getting a 4-inch nail?
2. What is the probability of getting a 5-inch nail?
3. If a particular application requires a nail that is 3 inches or shorter, what is the probability of getting a nail that will satisfy the requirements of the application?

Exercise 2-18

Last year, at Northern Manufacturing Company, 200 people had colds during the year. One hundred fifty- five people who did no exercising had colds, and the remainder of the people with colds were involved in a weekly exercise program.  Half of the 1,000 employees were involved in some type of exercise.

1. What is the probability that an employee will have a cold next year?
2. Given that an employee is involved in an exercise program, what is the probability that he or she will get a cold next year?
3. What is the probability that an employee who is not involved in an exercise program will get a cold next year?
4. Are exercising and getting a cold independent events? Explain your answer.

Exercise 2-27

In a sample of 1,000 representing a survey from the entire population, 650 people were from Laketown, and the rest of the people were from River City. Out of the sample, 19 people had some form of cancer.  Thirteen of these people were from Laketown.

1. Are the events of living in Laketown and having some sort of cancer independent?
2. Which city would you prefer to live in, assuming that your main objective was to avoid having cancer?

Exercise 1

An engineering company advertises a job in three papers, A, B and C. It is known that these papers attract undergraduate engineering readerships in the proportions 2:3:1. The probabilities that an engineering undergraduate sees and replies to the job advertisement in these papers are 0.002, 0.001 and 0.005 respectively. Assume that the undergraduate sees only one job advertisement.

1. If the engineering company receives only one reply to it advertisements, calculate the probability that the applicant has seen the job advertised in place A.
2. If the company receives two replies, what is the probability that both applicants saw the job advertised in paper A?

Exercise 2-33

Gary Schwartz is the top salesman for his company.  Records indicate that he makes a sale on 70% of his sales calls. If he calls on four potential clients, what is the probability that he makes exactly 3 sales? What is the probability that he makes exactly 4 sales?

Exercise 2

A special five football matches series will be played between UAE and KSA. The probability that UAE will win a match against KSA is 0.6.              

            (i). What is the probability that UAE will win at least 3 matches in the series?        

            (ii). What is the probability that UAE will lose all the matches of the series?          

            (iii). What is the probability that UAE will win at most 2 matches in the series?     

Exercise 2-34

Trowbridge Manufacturing produces cases for personal computers and other electronic equipment. The quality control inspector for this company believes that a particular process is out of control. Normally, only 5% of all cases are deemed defective due to   discolorations. If 6 such cases are sampled, what is the probability that there will be 0 defective cases if the process is operating correctly? What is the probability that there will be exactly 1 defective case?

Exercise 2-34

If 10% of all disk drives produced on an assembly line are defective, what is the probability that there will be exactly one defect in a random sample of 5 of these? What is the probability that there will be no defects in a random sample of 5?

Exercise 2-38

Steve Goodman, production foreman for the Florida Gold Fruit Company, estimates that the average sale of oranges is 4,700 and the standard deviation is 500 oranges. Sales follow a normal distribution.

(a)  What is the probability that sales will be greater than 5,500 oranges?

(b)  What is the probability that sales will be greater than 4,500 oranges?

(c)  What is the probability that sales will be less than 4,900 oranges?

(d) What is the probability that sales will be less than 4,300 oranges?

Exercise 2-41

The time to complete a construction project is normally distributed with a mean of 60 weeks and a standard deviation of 4 weeks.

(a)  What is the probability the project will be finished in 62 weeks or less?

(b) What is the probability the project will be finished in 66 weeks or less?

(c)  What is the probability the project will take longer than 65 weeks?

Exercise 2-42

A new integrated computer system is to be installed worldwide for a major corporation.  Bids on this project are being solicited, and the contract will be awarded to one of the bidders. As a part of the proposal for this project, bidders must specify how long the project will take. There will be a significant penalty for finishing late. One potential contractor determines that the average time to complete a project of this type is 40 weeks with a standard deviation of 5 weeks. The time required to complete this project is assumed to be normally distributed.

1. If the due date of this project is set at 40 weeks, what is the probability that the contractor will have to pay a penalty (i.e., the project will not be finished on schedule)?
2. If the due date of this project is set at 43 weeks what is the probability that the contractor will have to pay a penalty (i.e., the project will not be finished on schedule)?
3. If the bidder wishes to set the due date in the proposal so that there is only a 5% chance of being late (and consequently only a 5% chance of having to pay a penalty), what due date should be set?

Exercise 2-43

Patients arrive at the emergency room of Costa   Valley Hospital at an average of 5 per day. The demand for emergency room treatment at Costa Valley follows a Poisson distribution.

(a)  Using Appendix C, compute the probability of exactly 0, 1, 2, 3, 4, and 5 arrivals per day.

(b)  What is the sum of these probabilities, and why is the number less than 1?

Exercise 2-44

Using the data in Problem 2-43, determine the probability of more than 3 visits for emergency room service on any given day.

Answer 1