# Probability: Failures, Least Amounts and Joint Probability

1. There are five computers in an office. The probability of failure of a computer is 0.3. What is the probability that (a) there are 3 computers working, and (b) at least three computers working?

2. A computer service company is interested in the number of computers that fail. There are hundreds of computers in use in their service area. The probability of failure of a computer is 0.08. What is the probability that (a) there are 4 failures, and (b) no failure?

3. Flight time between points A and B is normally distributed with mean of 5 hours and a standard deviation of 2.5 hours. What is the probability that a given flight will take more than 7 hours?

9. The joint probability mass function of X and Y, p(x, y), is given by

p(1,1)= , p(2,1)= , p(3,1)= ,

p(1,2)= , p(2,2)=0, p(3,2)= ,

p(1,3)=0, p(2,3)= , p(3,3)=

Compute for 1, 2, 3.

https://brainmass.com/statistics/probability/probability-failures-least-amounts-joint-probability-227197

#### Solution Summary

This solution contains step-by-step calculations to determine the probability of failures, least amounts and joint probability of mass transfer function. All workings and equations are shown.

Quantitative Analysis - 50 Multiple choice Questions

1. Which of the following is not a reason for the failure of a particular

Quantitative analysis technique in solving a problem?

a. underestimating the total cost of using quantitative

techniques

b. failure to define the real problem

c. under-emphasis on theory and over-emphasis on application

d. underestimating the total time required to develop and

implement the most appropriate technique

e. resistance to change and reluctance of decision-makers to

trust and act upon the results obtained by using unfamiliar

techniques

2. Evaluating all possible values of a variable in a model is called

a. Trial and error.

b. complete enumeration.

c. an algorithm.

d. variablization.

e. none of the above

3. The classical method of determining probability is

a. subjective probability.

b. marginal probability.

c. objective probability.

d. joint probability.

e. conditional probability.

4. The expected value of a probability distribution is

a. the measure of the spread of the distribution.

b. the variance of the distribution.

c. the average value of the distribution.

d. the probability density function.

e. the range of continuous values from point A to point B,

inclusive.

5. At a university with 1,000 business majors, there are 200 business

students who are enrolled in an introductory statistics course. Of

these 200, 50 are also enrolled in an introductory accounting course.

There are an additional 250 business students who are enrolled in

accounting but are not enrolled in statistics. If a business student is

selected at random, what is the probability that the student is not

enrolled in accounting?

a. 0.20

b. 0.25

c. 0.30

d. 0.50

e. none of the above

6. Expected monetary value (EMV) is

a. the average or expected monetary outcome of a decision if it

can be repeated a large number of times.

b. the average or expected value of the decision if you know

what would happen ahead of time.

c. the average or expected value of information if it were

completely accurate.

d. the amount you would lose by not picking the best

alternative.

e. a decision criterion that places an equal weight on all states

of nature.

7. The maximax decision criterion

a. minimizes the maximum opportunity loss.

b. maximizes the minimum outcome.

c. yields the best of the worst possible outcomes when the

probability is favorable.

d. produces the alternative with the highest possible return.

e. none of the above

8. If product demand follows a normal distribution and we want to

apply marginal analysis, which of the following do we not need to

know?

a. the mean sales estimate

b. the standard deviation of the sales estimate

c. the marginal profit

d. the marginal loss

e. the standard deviation of the profit estimate

9. In constructing a utility curve,

a. a comparison is made with the different amounts of money at

different times.

b. the certainty of a certain amount is compared with the

willingness to gamble that amount on a larger amount.

c. one takes the risk out of gambling.

d. inflation plays a critical part in the evaluation.

e. none of the above

10. Payoff tables are more difficult to use than decision trees when

a. perfect information is available.

b. formulating a conditional values table.

c. the opportunity loss table is available.

d. a sequence of decisions must be made.

e. all possible outcomes and alternatives are not known.

11. When using Bayesian analysis

a. the actual probabilities of the states of nature are never

known.

b. it makes no difference whether one is employing EMV or

utility as a measure of outcome.

c. the joint probabilities must total less than 1.

d. the more accurate a survey, the more likely the survey is to

make a positive prediction.

e. none of the above

12. Which method is NOT a step used to develop a forecast using the

Decomposition Method?

a. Compute seasonal indices using CMAs.

b. Deseasonalize the data by dividing each number by its

seasonal index.

c. Proctoring the data by seasonal indexes and statistical

methods.

d. Multiply the trendline forecast by the appropriate seasonal

index .

e. Forecast for future periods using the trend line.

