1 Million+ Step-by-step solutions

a. Suppose that $68,000 is to be allocated for advertising, research, and investment in the ratio 8:6:3. How much money will be allocated for each

b. Computer Warehouse sells batteries ($2) and small boxes of mechanical pencils ($6). In July, total sales were $1056. Customers bought 5 times as many batteries as boxes of mechanical pencils. How many of each did Computer Warehouse sell?

b. Computer Warehouse sells batteries ($2) and small boxes of mechanical pencils ($6). In July, total sales were $1056. Customers bought 5 times as many batteries as boxes of mechanical pencils. How many of each did Computer Warehouse sell?

A quality characteristic of interest for a coffee-bag-filling process is the weight of the coffee in the individual bags. If the bags are under filled, two problems arise. First, customers may not be able to brew the coffee to be as strong as they wish. Second, the company may be in violation of the truth-in-labeling laws. In this example, the label weight on the package indicates that, on average, there are 5.45 grams of coffee in a bag. If the average amount of coffee in a bag exceeds the label weight, the company is giving away product. Getting an exact amount of coffee in a bag is problematic because of variation in the temperature and humidity inside the factory, differences in the density of the coffee, and the extremely fast filling operation of the machine (approximately 170 bags a minute). The following table provides the weight in grams of a sample of 50 bags produced in one hour by a single machine:

a. Compute the arithmetic mean and median.

b. Compute the first quartile and third quartile.

c. Compute the range, interquartile range, variance, standard deviation, and coefficient of variation.

d. Interpret the measures of central tendency within the context of this problem. Why should the company producing the coffee bags be concerned about the central tendency?

e. Interpret the measures of variation within the context of this problem. Why should the company producing the coffee bags be concerned about variation?

a. Compute the arithmetic mean and median.

b. Compute the first quartile and third quartile.

c. Compute the range, interquartile range, variance, standard deviation, and coefficient of variation.

d. Interpret the measures of central tendency within the context of this problem. Why should the company producing the coffee bags be concerned about the central tendency?

e. Interpret the measures of variation within the context of this problem. Why should the company producing the coffee bags be concerned about variation?

An apple juice bottling company maintains records concerning the number of unacceptable bottles of juice obtained from the filling and capping machines. Based on past data, the probability that a bottle came from machine I and was nonconforming is 0.05 and the probability that a bottle came from machine II and was nonconforming is 0.075. These probabilities represent the probability of one bottle out of the total sample having the specified characteristics. Half the bottles are filled on machine I and the other half are filled on machine II.

a. If a filled bottle of juice is selected at random, what is the probability that it is a nonconforming bottle?

b. If a filled bottle of juice is selected at random, what is the probability that it was filled on machine II?

c. If a filled bottle of juice is selected at random, what is the probability that it was filled on machine I and is a conforming bottle?

a. If a filled bottle of juice is selected at random, what is the probability that it is a nonconforming bottle?

b. If a filled bottle of juice is selected at random, what is the probability that it was filled on machine II?

c. If a filled bottle of juice is selected at random, what is the probability that it was filled on machine I and is a conforming bottle?

According to Investment Digest ("Diversification and the Risk/Reward Relationship", Winter 1994, 1-3), the mean of the annual return for common stocks from 1926 to 1992 was 15.4%, and the standard deviation of the annual return was 24.5%. During the same 67-year time span, the mean of the annual return for long-term government bonds was 5.5%, and the standard deviation was 6.0%. The article claims that the distributions of annual returns for both common stocks and long-term government bonds are bell-shaped and approximately symmetric. Assume that these distributions are distributed as normal random variables with the means and standard deviations given previously.

a. Find the probability that the return for common stocks will be greater than 0%.

b. Find the probability that the return for common stocks will be less than 20%.

a. Find the probability that the return for common stocks will be greater than 0%.

b. Find the probability that the return for common stocks will be less than 20%.

Compute a 95% confidence interval for the population mean, based on the sample 10, 12, 13, 14, 15, 16, and 49. Change the number from 49 to 16 and recalculate the confidence interval. Using the results, describe the effect of an outlier or extreme value on the confidence interval

The director of admissions at the University of Maryland, University College is concerned about the high cost of textbooks for the students each semester. A sample of 25 students enrolled in the university indicates that X (bar) = $315.4 and s = $43.20.

a. Using the 0.10 level of significance, is there evidence that the population mean is above $300?

b. What is your answer in (a) if s = $75 and the 0.05 level of significance is used?

c. What is your answer in (a) if X (bar) = $305.11 and s = $43.20?

d. Based on the information in part (a), what decision should the director make about the books used for the courses if the goal is to keep the cost below $300?

a. Using the 0.10 level of significance, is there evidence that the population mean is above $300?

b. What is your answer in (a) if s = $75 and the 0.05 level of significance is used?

c. What is your answer in (a) if X (bar) = $305.11 and s = $43.20?

d. Based on the information in part (a), what decision should the director make about the books used for the courses if the goal is to keep the cost below $300?

A large candy manufacturer is concerned that the mean weight of their bag of Gooey Sour Worms is not greater than 7.3 ounces. It can be assumed that the population standard deviation is .5 ounces based on past experience. A sample of 169 gummy worms is selected and the sample mean is 7.35 ounces. Using a level of significance of .10, is there evidence that the population mean weight of the candy bars is greater than 7.3? Fully explain your answer.

Each of the possible five outcomes of a random experiment is equally likely. The sample space is {a, b, c, d, e}.

