Statistics

James T. McClave, Terry T. Sincich

Chapter 3 Probability - all with Video Answers

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Chapter Questions

02:27

Problem 1

What is an experiment?

Anna Marie Morra

Numerade Educator

00:36

Problem 2

What are the most basic outcomes of an experiment called?

Lucas Finney

Numerade Educator

01:18

Problem 3

Define the sample space.

Lauren Shelton

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00:43

Problem 4

What is a Venn diagram?

Lucas Finney

Numerade Educator

00:51

Problem 5

Give two probability rules for sample points.

Lucas Finney

Numerade Educator

00:06

Problem 6

What is an event?

Amy Jiang

Numerade Educator

00:29

Problem 7

How do you find the probability of an event made up of several sample points?

Lucas Finney

Numerade Educator

00:44

Problem 8

Give a scenario where the combinations rule is appropriate for counting the number of sample points.

Lucas Finney

Numerade Educator

02:23

Lead bullets as forensic evidence. Chance (Summer 2004) published an article on the use of lead bullets as forensic evidence in a federal criminal case. Typically, the Federal Bureau of Investigation (FBI) will use a laboratory method to match the lead in a bullet found at a crime scene with unexpended lead cartridges found in the possession of a suspect. The value of this evidence depends on the chance of
a false positive $-$ that is, the probability that the FBI finds a match, given that the lead at the crime scene and the lead in the possession of the suspect are actually from two different "melts," or sources. To estimate the false positive rate, the FBI collected 1,837 bullets that the agency was confident all came from different melts. Then, using its established criteria, the FBI examined every possible pair of bullets and counted the number of matches. According to Chance, the FBI found 693 matches. Use this information to compute the chance of a false positive. Is this probability small enough for you to have confidence in the FBI's forensic evidence?

Lucas Finney

Numerade Educator

02:30

Problem 34

Matching socks. Consider the following question posed to Marilyn vos Savant in her weekly newspaper column, "Ask Marilyn":

I have two pairs of argyle socks, and they look nearly identical-one navy blue and the other black. $[$ When doing the laundry] my wife matches the socks incorrectly much more often than she does correctly.... If all four socks are in front of her, it seems to me that her chances are 50\% for a wrong match and $50 \%$ for a right match. What do you think?
Source: Parade Magazine, Feb. 27,1994.
Use your knowledge of probability to answer this question. [Hint: List the sample points in the experiment.]

Lucas Finney

Numerade Educator

02:52

Problem 35

Post-op nausea study. Nausea and vomiting after surgery are common side effects of anesthesia and painkillers. Six different drugs, varying in cost, were compared for their effectiveness in preventing nausea and vomiting (New England Journal of Medicine, June 10,2004 ). The medical researchers looked at all possible combinations of the drugs as treatments, including a single drug, as well as two-drug, three-drug, four-drug, five-drug, and six-drug combinations.
a. How many two-drug combinations of the six drugs are possible?
b. How many three-drug combinations of the six drugs are possible?
c. How many four-drug combinations of the six drugs are possible?
d. How many five-drug combinations of the six drugs are possible?
e. The researchers stated that a total of 64 drug combinations were tested as treatments for nausea. Verify that there are 64 ways that the six drugs can be combined. (Remember to include the one-drug and sixdrug combinations as well as the control treatment of no drugs.)

Lucas Finney

Numerade Educator

02:31

Problem 36

Dominant versus recessive traits. A gene is composed of two alleles. An allele can be either dominant or recessive. Suppose a husband and a wife, who are both carriers of the sickle cell anemia allele but do not have the disease, decide to have a child. Because both parents are carriers of the disease, each has one dominant normal cell allele
(S) and one recessive sickle cell allele $(s)$. Therefore, the genotype of each parent is $S s$. Each parent contributes one allele to his or her offspring with each allele being equally likely.
a. List the possible genotypes of their offspring.
b. What is the probability that the offspring will have sickle cell anemia? In other words, what is the probability that the offspring will have genotype $s s ?$ Interpret this probability.
c. What is the probability that the offspring will not have sickle cell anemia but will be a carrier (one normal cell allele and one sickle cell allele)? Interpret this probability.

