# Probability, Simulation, Demand, Sample Space, Decision

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18. Employees of a local company are classified according to gender and job type. The following table summarizes the number of people in each job category.

Male (M) Female (F)

Job

Administrative (AD) 110 10

Salaried staff (SS) 30 50

Hourly staff (HS) 60 40

If an employee is selected at random, what is the probability that the employee is female given that the employee is a salaried staff member.

a) .1667

b) .5

c) .6

d) .625

e) .7

19. In the set of all past due accounts, let the event A mean the account is between 31 and 60 days past due and the event B mean the account is that of a new customer. The complement of A is

a) all new customers.

b) all accounts fewer than 31 or more than 60 days past due.

c) all accounts from new customers and all accounts that are from 31 to 60 days past due.

d) all new customers whose accounts are between 31 and 60 days past due.

41. The number of cars arriving at Joe Kelly's oil change and tune-up place during the last 200 hours of operation is observed to be the following:

Number of cars arriving Frequency

3 or less 0

4 10

0 30

6 70

7 50

8 40

9 or more 0

Based on the above frequencies, use two digit random numbers, start with random number 00 and determine the random number ranges for the data set given above.

39. A researcher wants to simulate sunny and rainy days in her town for a 3-week period. What is the minimum number of digits the student must obtain from a random number table for each observation if it rained on two-fifths of the days over the past several years at this time of the year? Assume that days can be classified historically as either sunny or rainy.

a) 1

b) 2

c) 3

d) all of the above.

42. The relationship d = 5000 - 25p describes what happens to demand (d) as price (p) varies. Here, price can vary between $10 and $50.

a. How many units can be sold at the $10 price? How many can be sold at the $50 price?

b. Model the expression for total revenue.

c. Consider prices of $20, $30, and $40.

Which price alternative will maximize total revenue?

What are the values for demand and revenue at this price?

43. A package of candy contains 12 brown, 5 red, and 8 green candies. You grab three pieces from the package. Give the sample space of colors you could get. Order is not important.

44. There are two more assignments in a class before its end, and if you get an A on at least one of them, you will get an A for the semester. Your subjective assessment of your performance is

Event Probability

A on paper and A on exam .25

A on paper only .10

A on exam only .30

A on neither .35

a. What is the probability of getting an A on the paper?

b. What is the probability of getting an A on the exam?

c. What is the probability of getting an A in the course?

d. Are the grades on the assignments independent?

45. The high school GPA of applicants for admission to a college program are recorded and relative frequencies are calculated for the categories.

GPA F(x)

x < 2.0 .08

2.0 = x < 2.5 .12

2.5 = x < 3.0 .35

3.0 = x < 3.5 .30

3.5 = x

a. Complete the table to make this a valid probability distribution.

b. What is the probability an applicant's GPA will be below 3.0?

c. What is the probability an applicant's GPA will be 2.5 or above?

46. A calculus instructor uses computer aided instruction and allows students to take the midterm exam as many times as needed until a passing grade is obtained. Following is a record of the number of students in a class of 20 who took the test each number of times.

Students Number of tests

10 1

7 2

2 3

1 4

a. Use the relative frequency approach to construct a probability distribution and show that it satisfies the required condition.

b. Find the expected value of the number of tests taken.

c. Compute the variance.

d. Compute the standard deviation.

47. Lakewood Fashions must decide how many lots of assorted ski wear to order for its three stores. Information on pricing, sales, and inventory costs has led to the following payoff table, in thousands.

Demand

Order Size Low Medium High

1 lot 12 15 15

2 lots 9 25 35

3 lots 6 35 60

a. What decision should be made by the optimist?

b. What decision should be made by the conservative?

c. What decision should be made using minimax regret?

48. The table shows both prospective profits and losses for a company, depending on what decision is made and what state of nature occurs. Use the information to determine what the company should do.

s1 s2 s3

d1 30 80 -30

d2 100 30 -40

d3 -80 -10 120

d4 20 20 20

a. if an optimistic strategy is used.

b. if a conservative strategy is used.

c. if minimax regret is the strategy.

49. The drying rate in an industrial process is dependent on many factors and varies according to the following distribution.

Minutes Relative Frequency

3 0.22

4 0.36

5 0.28

6 0.10

7 0.04

Using these random numbers, simulate the drying time for 5 processes: 0.53; 0.95; 0.97; 0.96; and 0.07.

50. Use a four period moving average to forecast attendance at baseball games. Historical records show the attendance at ten consecutive baseball games as 5346, 7812, 6513, 5783, 5982, 6519, 6283, 5577, 6712, 7345.

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