Let A denote the event {a, b}, and let B denote the event {c, d, e}. Determine the following:

(a) P (A) (b) P (B)

(c) P (A’) (d) P (A U B)

(e) P (A ∩ B)

Let A denote the event {a, b}, and let B denote the event {c, d, e}. Determine the following:

(a) P (A) (b) P (B)

(c) P (A’) (d) P (A U B)

(e) P (A ∩ B)

The sample space of a random experiment is {a, b, c, d, e} with probabilities 0.1, 0.1, 0.2, 0.4, and 0.2, respectively. Let A denote the event {a, b, c}, and let B denote the event

{c, d, e}. Determine the following:

(a) P (A) (b) P (B)

(c) P (A’) (d) P (A U B)

(e) P (A ∩ B)

{c, d, e}. Determine the following:

(a) P (A) (b) P (B)

(c) P (A’) (d) P (A U B)

(e) P (A ∩ B)

A part selected for testing is equally likely to have been produced on any one of six cutting tools.

(a) What is the sample space?

(b) What is the probability that the part is from tool 1?

(c) What is the probability that the part is from tool 3 or tool 5?

(d) What is the probability that the part is not from tool 4?

(a) What is the sample space?

(b) What is the probability that the part is from tool 1?

(c) What is the probability that the part is from tool 3 or tool 5?

(d) What is the probability that the part is not from tool 4?

An injection-molded part is equally likely to be obtained from any one of the eight cavities on a mold.

(a) What is the sample space?

(b) What is the probability a part is from cavity 1 or 2?

(c) What is the probability that a part is neither from cavity 3 nor 4?

(a) What is the sample space?

(b) What is the probability a part is from cavity 1 or 2?

(c) What is the probability that a part is neither from cavity 3 nor 4?

A sample space contains 20 equally likely outcomes.

If the probability of event A is 0.3, how many outcomes are in event A?

If the probability of event A is 0.3, how many outcomes are in event A?

Orders for a computer are summarized by the optional features that are requested as follows:

Proportion of orders

No optional features 0.3

One optional feature 0.5

More than one optional feature 0.2

(a) What is the probability that an order requests at least one optional feature?

(b) What is the probability that an order does not request more than one optional feature?

Proportion of orders

No optional features 0.3

One optional feature 0.5

More than one optional feature 0.2

(a) What is the probability that an order requests at least one optional feature?

(b) What is the probability that an order does not request more than one optional feature?

If the last digit of a weight measurement is equally likely to be any of the digits 0 through 9,

(a) What is the probability that the last digit is 0?

(b) What is the probability that the last digit is greater than or equal to 5?

(a) What is the probability that the last digit is 0?

(b) What is the probability that the last digit is greater than or equal to 5?

A sample preparation for a chemical measurement is completed correctly by 25% of the lab technicians, completed with a minor error by 70%, and completed with a major error by 5%.

(a) If a technician is selected randomly to complete the preparation, what is the probability it is completed without error?

(b) What is the probability that it is completed with either a minor or a major error

(a) If a technician is selected randomly to complete the preparation, what is the probability it is completed without error?

(b) What is the probability that it is completed with either a minor or a major error

A credit card contains 16 digits between 0 and 9. However, only 100 million numbers are valid. If a number is entered randomly, what is the probability that it is a valid number?

Suppose your vehicle is licensed in a state that issues license plates that consist of three digits (between 0 and 9) followed by three letters (between A and Z). If a license number is selected randomly, what is the probability that yours is the one selected

A message can follow different paths through servers on a network. The senders message can go to one of five servers for the first step, each of them can send to five servers at the second step, each of which can send to four servers at the third step, and then the message goes to the recipients server.

(a) How many paths are possible?

(b) If all paths are equally likely, what is the probability that a message passes through the first of four servers at the third step?

(a) How many paths are possible?

(b) If all paths are equally likely, what is the probability that a message passes through the first of four servers at the third step?

Disks of polycarbonate plastic from a supplier are analyzed for scratch and shock resistance. The results from 100 disks are summarized as follows:

Shock resistance

High Low

Scratch high 70 9

Resistance low 16 5

Let A denote the event that a disk has high shock resistance, and let B denote the event that a disk has high scratch resistance.

If a disk is selected at random, determine the following probabilities:

(a) P (A) (b) P (A)

(c) P (A`) (d) P (A ∩ B)

(e) (A U B) (f) (A` U B)

Shock resistance

High Low

Scratch high 70 9

Resistance low 16 5

Let A denote the event that a disk has high shock resistance, and let B denote the event that a disk has high scratch resistance.

If a disk is selected at random, determine the following probabilities:

(a) P (A) (b) P (A)

(c) P (A`) (d) P (A ∩ B)

(e) (A U B) (f) (A` U B)

Samples of a Fast aluminum part are classified on the basis of surface finish (in micro inches) and edge finish. The results of 100 parts are summarized as follows:

Edge finish

Excellent Good

Surface excellent 80 2

Finish good 10 8

Let A denote the event that a sample has excellent surface finish, and let B denote the event that a sample has excellent length. If a part is selected at random,

determine the following probabilities:

(a) P (A) (b) P (A)

(c) P (A`) (d) P (A ∩ B)

(e) (A U B) (f) (A` U B)

Edge finish

Excellent Good

Surface excellent 80 2

Finish good 10 8

Let A denote the event that a sample has excellent surface finish, and let B denote the event that a sample has excellent length. If a part is selected at random,

determine the following probabilities:

(a) P (A) (b) P (A)

(c) P (A`) (d) P (A ∩ B)

(e) (A U B) (f) (A` U B)

Samples of emissions from three suppliers are classified for conformance to air-quality specifications. The results from 100 samples are summarized as follows:

Conforms

Yes No

1 22 8

Supplier 2 25 5

3 30 10

Let A denote the event that a sample is from supplier 1, and let B denote the event that a sample conforms to specifications. If a sample is selected at random, determine the following probabilities:

(a) P (A) (b) P (A)

(c) P (A`) (d) P (A ∩ B)

(e) (A U B) (f) (A` U B)

Conforms

Yes No

1 22 8

Supplier 2 25 5

3 30 10

Let A denote the event that a sample is from supplier 1, and let B denote the event that a sample conforms to specifications. If a sample is selected at random, determine the following probabilities:

(a) P (A) (b) P (A)

(c) P (A`) (d) P (A ∩ B)

(e) (A U B) (f) (A` U B)

Use the axioms of probability to show the following:

(a) For any event. E, P (E’) = 1 – P (E).

