Chapter 2

A computer system uses passwords that are six characters, and each character is one of the 26 letters \((a-z)\) or 10 integers \((0-9)\). Uppercase letters are not used. Let \(A\) denote the event that a password begins with a vowel (either \(a, e, i, o\), or \(u\) ), and let \(B\) denote the event that a password ends with an even number (either \(0,2,4,6,\) or 8 ). Suppose a hacker selects a password at random. Determine the following probabilities: (a) \(P(A)\) (b) \(P(B)\) (c) \(P(A \cap B)\) (d) \(P(A \cup B)\)

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 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? Three containers are selected, at random, without replacement, from the batch. (d) What is the probability that the third one selected is defective given that the first and second ones selected were defective? (e) 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? (f) What is the probability that all three are defective?

A batch of 350 samples of rejuvenated mitochondria contains 8 that are mutated (or defective). Two are selected from the batch, at random, without replacement. (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 computer system uses passwords that are exactly seven characters and each character is one of the 26 letters \((a-z)\) or 10 integers \((0-9)\). You maintain a password for this computer system. Let \(A\) denote the subset of passwords that begin with a vowel (either \(a, e, i, o,\) or \(u\) ) and let \(B\) denote the subset of passwords that end with an even number (either \(0,2,4,6,\) or 8 ). (a) Suppose a hacker selects a password at random. What is the probability that your password is selected? (b) Suppose a hacker knows that your password is in event \(A\) and selects a password at random from this subset. What is the probability that your password is selected? (c) Suppose a hacker knows that your password is in \(A\) and \(B\) and selects a password at random from this subset. What is the probability that your password is selected?

A computer system uses passwords that contain exactly eight characters, and each character is one of the 26 lowercase letters \((a-z)\) or 26 uppercase letters \((A-Z)\) or 10 integers \((0-9)\). Let \(\Omega\) denote the set of all possible passwords. Suppose that all passwords in \(\Omega\) are equally likely. Determine the probability for each of the following: (a) Password contains all lowercase letters given that it contains only letters (b) Password contains at least 1 uppercase letter given that it contains only letters (c) Password contains only even numbers given that is contains all numbers

Suppose that \(P(A \mid B)=0.4\) and \(P(B)=0.5 .\) Determine the following: (a) \(P(A \cap B\) (b) \(P\left(A^{\prime} \cap B\right)\)

Suppose that \(P(A \mid B)=0.2, P\left(A \mid B^{\prime}\right)=0.3,\) 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 pcriod 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 conncctors fail during the warranty period?

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

The edge roughness of slit paper products increases as knife blades wour, Only \(1 \%\) of products slit with ncw bladcs have rough edges, \(3 \%\) of products slit with blades of average sharpness exhibit roughness, and \(5 \%\) of products slit with worn bladcs cxhibit roughncss. If \(25 \%\) of the bladcs in manufacturing are ncw, \(60 \%\) are of average sharpncss, and \(15 \%\) are worn, what is the proportion of products that exhibit edge roughness?

In the 2012 presidential clection, exit polls from the critical state of Ohio provided the following results: $$\begin{array}{lcc}\text { Total } & \text { Obama } & \text { Romney } \\\\\text { No college degree }(60 \%) & 52 \% & 45 \% \\\\\text { Collcge graduate }(40\%) & 47 \% & 51 \%\end{array}$$ What is the probability a randomly selected respondent voted for Obama?

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.

Heart failures are due to either natural occurrences \((87 \%)\) or outside factors \((13 \%) .\) Outside factors are related to induced substances \((73 \%)\) or foreign objects \((27 \%) .\) Natural occurrences are caused by arterial blockage \((56 \%),\) disease \((27 \%),\) and infection (e.g., staph infection) (17\%). (a) Determine the probability that a failure is due to an induced substance (b) Determine the probability that a failure is due to disease or infection.

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.