13. A tracking signal was calculated for a particular set of demand

forecasts. This tracking signal was positive. This would indicate

that

a. demand is greater than the forecast.

b. demand is less than the forecast.

c. demand is equal to the forecast.

d. the MAD is negative.

e. none of the above

14. The EOQ model assumes,

a. a simple regression model would always be adequate.

b. a moving average model is uncertain.

c. all input values are fixed and known with certainty.

d. an exponential smoothing model would always be best.

e. none of the above

15. Which of the following factors is (are) not included in carrying cost?

a. spoilage

b. obsolescence

c. cost of capital

d. inspecting incoming inventory

e. warehousing overhead costs

16. The annual demand for a product is 1,000 units. The company

orders 200 units each time an order is placed. The lead time is 6

days, and the company has determined that 20 units should be held

as a safety stock. There are 250 working days per year. What is the

reorder point?

a. 20

b. 24

c. 44

d. 120

e. none of the above

17. To evaluate quantity discounts, one uses the EOQ model, and

modifies the model so that

a. holding cost is considered as a percentage of unit cost rather

than a separable dollar cost.

b. the EOQ is calculated to lie within one of the cost corridors.

c. holding cost is calculated on an "average basis" across the

discounts.

d. any one of the above approaches could be used.

e. none of the above

18. An optimal solution to a linear program

a. will always lie at an extreme point of the feasible region.

b. could be any point in the feasible region of the problem.

c. will always be unique (only one optimal solution possible for

any one problem).

d. will always include at least some of each product or variable.

e. must always be in whole numbers (integers).

19. Two models of a product - Regular (X) and Deluxe (Y) - are produced

by a company. A linear programming model is used to determine the

production schedule. The formulation is as follows:

Maximize profit = 50X + 60 Y

Subject to: 8X + 10Y ≤ 800 (labor hours)

X + Y ≤ 120 (total units demanded)

4X + 5Y ≤ 500 (raw materials)

all variables ≥ 0

The optimal solution is X = 100 Y = 0.

How many units of the raw materials would be used to produce this

number of units?

a. 400

b. 200

c. 500

d. 120

e. none of the above

20. In order for a linear programming problem to have a unique

solution, the solution must exist

a. at the intersection of the non-negativity constraints.

b. at the intersection of a non-negativity constraint and a

resource constraint.

c. at the intersection of the objective function and a constraint.

d. at the intersection of two or more constraints.

e. none of the above

21. The following does not represent a factor a manager might consider

when employing linear programming for a production scheduling:

a. labor capacity

b. space limitations

c. product demand

d. risk assessment

e. inventory costs

22. Production scheduling is amenable to solution by linear

Programming because

a. the optimal product combination will minimize production

risk.

b. effective resource use will lead to optimal profits.

c. linear programming will allow investment losses to be

minimized.

d. scheduling requires specific, narrowly defined constraints.

e. objective functions and constraints can be readily developed

and are relatively stable over time.

23. A type of linear programming problem that is used in marketing is

called the

a. media selection problem.

b. Madison Avenue problem.

c. marketing allocation problem.

d. All of the above are examples of marketing linear

programming problems.

e. None of the above are examples of marketing linear

programming problems.

24. In applying the simplex solution procedure to a maximization

problem to determine which variable enters the solution mix

a. pick the one with the smallest Cj − Zj.

b. pick the one with the largest Cj − Zj.

c. pick the one with the largest Cj.

d. pick the one with the smallest Zj.

e. pick the smallest nonnegative number formed by dividing

each amount in the quantity column by the appropriate

column at the exiting variable.

25. Sensitivity analysis in linear programming

a. centers around applications of computer routines to solve LP

problems.

b. concerns studies of the extra information associated with the

dual LP.

c. concerns changes in the data used to build the LP model and

the effects on the optimal solution.

d. all of the above

26. The presence of one or more constraints that do not affect the

feasible solution region is:

a. redundancy.

b. inequality.

c. unboundedness.

d. a constraint.

e. none of the above

27. In applying Vogel's approximation method to a cost minimization

problem, row and column penalties are determined by

a. finding the largest unit cost in each row or column.

b. finding the smallest unit cost in each row or column.

c. finding the sum of the unit costs in each row or column.

d. finding the difference between the two lowest unit costs in

each row and column.

e. finding the difference between the two highest unit costs in

each row and column.

28. In solving maximization assignment problems,

a. just reverse all the decision rules used in the minimizing

algorithm (if it says subtract, now add, etc.).

b. use the Australian transformation process and convert the

data.

c. convert the problem to an equivalent minimization problem.

d. all of the above

29. The only restriction we place on the initial solution of a

transportation problem is that

a. we must have nonzero quantities in a majority of the boxes.

b. all constraints must be satisfied.

c. demand must be less than supply.

d. we must have a number (equal to the number of rows plus the

number of columns minus one) of boxes which contain

nonzero quantities.

e. none of the above

30. Which of the following is not a technique used in solving nonlinear

program problems?

a. classical optimization

b. gradient method

c. diorama algorithm

d. steepest ascent method

e. separable programming

31. In a goal programming problem with two goals at the same priority

level, all the deviational variables are equal to zero in the optimal

solution. This means

a. there is no feasible solution to the problem.

b. all goals are fully achieved.

c. nonlinear programming must be used to solve this.

d. this problem was an integer programming problem.

e. none of the above

32. If we wish to select a location for a new manufacturing plant, the

best approach would be to use

a. zero-one integer programming.

b. mixed-integer programming.

c. any linear programming model.

d. an assignment model.

e. a goal programming model.