(b) P (Ø) = 0

(c) If A is contained in B, then P (A) < P (B)

(a) For any event. E, P (E’) = 1 – P (E).

(b) P (Ø) = 0

(c) If A is contained in B, then P (A) < P (B)

If P (A) = 0.3, P (B) = 0.2, and P (A U B) = 0.1, determine the following probabilities:

(a) P (A’)

(b) P (A U B)

(c) P (A` ∩ B)

(d) P (A ∩ B’)

(e) P [(A U B’)]

(f) P (A` U B)

(a) P (A’)

(b) P (A U B)

(c) P (A` ∩ B)

(d) P (A ∩ B’)

(e) P [(A U B’)]

(f) P (A` U B)

If A, B, and C are mutually exclusive events with P(A) = 0.2, P (B) =0.3, and P(C) = 0.4, determine the following probabilities:

(a) P (A U B U C)

(b) P (A ∩ B ∩ C)

(c) P (A ∩ B)

(d) P [(A U B) ∩ C]

(e) P (A’ ∩ B’ ∩ C’)

(a) P (A U B U C)

(b) P (A ∩ B ∩ C)

(c) P (A ∩ B)

(d) P [(A U B) ∩ C]

(e) P (A’ ∩ B’ ∩ C’)

If A, B, and C are mutually exclusive events, is it possible for P (A) = 0.3, P (B) = 0.4, and P (C) = 0.5? Why or why not?

Disks of polycarbonate plastic from a supplier are analyzed for scratch and shock resistance. The results from 100 disks are summarized as follows:

Shock resistance

High Low

Scratch high 70 9

Resistance low 16 5

(a) If a disk is selected at random, what is the probability that its scratch resistance is high and its shock resistance is high?

(b) If a disk is selected at random, what is the probability that its scratch resistance is high or its shock resistance is high?

(c) Consider the event that a disk has high scratch resistance and the event that a disk has high shock resistance. Are these two events mutually exclusive?

Shock resistance

High Low

Scratch high 70 9

Resistance low 16 5

(a) If a disk is selected at random, what is the probability that its scratch resistance is high and its shock resistance is high?

(b) If a disk is selected at random, what is the probability that its scratch resistance is high or its shock resistance is high?

(c) Consider the event that a disk has high scratch resistance and the event that a disk has high shock resistance. Are these two events mutually exclusive?

The analysis of shafts for a compressor is summarized by conformance to specifications.

Roundness conforms

Yes No

Surface finish yes 345 5

Conforms no 12 8

(a) If a shaft is selected at random, what is the probability that the shaft conforms to surface finish requirements?

(b) What is the probability that the selected shaft conforms to surface finish requirements or to roundness requirements?

(c) What is the probability that the selected shaft either conforms to surface finish requirements or does not conform to roundness requirements?

(d) What is the probability that the selected shaft conforms to both surface finish and roundness requirements

Roundness conforms

Yes No

Surface finish yes 345 5

Conforms no 12 8

(a) If a shaft is selected at random, what is the probability that the shaft conforms to surface finish requirements?

(b) What is the probability that the selected shaft conforms to surface finish requirements or to roundness requirements?

(c) What is the probability that the selected shaft either conforms to surface finish requirements or does not conform to roundness requirements?

(d) What is the probability that the selected shaft conforms to both surface finish and roundness requirements

Cooking oil is produced in two main varieties: mono-and polyunsaturated. Two common sources of cooking oil are corn and canola. The following table shows the number of bottles of these oils at a supermarket:

Type of oil _________________________________Canola Corn

Type of…………….mono…………………….7……………...13

Un-saturation………poly……………………...93……………77

(a) If a bottle of oil is selected at random, what is the probability that it belongs to the polyunsaturated category?

(b) What is the probability that the chosen bottle is monounsaturated canola oil?

Type of oil _________________________________Canola Corn

Type of…………….mono…………………….7……………...13

Un-saturation………poly……………………...93……………77

(a) If a bottle of oil is selected at random, what is the probability that it belongs to the polyunsaturated category?

(b) What is the probability that the chosen bottle is monounsaturated canola oil?

A manufacturer of front lights for automobiles tests lamps under a high humidity, high temperature environment using intensity and useful life as the responses of interest. The following table shows the performance of 130 lamps:

Useful life _________________________Satisfactory…………..Unsatisfactory Intensity satisfactory…………………………….117……………………3

Unsatisfactory ……………………………………8……………………..2

(a) Find the probability that a randomly selected lamp will yield unsatisfactory results under any criteria.

(b) The customers for these lamps demand 95% satisfactory results. Can the lamp manufacturer meet this demand?

Useful life _________________________Satisfactory…………..Unsatisfactory Intensity satisfactory…………………………….117……………………3

Unsatisfactory ……………………………………8……………………..2

(a) Find the probability that a randomly selected lamp will yield unsatisfactory results under any criteria.

(b) The customers for these lamps demand 95% satisfactory results. Can the lamp manufacturer meet this demand?

The shafts in Exercise 2-53 are further classified in terms of the machine tool that was used for manufacturing the shaft.