Lucas Finney

Numerade Educator

01:47

Problem 37

Drug testing of firefighters. Hillsborough County, Florida, has 42 fire stations that employ 980 career firefighters and 160 volunteers. Both types of firefighters are drug tested every six months. However, the career firefighters and volunteers are treated as separate populations. For drug testing, $50 \%$ of the career firefighters are randomly selected and $50 \%$ of the volunteers are randomly selected. Some firefighters at a particular station argue that due to the smaller number of volunteers, an individual volunteer is more likely to be selected in the current sampling scheme than if all the career firefighters and volunteers were combined into a single population and $50 \%$ sampled. Do you agree? Explain your reasoning.
$$
\left[\operatorname{Hint}:\left(\begin{array}{c}
N-1 \\
n-1
\end{array}\right) /\left(\begin{array}{c}
N \\
n
\end{array}\right)=n / N\right]
$$

Kari Hasz

Numerade Educator

00:53

Problem 38

Define in words mutually exclusive events.

Study of analysts' forecasts. The Journal of Accounting Research (Mar. 2008) published a study on relationship incentives and degree of optimism among analysts' forecasts. Participants were analysts at either a large or small brokerage firm who made their forecasts either early or late in the quarter. Also, some analysts were only concerned with making an accurate forecast, while others were also interested in their relationship with management. Suppose one of these analysts is randomly selected. Consider the following events:
$A=\{$ The analyst is only concerned with making an accurate forecast.\}
$B=\{$ The analyst makes the forecast early in the quarter. $\}$ $C=\{$ The analyst is from a small brokerage firm. $\}$

Employee behavior problems. The Organizational Development Journal (Summer 2006) reported on the results of a survey of human resource officers (HROs) at major firms. The focus of the study was employee behavior, namely absenteeism, promptness to work, and turnover. The study found that $55 \%$ of the HROs had problems with absenteeism. Also, $41 \%$ of the HROs had problems with turnover. Suppose that $22 \%$ of the HROs had problems with both absenteeism and turnover.
a. Find the probability that a human resource officer selected from the group surveyed had problems with employee absenteeism or employee turnover.
b. Find the probability that a human resource officer selected from the group surveyed did not have problems with employee absenteeism.
c. Find the probability that a human resource officer selected from the group surveyed did not have problems with employee absenteeism nor with employee turnover.

Lucas Finney

Numerade Educator

05:17

Problem 65

Cloning credit or debit cards. Wireless identity theft is a technique of stealing an individual's personal information from radio-frequency-enabled cards (e.g., credit or debit cards). Upon capturing this data, thieves are able to program their own cards to respond in an identical fashion via cloning. A method for detecting cloning attacks in radio-frequency identification (RFID) applications was explored in IEEE Transactions on Information Forensics and Security (Mar. 2013). The method was illustrated using a simple ball drawing game. Consider a group of 10 balls, 5 representing genuine RFID cards and 5 representing clones of one or more of these cards. A coloring system was used to distinguish among the different genuine cards. Because there were 5 genuine cards, 5 colors - yellow, blue, red, purple, and orange-were used. Balls of the same color represent either the genuine card or a clone of the card. Suppose the 10 balls are colored as follows: 3 yellow, 2 blue, 1 red, 3 purple, 1 orange. (See figure on p. $172 .)$ Note that the singleton red and orange balls must represent the genuine cards (i.e., there are no clones of these cards). If two balls of the same color are drawn (without replacement) from the 10 balls, then a cloning attack is detected. For this example, find the probability of detecting a cloning attack.

Eric Tran

Numerade Educator

06:13

Problem 66

Galileo's passe-dix game. Passe-dix is a game of chance played with three fair dice. Players bet whether the sum of the faces shown on the dice will he above or helow 10 . During the late 16 th century, the astronomer and mathematician Galileo Galilei was asked by the Grand Duke of Tuscany to explain why "the chance of throwing a total of
9 with three fair dice was less than that of throwing a total of 10." (Interstat, Jan. 2004). The Grand Duke believed that the chance should be the same, since "there are an equal number of partitions of the numbers 9 and 10 ."
Find the flaw in the Grand Duke's reasoning and answer the question posed to Galileo.

