Chapter 2

Provide a reasonable description of the sample space for each of the random experiments in Exercises \(2-1\) to \(2-17\). There can be more than one acceptable interpretation of each experiment. Describe any assumptions you make. Each of three machined parts is classified as either above or below the target specification for the part.

The following two questions appear on an employee survey questionnaire. Each answer is chosen from the fivepoint scale 1 (never), 2,3,4,5 (always). Is the corporation willing to listen to and fairly evaluate new ideas? How often are my coworkers important in my overall job performance?

Provide a reasonable description of the sample space for each of the random experiments in Exercises \(2-1\) to \(2-17\). There can be more than one acceptable interpretation of each experiment. Describe any assumptions you make. The time until a service transaction is requested of a computer to the nearest millisecond.

Provide a reasonable description of the sample space for each of the random experiments in Exercises \(2-1\) to \(2-17\). There can be more than one acceptable interpretation of each experiment. Describe any assumptions you make. The pH reading of a water sample to the nearest tenth of a unit.

An order for an automobile can specify either an automatic or a standard transmission, either with or without air conditioning, and with any one of the four colors red, blue, black, or white. Describe the set of possible orders for this experiment.

An order for a computer system can specify memory of 4 \(8,\) or 12 gigabytes and disk storage of \(200,300,\) or 400 gigabytes. Describe the set of possible orders.

Three attempts are made to read data in a magnetic storage device before an error recovery procedure that repositions the magnetic head is used. The error recovery procedure attempts three repositionings before an "abort" message is sent to the operator. Let \(s\) denote the success of a read operation \(f\) denote the failure of a read operation \(S\) denote the success of an error recovery procedure \(F\) denote the failure of an error recovery procedure \(A\) denote an abort message sent to the operator Describe the sample space of this experiment with a tree diagram.

A digital scale that provides weights to the nearest gram is used. (a) What is the sample space for this experiment? Let \(A\) denote the event that a weight exceeds 11 grams, let \(B\) denote the event that a weight is less than or equal to 15 grams, and let \(C\) denote the event that a weight is greater than or equal to 8 grams and less than 12 grams. Describe the following events. (b) \(A \cup B\) (c) \(A \cap B\) (d) \(A^{\prime}\) (e) \(A \cup B \cup C\) (f) \((A \cup C)^{\prime}\) (g) \(A \cap B \cap C\) (h) \(B^{\prime} \cap C\) (i) \(A \cup(B \cap C)\)

Four bits are transmitted over a digital communications channel. Each bit is either distorted or received without distortion. Let \(A i\) denote the event that the ith bit is distorted, \(i=1, \ldots .4\) (a) Describe the sample space for this experiment. (b) Are the \(A\) 's mutually exclusive? Describe the outcomes in each of the following events: (c) \(A_{1}\) (d) \(A_{1}^{\prime}\) (e) \(A_{1} \cap A_{2} \cap A_{3} \cap A_{4}\) (f) \(\left(A_{1} \cap A_{2}\right) \cup\left(A_{3} \cap A_{4}\right)\)

In light-dependent photosynthesis, light quality refers to the wavelengths of light that are important. The wavelength of a sample of photosynthetically active radiations (PAR) is measured to the nearest nanometer. The red range is \(675-700\) \(\mathrm{nm}\) and the blue range is \(450-500 \mathrm{nm}\). Let \(A\) denote the event that PAR occurs in the red range, and let \(B\) denote the event that PAR occurs in the blue range. Describe the sample space and indicate each of the following events: (a) \(A\) (b) \(B\) (c) \(A \cap B\) (d) \(A \cup B\)

In control replication, cells are replicated over a period of two days. Not until mitosis is completed can freshly synthesized DNA be replicated again. Two control mechanisms have been identified-one positive and one negative. Suppose that a replication is observed in threc cells. Let \(A\) denote the event that all cells are identified as positive, and let \(B\) denote the event that all cells are negative. Describe the sample space graphically and display each of the following events: (a) A (b) \(B\) (c) \(A \cap B\) (d) \(A \cup B\)

The rise time of a reactor is measured in minutes (and fractions of minutes). Let the sample space be positive, real numbers. Define the events \(A\) and \(B\) as follows: \(A=\\{x \mid x<72.5\\}\) and \(B=\\{x \mid x>52.5\\}\) Describe each of the following events: (a) \(A^{\prime}\) (b) \(B^{\prime}\) (c) \(A \cap B\) (d) \(A \cup B\)

A sample of two items is selected without replacement from a batch. Describe the (ordered) sample space for each of the following batches: (a) The batch contains the items \(\\{a, b, c, d\\}\). (b) The batch contains the items \(\\{a, b, c, d, e, f, g\\}\). (c) The batch contains 4 defective items and 20 good items. (d) The batch contains 1 defective item and 20 good items.