Disks of polycarbonate plastic from a supplier are analyzed for scratch and shock resistance. The results from 100 disks are summarized as follows:

Samples of a cast aluminum part are classified on the basis of surface finish (in micro inches) and length measurements. The results of 100 parts are summarized as follows:

The analysis of shafts for a compressor is summarized by conformance to specifications:

The following table summarizes the analysis of samples of galvanized steel for coating weight and surface roughness

Consider the data on wafer contamination and location in the sputtering tool shown in Table 2-2. Assume that one wafer is selected at random from this set. Let A denote the event that a wafer contains four or more particles, and let B denote the event that a wafer is from the center of the sputtering tool. Determine:

(a) P (A)

(b) P (A|B)

(c) P (B)

(d) P (B|C)

(e) P (A ∩ B)

(f) P (A U B)

(a) P (A)

(b) P (A|B)

(c) P (B)

(d) P (B|C)

(e) P (A ∩ B)

(f) P (A U B)

A lot of 100 semiconductor chips contains 20 that are defective. Two are selected randomly, without replacement, from the lot.

(a) What is the probability that the first one selected is defective?

(b) What is the probability that the second one selected is defective given that the first one was defective?

(c) What is the probability that both are defective?

(d) How does the answer to part (b) change if chips selected were replaced prior to the next selection?

(a) What is the probability that the first one selected is defective?

(b) What is the probability that the second one selected is defective given that the first one was defective?

(c) What is the probability that both are defective?

(d) How does the answer to part (b) change if chips selected were replaced prior to the next selection?

A lot contains 15 castings from a local supplier and 25 castings from a supplier in the next state. Two castings are selected randomly, without replacement, from the lot of 40.

Let A be the event that the first casting selected is from the local supplier, and let B denote the event that the second casting is selected from the local supplier. Determine:

(a) P (A)

(b) P (B|A)

(c) P (A ∩ B)

(D) P (A U B)

Let A be the event that the first casting selected is from the local supplier, and let B denote the event that the second casting is selected from the local supplier. Determine:

(a) P (A)

(b) P (B|A)

(c) P (A ∩ B)

(D) P (A U B)

Continuation of Exercise 2-63. Suppose three castings are selected at random, without replacement, from the lot of 40. In addition to the definitions of events A and B, let C denote the event that the third casting selected is from the local supplier. Determine:

(a) P (A ∩ B ∩ C)

(b) P (A ∩ B ∩ C’)

(a) P (A ∩ B ∩ C)

(b) P (A ∩ B ∩ C’)

A batch of 500 containers for frozen orange juice contains 5 that are defective. Two are selected, at random, without replacement from the batch.

(a) What is the probability that the second one selected is defective given that the first one was defective?

(b) What is the probability that both are defective?

(c) What is the probability that both are acceptable?

(a) What is the probability that the second one selected is defective given that the first one was defective?

(b) What is the probability that both are defective?

(c) What is the probability that both are acceptable?

Continuation of Exercise 2-65. Three containers are selected, at random, without replacement, from the batch.

(a) What is the probability that the third one selected is defective given that the first and second one selected were defective?

(b) What is the probability that the third one selected is defective given that the first one selected was defective and the second one selected was okay?

(c) What is the probability that all three are defective?

(a) What is the probability that the third one selected is defective given that the first and second one selected were defective?

(b) What is the probability that the third one selected is defective given that the first one selected was defective and the second one selected was okay?

(c) What is the probability that all three are defective?

A maintenance firm has gathered the following information regarding the failure mechanisms for air conditioning systems:

If P (A|B) = 1, must A = B? Draw a Venn diagram to explain your answer.

Suppose A and B are mutually exclusive events. Construct a Venn diagram that contains the three events A, B, and C such that P (A|C) _ 1 and P (B) C) = 0?

Suppose that P (A|B) = 0.4 and P(B) = 0.5. Determine the following:

(a) P (A ∩ B)

(b) P (A’ ∩ B)

(a) P (A ∩ B)

(b) P (A’ ∩ B)

Suppose that and P(A)|B) = 0.2, (PA|B’) = 0.3, and P(B) = 0.8. What is P (A)?

The probability is 1% that an electrical connector that is kept dry fails during the warranty period of a portable computer. If the connector is ever wet, the probability of a failure during the warranty period is 5%. If 90% of the connectors are kept dry and 10% are wet, what proportion of connectors fail during the warranty period?

Suppose 2% of cotton fabric rolls and 3% of nylon fabric rolls contain flaws. Of the rolls used by a manufacturer, 70% are cotton and 30% are nylon. What is the probability that a randomly selected roll used by the manufacturer contains flaws?

In the manufacturing of a chemical adhesive, 3% of all batches have raw materials from two different lots. This occurs when holding tanks are replenished and the remaining portion of a lot is insufficient to fill the tanks. Only 5% of batches with material from a single lot require reprocessing. However, the viscosity of batches consisting of two or more lots of material is more difficult to control, and 40% of such batches require additional processing to achieve the required viscosity. Let A denote the event that a batch is formed from two different lots, and let B denote the event that a lot requires additional processing. Determine the following probabilities:

(a) P (A)

(b) P (A’)

(c) P (B|A)

(d) P (B|A’)

(e) P (A ∩ B)

(f) p (A ∩ B’)

(g) P (B)

(a) P (A)

(b) P (A’)

(c) P (B|A)

(d) P (B|A’)

(e) P (A ∩ B)

(f) p (A ∩ B’)

(g) P (B)

The edge roughness of slit paper products increases as knife blades wear. Only 1% of products slit with new blades have rough edges, 3% of products slit with blades of average sharpness exhibit roughness, and 5% of products slit with worn blades exhibit roughness. If 25% of the blades in manufacturing are new, 60% are of average sharpness, and 15% are worn, what is the proportion of products that exhibit edge roughness?

Samples of laboratory glass are in small, light packaging or heavy, large packaging. Suppose that 2 and 1% of the sample shipped in small and large packages, respectively, break during transit. If 60% of the samples are shipped in large packages and 40% are shipped in small packages, what proportion of samples break during shipment?