An article in the British Medical Journal ["Comparison of treatment of renal calculi by operative surgery, percutaneous nephrolithotomy, and extracorporeal shock wave lithotripsy" (1986, Vol. 82, pp. \(879-892\) ) ] provided the following discussion of success rates in kidney stone removals. Open surgery had a success rate of \(78 \%(273 / 350)\) and a newer method, percutaneous nephrolithotomy (PN), had a success rate of \(83 \%(289 / 350)\). This newer method looked better, but the results changed when stone diameter was considered. For stones with diameters less than 2 centimeters, \(93 \%(81 / 87)\) of cases of open surgery were successful compared with only \(83 \%(234 / 270)\) of cases of PN. For stones greater than or equal to 2 centimeters, the success rates were \(73 \%(192 / 263)\) and \(69 \%(55 / 80)\) for open surgery and PN, respectively. Open surgery is better for both stone sizes, but less successful in total. In \(1951,\) E. H. Simpson pointed out this apparent contradiction (known as Simpson's paradox), and the hazard still persists today. Explain how open surgery can be better for both stone sizes but worse in total.

A Web ad can be designed from four different colors, three font types, five font sizes, three images, and five text phrases. A specific design is randomly generated by the Web server when you visit the site. Determine the probability that the ad color is red and the font size is not the smallest one.

A computer system uses passwords that contain exactly eight characters, and each character is one of the 26 lowercase letters \((a-z)\) or 26 uppercase letters \((A-Z)\) or 10 integers \((0-9)\). Let \(\Omega\) denote the set of all possible password, and let \(A\) and \(B\) denote the events that consist of passwords with only letters or only integers, respectively. Suppose that all passwords in \(\Omega\) are equally likely. Determine the following robabilities: (a) \(P\left(A \mid B^{\prime}\right)\) (b) \(P\left(A^{\prime} \cap B\right)\) (c) \(P\) (password contains exactly 2 integers given that it contains at least 1 integer)

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

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

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

A batch of 500 containers of 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 containers selected are defective, respectively. (a) Are \(A\) and \(B\) independent events? (b) If the sampling were done with replacement, would \(A\) and \(B\) be independent?

Redundant array of inexpensive disks (RAID) is a technology that uses multiple hard drives to increase the speed of data transfer and provide instant data backup. Suppose that the probability of any hard drive failing in a day is 0.001 and the drive failures are independent. (a) A RAID 0 scheme uses two hard drives, each containing a mirror image of the other. What is the probability of data loss? Assume that data loss occurs if both drives fail within the same day. (b) A RAID 1 scheme splits the data over two hard drives. What is the probability of data loss? Assume that data loss occurs if at least one drive fails within the same day.

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 contain 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 \(1 \mathrm{~s}\) ? (b) What is the probability that all bits are 0 s? (c) What is the probability that exactly 5 bits are \(1 \mathrm{~s}\) and 5 bits are \(0 \mathrm{~s} ?\)

Six tissues are extracted from an ivy plant infested by spider mites. The plant in infested in \(20 \%\) of its area. Each tissue is chosen from a randomly selected area on the ivy plant. (a) What is the probability that four successive samples show the signs of infestation? (b) What is the probability that three out of four successive samples show the signs of infestation?

A player of a video game is confronted with a series of four opponents and an \(80 \%\) probability of defeating each opponent. Assume that the results from opponents are independent (and that when the player is defeated by an opponent the game ends). (a) What is the probability that a player defeats all four opponents in a game? (b) What is the probability that a player defeats at least two opponents in a game? (c) If the game is played three times, what is the probability that the player defeats all four opponents at least once?

A credit card contains 16 digits. It also contains the month and year of expiration. Suppose there are 1 million credit card holders with unique card numbers. A hacker randomly selects a 16-digit credit card number. (a) What is the probability that it belongs to a user? (b) Suppose a hacker has a \(25 \%\) chance of correctly guessing the year your card expires and randomly selects 1 of the 12 months. What is the probability that the hacker correctly selects the month and year of expiration?

A Web ad can be designed from four different colors, three font types, five font sizes, three images, and five text phrases. A specific design is randomly generated by the Web server when you visit the site. Let \(A\) denote the event that the design color is red, and let \(B\) denote the event that the font size is not the smallest one. Are \(A\) and \(B\) independent events? Explain why or why not.