A sample of two printed circuit boards is selected without replacement from a batch. Describe the (ordered) sample space for each of the following batches: (a) The batch contains 90 boards that are not defective, 8 boards with minor defects, and 2 boards with major defects. (b) The batch contains 90 boards that are not defective, 8 boards with minor defects, and 1 board with major defects.

Counts of the Web pages provided by each of two computer servers in a selected hour of the day are recorded. Let \(A\) denote the event that at least 10 pages are provided by server 1 , and let \(B\) denote the event that at least 20 pages are provided by server \(2 .\) Describe the sample space for the numbers of pages for the two servers graphically in an \(x-y\) plot. Show each of the following events on the sample space graph: (a) \(A\) (b) \(B\) (c) \(A \cap B\) (d) \(A \cup B\)

A reactor's rise time is measured in minutes (and fractions of minutes). Let the sample space for the rise time of each batch be positive, real numbers. Consider the rise times of two batches. Let \(A\) denote the event that the rise time of batch 1 is less than 72.5 minutes, and let \(B\) denote the event that the rise time of batch 2 is greater than 52.5 minutes. Describe the sample space for the rise time of two batches graphically and show each of the following events on a twodimensional plot: (a) \(A\) (b) \(B^{\prime}\) (c) \(A \cap B\) (d) \(A \cup B\)

A wireless garage door opener has a code determined by the up or down setting of 12 switches. How many outcomes are in the sample space of possible codes?

An order for a computer can specify any one of five memory sizes, any one of three types of displays, and 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?

A manufacturing operation consists of 10 operations. However, five machining operations must be completed before any of the remaining five assembly operations can begin. Within each set of five, operations can be completed in any order. How many different production sequences are possible?

In a sheet metal operation, three notches and four bends are required. If the operations can be done in any order, how many different ways of completing the manufacturing are possible?

A batch of 140 semiconductor chips is inspected by choosing a sample of 5 chips. Assume 10 of the chips do not conform to customer requirements. (a) How many different samples are possible? (b) How many samples of five contain exactly one nonconforming chip? (c) How many samples of five contain at least one nonconforming chip?

In the laboratory analysis of samples from a chemical process, five samples from the process are analyzed daily. In addition, a control sample is analyzed twice each day to check the calibration of the laboratory instruments. (a) How many different sequences of process and control samples are possible each day? Assume that the five process samples are considered identical and that the two control samples are considered identical. (b) How many different sequences of process and control samples are possible if we consider the five process samples to be different and the two control samples to be identical? (c) For the same situation as part (b), how many sequences are possible if the first test of each day must be a control sample?

In the design of an electromechanical product, 12 components are to be stacked into a cylindrical casing in a manner that minimizes the impact of shocks. One end of the casing is designated as the bottom and the other end is the top. (a) If all components are different, how many different designs are possible? (b) If seven components are identical to one another, but the others are different, how many different designs are possible? (c) If three components are of one type and identical to one another, and four components are of another type and identical to one another, but the others are different, how many different designs are possible?

Consider the design of a communication system. (a) How many three-digit phone prefixes that are used to represent a particular geographic area (such as an area code) can be created from the digits 0 through \(9 ?\) (b) As in part (a), how many three-digit phone prefixes are possible that do not start with 0 or \(1,\) but contain 0 or 1 as the middle digit? (c) How many three-digit phone prefixes are possible in which no digit appears more than once in each prefix?

A byte is a sequence of eight bits and each bit is either 0 or 1 . (a) How many different bytes are possible? (b) If the first bit of a byte is a parity check, that is, the first byte is determined from the other seven bits, how many different bytes are possible?

In a chemical plant, 24 holding tanks are used for final product storage. Four tanks are selected at random and without replacement. Suppose that six of the tanks contain material in which the viscosity exceeds the customer requirements. (a) What is the probability that exactly one tank in the sample contains high- viscosity material? (b) What is the probability that at least one tank in the sample contains high-viscosity material? (c) In addition to the six tanks with high viscosity levels, four different tanks contain material with high impurities. What is the probability that exactly one tank in the sample contains high-viscosity material and exactly one tank in the sample contains material with high impurities?