33. The minimal-spanning technique would best be used

a. by an architect to lay out corridors between offices in a new

office building.

b. by a telephone company attempting to lay out wires in a new

housing development.

c. by an airline laying out flight routes.

d. none of the above

e. all of the above

34. When using the shortest-route technique, the second step is to

a. find the next-nearest node to the origin and put the distance

in a box by the node.

b. trace the path from the warehouse to the plant.

c. determine the average distance traveled from source to end.

d. find the nearest node to the origin and put a distance box by

the node.

e. none of the above

35. Find the shortest route from Node 1 to Node 5 using the shortest route

Technique.

From Node/ To Node /Distance

1 2 200

1 3 150

2 3 50

2 4 300

3 4 250

3 5 200

4 5 150

a. 350

b. 400

c. 450

d. 600

e. None of the above

36. In a PERT network, the earliest (activity) start time is the

a. earliest time that an activity can be finished without delaying

the entire project.

b. latest time that an activity can be started without delaying

the entire project.

c. earliest time that an activity can start without violation of

precedence requirements.

d. latest time that an activity can be finished without delaying

the entire project.

e. none of the above

37. Consider a project that has an expected completion time of 60 weeks

and a standard deviation of five weeks. What is the probability that

the project is finished in 70 weeks or less (round to two decimals)?

a. 0.98

b. 0.48

c. 0.50

d. 0.02

e. none of the above

38. In CPM,

a. an activity may start before its immediate predecessors have

finished.

b. no more than two activities may be performed

simultaneously.

c. the total cost of completing an activity in the crash time is

higher than the normal cost.

d. when we crash an activity, we complete the activity in the

minimum possible time.

e. none of the above

39. Which of the following is not an assumption in common queuing

mathematical models?

a. Arrivals come from an infinite, or very large, population.

b. Arrivals are Poisson distributed.

c. Arrivals are treated on a first-in, first-out basis and do not

balk or renege.

d. Service times follow the negative exponential distribution.

e. The average arrival rate is faster than the average service

rate.

40. Customers enter the waiting line at a cafeteria on a first come, first

served basis. The arrival rate follows a Poisson distribution, while

service times follow an exponential distribution. If the average

number of arrivals is six per minute and the average service rate of

a single server is eight per minute, what is the average number of

customers in the system?

a. 0.50

b. 0.75

c. 2.25

d. 3.00

e. none of the above

41. In queuing problems, the size of the calling population is important

because

a. we have models only for problems with infinite calling

populations.

b. we have models only for problems with finite calling

populations.

c. the size of the calling population determines whether or not

the arrival of one customer influences the probability of

arrival of the next customer.

d. we will have to consider the amount of space available for the

queue.

e. none of the above

42. Which of the following is not always a step in Monte Carlo

Simulation:

a. establishing probability distributions

b. building a cumulative probability distribution for each

variable

c. setting random number intervals

d. starting an exponential generator

e. simulating the experiment

43. Using simulation for a queuing problem

a. would be rare in a realistic situation.

b. is an unreasonable alternative if the arrival rate is not

Poisson distributed but can be plotted on a curve.

c. could be appropriate if the service time was not exponential or

constant.

d. all of the above

44. Special purpose simulation languages include

a. FORTRAN.

b. BASIC.

c. GPSS.

d. PL/1.

e. all of the above

45. Of the following, this is NOT a step in modeling

a. developing a solution.

b. defining the problem.

c. training and development.

d. implementing the results.

e. testing the solution.

46. A technique that allows a decision maker to plan, schedule, monitor,

and control project costs is:

a. PERT/Cost.

b. PERT/Monitoring.

c. PERT/Planning.

d. PERT/Scheduling.

47. The length of a waiting line can be

a. finite

b. limited or unlimited

c. studied indefinitely

d. infinite

e. none of the above

48. A probability distribution has been developed, and the probability of

2 arrivals in the next hour is 0.20. A random number interval is to

be assigned to this. Which of the following would NOT be an

appropriate interval?

a. 01-20

b. 21-40

c. 00-19

d. 00-20

49. Which of the following should cause a process control system to

sound the alarm?

a. natural variations

b. a mean is greater than a standard deviation

c. control variations

d. assignable variations

e. none of the above

50. The Poisson distribution forms the basis for which of the following

charts?

a. c-chart

b. x -chart

c. p-chart

d. all of the above

e. none of the above