An integrated circuit contains 10 million logic gates (each can be a logical AND or OR circuit). Assume the probability of a gate failure is \(p\) and that the failures are independent. The integrated circuit fails to function if any gate fails. Determine the value for \(p\) so that the probability that the integrated circuit functions is \(0.95 .\)

An integrated circuit contains 10 million logic gates (each can be a logical AND or OR circuit). Assume the probability of a gate failure is \(p\) and that the failures are independent. The integrated circuit fails to function if any gate fails. Determine the value for \(p\) so that the probability that the integrated circuit functions is \(0.95 .\)

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.

Decide whether a discrete or continuous random variable is the best model for each of the following variables: (a) The number of cracks exceeding one-half inch in 10 miles of an interstate highway. (b) The weight of an injection-molded plastic part. (c) The number of molecules in a sample of gas. (d) The concentration of output from a reactor. (e) The current in an electronic circuit.

Decide whether a discrete or continuous random variable is the best model for each of the following variables: (a) The time for a computer algorithm to assign an image to a category. (b) The number of bytes used to store a file in a computer. (c) The ozone concentration in micrograms per cubic meter. (d) The ejection fraction (volumetric fraction of blood pumped from a heart ventricle with each beat). (e) The fluid flow rate in liters per minute.

Samples of laboratory glass are in small, light packaging or heavy, large packaging. Suppose that \(2 \%\) and \(1 \%\), respectively, 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?

A sample of three calculators is selected from a manufacturing line, and each calculator is classified as either defective or acceptable. Let \(A, B,\) and \(C\) denote the events that the first, second, and third calculators, respectively, are defective. (a) Describe the sample space for this experiment with a tree diagram. Use the tree diagram to describe each of the following events: (b) \(A\) (c) \(B\) (d) \(A \cap B\) (e) \(B \cup C\)

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?

A researcher receives 100 containers of oxygen. Of those containers, 20 have oxygen that is not ionized, and the rest are ionized. Two samples are randomly selected, without replacement, from the lot. (a) What is the probability that the first one selected is not ionized? (b) What is the probability that the second one selected is not ionized given that the first one was ionized? (c) What is the probability that both are ionized? (d) How does the answer in part (b) change if samples 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) \(P(B \mid A)\) (c) \(P(A \cap B)\) (d) \(P(A \cup B)\) Suppose that 3 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: (e) \(P(A \cap B \cap C)\) (f) \(P\left(A \cap B \cap C^{\prime}\right)\)

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 dif- ferent lots, and let \(B\) denote the event that a lot requires additional processing. Determine the following probabilities: (a) \(P(A)\) (b) \(P\left(A^{\prime}\right)\) (c) \(P(B \mid A)\) (d) \(P\left(B \mid A^{\prime}\right)\) (e) \(P(A \cap B)\) (f) \(P\left(A \cap B^{\prime}\right)\) (g) \(P(B)\)

Incoming calls to a customer service center are classified as complaints \((75 \%\) of calls) 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; 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 congested computer network has a 0.002 probability of losing a data packet, and packet losses are independent events. A lost packet must be resent. (a) What is the probability that an e-mail message with 100 packets will need to be resent? (b) What is the probability that an e-mail message with 3 packets will need exactly 1 to be resent? (c) If 10 e-mail messages are sent, each with 100 packets, what is the probability that at least 1 message will need some packets to be resent?

Go Tutorial An optical storage device uses an 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?

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) \(P(A \mid B)\) (c) \(P\left(A \mid B^{\prime}\right)\) (d) \(P(A \cap B)\) (e) \(P\left(A \cap B^{\prime}\right)\) (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 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 that it is completed without error? (b) What is the probability that it is completed with either a minor or a major error?

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 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?

Transactions to a computer database are either new items or changes to previous items. The addition of an item can be completed in 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. 4 bolts are selected at random, without replacement, to be checked for torque. (a) What is the probability that all 4 of the selected bolts are torqued to the proper limit? (b) What is the probability that at least 1 of the selected bolts is not torqued to the proper limit?

The probability that concert tickets are available by telephone is \(0.92 .\) For the same event, the probability that tickets are available through a 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 Web and by phone will obtain tickets?