Incoming calls to a customer service center are classified as complaints (75% of call) or requests for information (25% of calls). Of the complaints, 40% deal with computer equipment that does not respond and 57% deal with incomplete software installation; and in the remaining 3% of complaints the user has improperly followed the installation instructions. The requests for information are evenly divided on technical questions (50%) and requests to purchase more products (50%).

(a) What is the probability that an incoming call to the customer service center will be from a customer who has not followed installation instructions properly?

(b) Find the probability that an incoming call is a request for purchasing more products.

(a) What is the probability that an incoming call to the customer service center will be from a customer who has not followed installation instructions properly?

(b) Find the probability that an incoming call is a request for purchasing more products.

Computer keyboard failures are due to faulty electrical connects (12%) or mechanical defects (88%). Mechanical defects are related to loose keys (27%) or improper assembly (73%). Electrical connect defects are caused by defective wires (35%), improper connections (13%), or poorly welded wires (52%).
(a) Find the probability that a failure is due to loose keys.

(b) Find the probability that a failure is due to improperly connected or poorly welded wires.

(b) Find the probability that a failure is due to improperly connected or poorly welded wires.

A batch of 25 injection-molded parts contains 5 that have suffered excessive shrinkage.

(a) If two parts are selected at random, and without replacement, what is the probability that the second part selected is one with excessive shrinkage?

(b) If three parts are selected at random, and without replacement, what is the probability that the third part selected is one with excessive shrinkage?

(a) If two parts are selected at random, and without replacement, what is the probability that the second part selected is one with excessive shrinkage?

(b) If three parts are selected at random, and without replacement, what is the probability that the third part selected is one with excessive shrinkage?

A lot of 100 semiconductor chips contains 20 that are defective.

(a) Two are selected, at random, without replacement, from the lot. Determine the probability that the second chip selected is defective.

(b) Three are selected, at random, without replacement, from the lot. Determine the probability that all are defective.

(a) Two are selected, at random, without replacement, from the lot. Determine the probability that the second chip selected is defective.

(b) Three are selected, at random, without replacement, from the lot. Determine the probability that all are defective.

If P (A|B) = 0.4, P (B) = 0.8, P (A) = 0.5, are the events A and B independent?

If P (A|B) = 0.3, P (B) = 0.8, P (A) = 0.3, and are the events B and the complement of A independent?

Disks of polycarbonate plastic from a supplier are analyzed for scratch and shock resistance. The results from 100 disks are summarized as follows:

Shock resistance

High Low

Scratch high 70 9

Resistance low 16 5

Let A denote the event that a disk has high shock resistance, and let B denote the event that a disk has high scratch resistance. Are events A and B independent?

Shock resistance

High Low

Scratch high 70 9

Resistance low 16 5

Let A denote the event that a disk has high shock resistance, and let B denote the event that a disk has high scratch resistance. Are events A and B independent?

Samples of a cast aluminum part are classified on the basis of surface finish (in micro inches) and length measurements. The results of 100 parts are summarized as follows:

Length

Excellent Good

Surface excellent 80 2

Finish good 10 8

Let A denote the event that a sample has excellent surface finish, and let B denote the event that a sample has excellent length. Are events A and B independent?

Length

Excellent Good

Surface excellent 80 2

Finish good 10 8

Let A denote the event that a sample has excellent surface finish, and let B denote the event that a sample has excellent length. Are events A and B independent?

Samples of emissions from three suppliers are classified for conformance to air-quality specifications. The results from 100 samples are summarized as follows:

Conforms

Yes no

1 22 8

Supplier 2 25 5

3 30 10

Let A denote the event that a sample is from supplier 1, and let B denote the event that a sample conforms to specifications.

(a) Are events A and B independent?

(b) Determine

Conforms

Yes no

1 22 8

Supplier 2 25 5

3 30 10

Let A denote the event that a sample is from supplier 1, and let B denote the event that a sample conforms to specifications.

(a) Are events A and B independent?

(b) Determine

If P (A) = 0.2, P (B) = 0.2, and A and B are mutually exclusive, are they independent?

The probability that a lab specimen contains high levels of contamination is 0.10. Five samples are checked, and the samples are independent.

(a) What is the probability that none contains high levels of contamination?

(b) What is the probability that exactly one contains high levels of contamination?

(c) What is the probability that at least one contains high levels of contamination?

(a) What is the probability that none contains high levels of contamination?

(b) What is the probability that exactly one contains high levels of contamination?

(c) What is the probability that at least one contains high levels of contamination?

In a test of a printed circuit board using a random test pattern, an array of 10 bits is equally likely to be 0 or 1.

Assume the bits are independent.

(a) What is the probability that all bits are 1s?

(b) What is the probability that all bits are 0s?

(c) What is the probability that exactly five bits are 1s and five bits are 0s?

Assume the bits are independent.

(a) What is the probability that all bits are 1s?

(b) What is the probability that all bits are 0s?

(c) What is the probability that exactly five bits are 1s and five bits are 0s?

Eight cavities in an injection-molding tool produce plastic connectors that fall into a common stream. A sample is chosen every several minutes. Assume that the samples are independent.

(a) What is the probability that five successive samples were all produced in cavity one of the mold?

(b) What is the probability that five successive samples were all produced in the same cavity of the mold?

(c) What is the probability that four out of five successive samples were produced in cavity one of the mold?

(a) What is the probability that five successive samples were all produced in cavity one of the mold?

(b) What is the probability that five successive samples were all produced in the same cavity of the mold?

(c) What is the probability that four out of five successive samples were produced in cavity one of the mold?

The following circuit operates if and only if there is a path of functional devices from left to right. The probability that each device functions is as shown. Assume that the probability that a device is functional does not depend on whether or not other devices are functional. What is the probability that the circuit operates?