Plastic parts produced by an injection-molding operation are checked for conformance to specifications. Each tool contains 12 cavities in which parts are produced, and these parts fall into a conveyor when the press opens. An inspector chooses 3 parts from among the 12 at random. Two cavities are affected by a temperature malfunction that results in parts that do not conform to specifications. (a) How many samples contain exactly 1 nonconforming part? (b) How many samples contain at least 1 nonconforming part?

A bin of 50 parts contains 5 that are defective. A sample of 10 parts is selected at random, without replacement. How many samples contain at least four defective parts?

A Web ad can be designed from four different colors, three font types, five font sizes, three images, and five text phrases. How many different designs are possible?

A computer system uses passwords that contain exactly eight characters, and each character is 1 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, and let \(A\) and \(B\) denote the events that consist of passwords with only letters or only integers, respectively. Determine the number of passwords in each of the following events. (a) \(\Omega\) (b) \(A\) (c) \(A^{\prime} \cap B^{\prime}\) (d) Passwords that contain at least 1 integer (e) Passwords that contain exactly 1 integer

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) \(P\left(A^{\prime}\right)\) (d) \(P(A \cup B)\) (e) \(P(A \cap B)\)

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) \(P\left(A^{\prime}\right)\) (d) \(P(A \cup B)\) (e) \(P(A \cap B)\)

Orders for a computer are summarized by the optional features that are requested as follows: $$ \begin{array}{|lc|} \hline & \text { Proportion of Orders } \\ \hline \text { No optional features } & 0.3 \\ \text { One optional feature } & 0.5 \\ \text { More than one optional feature } & 0.2 \\ \hline \end{array} $$ (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 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 ?

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 that a part is from cavity 1 or 2 ? (c) What is the probability that a part is from neither cavity 3 nor 4 ?

In an acid-base titration, a base or acid is gradually added to the other until they have completely neutralized each other. Because acids and bases are usually colorless (as are the water and salt produced in the neutralization reaction), \(\mathrm{pH}\) is measured to monitor the reaction. Suppose that the equivalence point is reached after approximately \(100 \mathrm{~mL}\) of an \(\mathrm{NaOH}\) solution has been added (enough to react with all the acetic acid present) but that replicates are equally likely to indicate from 95 to \(104 \mathrm{~mL}\) to the nearest mL. Assume that volumes are measured to the nearest \(\mathrm{mL}\) and describe the sample space. (a) What is the probability that equivalence is indicated at \(100 \mathrm{~mL} ?\) (b) What is the probability that equivalence is indicated at less than \(100 \mathrm{~mL} ?\) (c) What is the probability that equivalence is indicated between 98 and \(102 \mathrm{~mL}\) (inclusive)?

In a NiCd battery, a fully charged cell is composed of nickelic hydroxide. Nickel is an element that has multiple oxidation states and that is usually found in the following states: $$ \begin{array}{|cc|} \hline \text { Nickel Charge } & \text { Proportions Found } \\ \hline 0 & 0.17 \\ +2 & 0.35 \\ +3 & 0.33 \\ +4 & 0.15 \\ \hline \end{array} $$ (a) What is the probability that a cell has at least one of the positive nickel-charged options? (b) What is the probability that a cell is not composed of a positive nickel charge greater than \(+3 ?\)

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 sender's 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 those can send to four servers at the third step; and then the message goes to the recipient's 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?

Use the axioms of probability to show the following: (a) For any event \(E, P\left(E^{\prime}\right)=1-P(E)\). (b) \(P(\varnothing)=0\) (c) If \(A\) is contained in \(B,\) then \(P(A) \leq P(B)\).

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. If you visit the site five times, what is the probability that you will not see the same design?

A computer system uses passwords that contain exactly eight characters, and each character is one of 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, 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 probability of each of the following: (a) A (b) \(B\) (c) A password contains at least 1 integer. (d) A password contains exactly 2 integers.

\(+\) If \(P(A)=0.3, P(B)=0.2,\) and \(P(A \cap B)=0.1,\) deter- mine the following probabilities: (a) \(P\left(A^{\prime}\right)\) (b) \(P(A \cup B)\) (c) \(P\left(A^{\prime} \cap B\right)\) (d) \(P\left(A \cap B^{\prime}\right)\) (e) \(P\left[(A \cup B)^{\prime}\right]\) (f) \(P\left(A^{\prime} \cup B\right)\)

\(+\) 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 follow. ing probabilities: (a) \(P(A \cup B \cup C)\) (b) \(P(A \cap B \cap C)\) (c) \(P(A \cap B)\) (d) \(P[(A \cup B) \cap C]\) (e) \(P\left(A^{\prime} \cap B^{\prime} \cap C^{\prime}\right)\)

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: (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?