The following circuit operates if and only if there is a path of functional devices from left to right. The probability each device functions is as shown. Assume that the probability that a device functions does not depend on whether or not other devices are functional. What is the probability that the circuit operates?

An optical storage device uses and error recovery Procedure that requires an immediate satisfactory readback of any written data. If the readback is not successful after three writing operations, that sector of the disk is eliminated as unacceptable for data storage. On an acceptable portion of the disk, the probability of a satisfactory readback is 0.98. Assume the readbacks are independent. What is the probability that an acceptable portion of the disk is eliminated as unacceptable for data storage?

A batch of 500 containers for frozen orange juice contains 5 that are defective. Two are selected, at random, without replacement, from the batch. Let A and B denote the events that the first and second container selected is defective, respectively.

(a) Are A and B independent events?

(b) If the sampling were done with replacement, would A and B be independent?

(a) Are A and B independent events?

(b) If the sampling were done with replacement, would A and B be independent?

Suppose that P (A|B) = 0.7, P (A) = 0.5, and P (B) = 0.2. Determine P (B|A).

Software to detect fraud in consumer phone cards tracks the number of metropolitan areas where calls originate each day. It is found that 1% of the legitimate users originate calls from two or more metropolitan areas in a single day. However, 30% of fraudulent users originate calls from two or more metropolitan areas in a single day. The proportion of fraudulent users is 0.01%. If the same user originates calls from two or more metropolitan areas in a single day, what is the probability that the user is fraudulent?

Semiconductor lasers used in optical storage products require higher power levels for write operations than for read operations. High-power-level operations lower the useful life of the laser. Lasers in products used for backup of higher speed magnetic disks primarily write, and the probability that the useful life exceeds five years is 0.95. Lasers that are in products that are used for main storage spend approximately an equal amount of time reading and writing, and the probability that the useful life exceeds five years is 0.995. Now, 25% of the products from a manufacturer are used for backup and 75% of the products are used for main storage. Let A denote the event that a laser’s useful life exceeds five years, and let B denote the event that a laser is in a product that is used for backup. Use a tree diagram to determine the following:

(a) P (B)

(b) P (A|B)

(c) P (A|B’)

(d) P (A ∩ B)

(e) P (A ∩ B’)

(f) P (A)

(g) What is the probability that the useful life of a laser exceeds five years?

(h) What is the probability that a laser that failed before five years came from a product used for backup?

(a) P (B)

(b) P (A|B)

(c) P (A|B’)

(d) P (A ∩ B)

(e) P (A ∩ B’)

(f) P (A)

(g) What is the probability that the useful life of a laser exceeds five years?

(h) What is the probability that a laser that failed before five years came from a product used for backup?

Customers are used to evaluate preliminary product designs. In the past, 95% of highly successful products received good reviews, 60% of moderately successful products received good reviews, and 10% of poor products received good reviews. In addition, 40% of products have been highly successful, 35% have been moderately successful, and 25% have been poor products.

(a) What is the probability that a product attains a good review?

(b) If a new design attains a good review, what is the probability that it will be a highly successful product?

(c) If a product does not attain a good review, what is the probability that it will be a highly successful product?

(a) What is the probability that a product attains a good review?

(b) If a new design attains a good review, what is the probability that it will be a highly successful product?

(c) If a product does not attain a good review, what is the probability that it will be a highly successful product?

An inspector working for a manufacturing company has a 99% chance of correctly identifying defective items and a 0.5% chance of incorrectly classifying a good item as defective. The company has evidence that its line produces 0.9% of nonconforming items.

(a) What is the probability that an item selected for inspection is classified as defective?

(b) If an item selected at random is classified as nondefective, what is the probability that it is indeed good?

(a) What is the probability that an item selected for inspection is classified as defective?

(b) If an item selected at random is classified as nondefective, what is the probability that it is indeed good?

A new analytical method to detect pollutants in water is being tested. This new method of chemical analysis is important because, if adopted, it could be used to detect three different contaminants—organic pollutants, volatile solvents, and chlorinated compounds—instead of having to use a single test for each pollutant. The makers of the test claim that it can detect high levels of organic pollutants with 99.7% accuracy, volatile solvents with 99.95% accuracy, and chlorinated compounds with 89.7% accuracy. If a pollutant is not present, the test does not signal. Samples are prepared for the calibration of the test and 60% of them are contaminated with organic pollutants, 27% with volatile solvents, and 13% with traces of chlorinated compounds.

A test sample is selected randomly. (a) What is the probability that the test will signal?

(b) If the test signals, what is the probability that chlorinated compounds are present?

A test sample is selected randomly. (a) What is the probability that the test will signal?

(b) If the test signals, what is the probability that chlorinated compounds are present?

Decide whether a discrete or continuous random variable is the best model for each of the following variables:

(a) The time until a projectile returns to earth.

(b) The number of times a transistor in a computer memory changes state in one operation.

(c) The volume of gasoline that is lost to evaporation during the filling of a gas tank.

(d) The outside diameter of a machined shaft.

(e) The number of cracks exceeding one-half inch in 10 miles of an interstate highway.

(f) The weight of an injection-molded plastic part.

(g) The number of molecules in a sample of gas.

(h) The concentration of output from a reactor.

(i) The current in an electronic circuit.

(a) The time until a projectile returns to earth.

(b) The number of times a transistor in a computer memory changes state in one operation.

(c) The volume of gasoline that is lost to evaporation during the filling of a gas tank.

(d) The outside diameter of a machined shaft.

(e) The number of cracks exceeding one-half inch in 10 miles of an interstate highway.

(f) The weight of an injection-molded plastic part.

(g) The number of molecules in a sample of gas.

(h) The concentration of output from a reactor.

(i) The current in an electronic circuit.

In circuit testing of printed circuit boards, each board either fails or does not fail the test. A board that fails the test is then checked further to determine which one of five defect types is the primary failure mode. Represent the sample space for this experiment.

The data from 200 machined parts are summarized as follows:

Depth of bore

Above Below

Edge condition Target Target

Coarse 15 10

Moderate 25 20

Smooth 50 80

(a) What is the probability that a part selected has a moderate edge condition and a below-target bore depth?

(b) What is the probability that a part selected has a moderate edge condition or a below-target bore depth?

(c) What is the probability that a part selected does not have a moderate edge condition or does not have a below-target bore depth?

(d) Construct a Venn diagram representation of the events in this sample space.

Depth of bore

Above Below

Edge condition Target Target

Coarse 15 10

Moderate 25 20

Smooth 50 80

(a) What is the probability that a part selected has a moderate edge condition and a below-target bore depth?

(b) What is the probability that a part selected has a moderate edge condition or a below-target bore depth?

(c) What is the probability that a part selected does not have a moderate edge condition or does not have a below-target bore depth?

(d) Construct a Venn diagram representation of the events in this sample space.

Computers in a shipment of 100 units contain a portable hard drive, CD RW drive, or both according to the following table:

Portable hard drive

Yes No

CD RW

Yes 15 80

No 4 1

Let A denote the events that a computer has a portable hard drive and let B denote the event that a computer has a CD RW drive. If one computer is selected randomly,

compute (a) P (A)

(b) P (A ∩ B)

(c) P (A U B)

(d) P (A’ ∩ B)

(e) P (A|B)

Portable hard drive

Yes No

CD RW

Yes 15 80

No 4 1

Let A denote the events that a computer has a portable hard drive and let B denote the event that a computer has a CD RW drive. If one computer is selected randomly,

compute (a) P (A)

(b) P (A ∩ B)

(c) P (A U B)

(d) P (A’ ∩ B)

(e) P (A|B)

The probability that a customer’s order is not shipped on time is 0.05. A particular customer places three orders, and the orders are placed far enough apart in time that they can be considered to be independent events.

(a) What is the probability that all are shipped on time?

(b) What is the probability that exactly one is not shipped on time?

(c) What is the probability that two or more orders are not shipped on time?

(a) What is the probability that all are shipped on time?

(b) What is the probability that exactly one is not shipped on time?

(c) What is the probability that two or more orders are not shipped on time?

Let E1, E2, and E3 denote the samples that conform to a percentage of solids specification, a molecular weight specification, and a color specification, respectively. A total of 240 samples are classified by the E1, E2, and E3 specifications, where yes indicates that the sample conforms.

Transactions to a computer database are either new items or changes to previous items. The addition of an item can be completed less than 100 milliseconds 90% of the time, but only 20% of changes to a previous item can be completed in less than this time. If 30% of transactions are changes, what is the probability that a transaction can be completed in less than 100 milliseconds?

A steel plate contains 20 bolts. Assume that 5 bolts are not torqued to the proper limit. Four bolts are selected at random, without replacement, to be checked for torque.

(a) What is the probability that all four of the selected bolts are torqued to the proper limit?

(b) What is the probability that at least one of the selected bolts is not torqued to the proper limit?

(a) What is the probability that all four of the selected bolts are torqued to the proper limit?

(b) What is the probability that at least one of the selected bolts is not torqued to the proper limit?

The following circuit operates if and only if there is a path of functional devices from left to right. Assume devices fail independently and that the probability of failure of each device is as shown. What is the probability that the circuit operates?

The probability of getting through by telephone to buy concert tickets is 0.92. For the same event, the probability of accessing the vendor’s Web site is 0.95. Assume that these two ways to buy tickets are independent. What is the probability that someone who tries to buy tickets through the Internet and by phone will obtain tickets?

The British government has stepped up its information campaign regarding foot and mouth disease by mailing brochures to farmers around the country. It is estimated that 99% of Scottish farmers who receive the brochure possess enough information to deal with an outbreak of the disease, but only 90% of those without the brochure can deal with an outbreak. After the first three months of mailing, 95% of the farmers in Scotland received the informative brochure. Compute the probability that a randomly selected farmer will have enough information to deal effectively with an outbreak of the disease.

In an automated filling operation, the probability of an incorrect fill when the process is operated at a low speed is 0.001. When the process is operated at a high speed, the probability of an incorrect fill is 0.01. Assume that 30% of the containers are filled when the process is operated at a high speed and the remainder are filled when the process is operated at a low speed.

(a) What is the probability of an incorrectly filled container?

(b) If an incorrectly filled container is found, what is the probability that it was filled during the high-speed operation?

(a) What is the probability of an incorrectly filled container?

(b) If an incorrectly filled container is found, what is the probability that it was filled during the high-speed operation?

An encryption-decryption system consists of three elements: encode, transmit, and decode. A faulty encode occurs in 0.5% of the messages processed, transmission errors occur in 1% of the messages, and a decode error occurs in 0.1% of the messages. Assume the errors are independent.

(a) What is the probability of a completely defect-free message?

(b) What is the probability of a message that has either an encode or a decode error?

(a) What is the probability of a completely defect-free message?

(b) What is the probability of a message that has either an encode or a decode error?

It is known that two defective copies of a commercial software program were erroneously sent to a shipping lot that has now a total of 75 copies of the program. A sample of copies will be selected from the lot without replacement.

(a) If three copies of the software are inspected, determine the probability that exactly one of the defective copies will be found.

(b) If three copies of the software are inspected, determine the probability that both defective copies will be found.

(c) If 73 copies are inspected, determine the probability that both copies will be found. Hint: Work with the copies that remain in the lot.

(a) If three copies of the software are inspected, determine the probability that exactly one of the defective copies will be found.

(b) If three copies of the software are inspected, determine the probability that both defective copies will be found.

(c) If 73 copies are inspected, determine the probability that both copies will be found. Hint: Work with the copies that remain in the lot.

A robotic insertion tool contains 10 primary components. The probability that any component fails during the warranty period is 0.01. Assume that the components fail independently and that the tool fails if any component fails. What is the probability that the tool fails during the warranty period?

An e-mail message can travel through one of two server routes. The probability of transmission error in each of the servers and the proportion of messages that travel each route are shown in the following table. Assume that the servers are independent.

Probability of error

Percentage Of messages server 1 server 2 server 3

server 4

Route 1 30 0.01 0.015

Route 2 70 0.02

0.003 (a) What is the probability that a message will arrive without error?

(b) If a message arrives in error, what is the probability it was sent through route 1?

Probability of error

Percentage Of messages server 1 server 2 server 3

server 4

Route 1 30 0.01 0.015

Route 2 70 0.02

0.003 (a) What is the probability that a message will arrive without error?

(b) If a message arrives in error, what is the probability it was sent through route 1?

A machine tool is idle 15% of the time. You request immediate use of the tool on five different occasions during the year. Assume that your requests represent independent events.

(a) What is the probability that the tool is idle at the time of all of your requests?

(b) What is the probability that the machine is idle at the time of exactly four of your requests?

(c) What is the probability that the tool is idle at the time of at least three of your requests?

(a) What is the probability that the tool is idle at the time of all of your requests?

(b) What is the probability that the machine is idle at the time of exactly four of your requests?

(c) What is the probability that the tool is idle at the time of at least three of your requests?

A lot of 50 spacing washers contains 30 washers that are thicker than the target dimension. Suppose that three washers are selected at random, without replacement, from the lot.

(a) What is the probability that all three washers are thicker than the target?

(b) What is the probability that the third washer selected is thicker than the target if the first two washers selected are thinner than the target?

(c) What is the probability that the third washer selected is thicker than the target?

(a) What is the probability that all three washers are thicker than the target?

(b) What is the probability that the third washer selected is thicker than the target if the first two washers selected are thinner than the target?

(c) What is the probability that the third washer selected is thicker than the target?

Continuation of Exercise 2-117. Washers are selected from the lot at random, without replacement.

(a) What is the minimum number of washers that need to be selected so that the probability that all the washers are thinner than the target is less than 0.10?

(b) What is the minimum number of washers that need to be selected so that the probability that one or more washers are thicker than the target is at least 0.90?

(a) What is the minimum number of washers that need to be selected so that the probability that all the washers are thinner than the target is less than 0.10?

(b) What is the minimum number of washers that need to be selected so that the probability that one or more washers are thicker than the target is at least 0.90?

The following table lists the history of 940 orders for features in an entry-level computer product.

Extra memory

No Yes

Optional high- no 514 68

Speed processor yes 112 246

Let A be the event that an order requests the optional high speed processor, and let B be the event that an order requests extra memory. Determine the following probabilities:

(a) P (A U B)

(b) P (A ∩ B)

(c) P (A’ U B)

(d) P (A’ ∩ B’)

(e) What is the probability that an order requests an optional high-speed processor given that the order requests extra memory?

(f) What is the probability that an order requests extra memory given that the order requests an optional high-speed processor?

Extra memory

No Yes

Optional high- no 514 68

Speed processor yes 112 246

Let A be the event that an order requests the optional high speed processor, and let B be the event that an order requests extra memory. Determine the following probabilities:

(a) P (A U B)

(b) P (A ∩ B)

(c) P (A’ U B)

(d) P (A’ ∩ B’)

(e) What is the probability that an order requests an optional high-speed processor given that the order requests extra memory?

(f) What is the probability that an order requests extra memory given that the order requests an optional high-speed processor?

The alignment between the magnetic tape and head in a magnetic tape storage system affects the performance of the system. Suppose that 10% of the read operations are degraded by skewed alignments, 5% of the read operations are degraded by off-center alignments, and the remaining read operations are properly aligned. The probability of a read error is 0.01 from a skewed alignment, 0.02 from an off-center alignment, and 0.001 from a proper alignment.

(a) What is the probability of a read error?

(b) If a read error occurs, what is the probability that it is due to a skewed alignment?

(a) What is the probability of a read error?

(b) If a read error occurs, what is the probability that it is due to a skewed alignment?

The following circuit operates if and only if there is a path of functional devices from left to right. Assume that devices fail independently and that the probability of failure of each device is as shown. What is the probability that the circuit does not operate?

A company that tracks the use of its web site determined that the more pages a visitor views, the more likely the visitor are to provide contact information. Use the following tables to answer the questions:

Number of

Pages viewed: 1 2 3 4 or more

Percentage of Visitors: 40 30 20 10

Percentage of visitors In each page-view

Category that provide

Contact information: 10 10 20 40

(a) What is the probability that a visitor to the web site provides contact information?

(b) If a visitor provides contact information, what is the probability that the visitor viewed four or more pages?

Number of

Pages viewed: 1 2 3 4 or more

Percentage of Visitors: 40 30 20 10

Percentage of visitors In each page-view

Category that provide

Contact information: 10 10 20 40

(a) What is the probability that a visitor to the web site provides contact information?

(b) If a visitor provides contact information, what is the probability that the visitor viewed four or more pages?

An order for a personal digital assistant can specify any one of five memory sizes, any one of three types of displays, any one of four sizes of a hard disk, and can either include or not include a pen tablet. How many different systems can be ordered?

In a manufacturing operation, a part is produced by machining, polishing, and painting. If there are three machine tools, four polishing tools, and three painting tools, how many different routings (consisting of machining, followed by polishing, and followed by painting) for a part are possible?

New designs for a wastewater treatment tank have proposed three possible shapes, four possible sizes, three locations for input valves, and four locations for output valves. How many different product designs are possible?

A manufacturing process consists of 10 operations that can be completed in any order. How many different production sequences are possible?

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