Appraisals and negative emotions. According to psychological theory, each negative emotion one has is linked to some thought (appraisal) about the environment. This theory was tested in the journal Cognition and Emotion (Vol. $24,$
2010). Undergraduate students were asked to respond to events that occurred within the past 15 minutes. Each student rated the negative emotions he or she felt (e.g., anger, sadness, fear, or guilt) and gave an appraisal of the event (e.g., unpleasant or unfair). A total of 10,797 responses were evaluated in the study, of which 594 indicated a perceived unfairness appraisal. Of these unfairness appraisals, the students reported feeling angry in 127 of them. Consider a response randomly selected from the study.
a. What is the probability that the response indicates a perceived unfairness appraisal?
b. Given a perceived unfairness appraisal, what is the probability of an angry emotion?

Lucas Finney

Numerade Educator

01:38

Problem 89

Health risks to beachgoers. Refer to the University of Florida study of health risks to beachgoers, Exercise 3.24 (p. 158). According to the study, 6 out of 1,000 people exposed to wet sand for a 10-minute period will acquire gastroenteritis and 7 out of 1,000 people exposed to ocean water for a 10 -minute period will acquire gastroenteritis. Suppose you and a friend go to the beach. Over the next 10 minutes, you play in the wet sand while your friend swims in the ocean. What is the probability that at least one of you acquires gastroenteritis? (What assumption did you need to make in order to find the probability?)

Christopher Stanley

Numerade Educator

05:03

Problem 90

Sleep apnea and sleep stage transitioning. Refer to the sleep apnea study published in Chance (Winter 2009), Exercise 3.57 (p. 169). Recall that the various stages of sleep for a large group of sleep apnea patients were monitored in 30 -second "epochs," and the sleep stage for the previous and current epoch was determined. A summary table is reproduced in the next column. One of the epochs in the study is selected.
a. Given that the previous sleep stage for the epoch was the Wake state, what is the probability that the current sleep stage is REM?
b. Given that the current sleep stage for the epoch is REM, what is the probability that the previous sleep stage was not the Wake state?
c. Are the events \{previous stage is REM\} and \{current
stage is REM\} mutually exclusive?
d. Are the events \{previous stage is REM\} and \{current
stage is REM\} independent?
e. Are the events \{previous stage is Wake $\}$ and \{current stage is Wake\} independent?

Lucas Finney

Numerade Educator

05:22

Problem 91

Firefighters' use of gas detection devices. Two deadly gases that can be present in fire smoke are hydrogen cyanide and carbon monoxide. Fire Engineering (Mar.
2013) reported the results of a survey of 244 firefighters conducted by the Fire Smoke Coalition. The purpose of the survey was to assess the base level of knowledge of firefighters regarding the use of gas detection devices at the scene of a fire. The survey revealed the following:
Eighty percent of firefighters had no standard operating procedures (SOP) for detecting/monitoring hydrogen cyanide in fire smoke; $49 \%$ had no SOP for detecting/ monitoring carbon monoxide in fire smoke. Assume that $94 \%$ of firefighters had no SOP for detecting either hydrogen cyanide or carbon monoxide in fire smoke. What is the probability that a firefighter has no SOP for detecting hydrogen cyanide and no SOP for detecting carbon monoxide in fire smoke?

P Krishnamurthy

Numerade Educator

05:07

Problem 92

Compensatory advantage in education. According to Sociology of Education (Apr. 2014), compensatory advantage occurs when individuals from privileged backgrounds are less dependent on prior negative outcomes than individuals from disadvantaged families. The researcher applied this theory to educational success. Let $S_{t}$ represent the event that a student is successfully promoted to the next grade level in year $t,$ and let $F_{t}$ represent the event that a student fails to be promoted to the next grade level in year $t$. Now consider two probabilities: $P_{1}=$ the probability that a student is successfully promoted to the next grade level in year $t+1$ given that the student was successfully promoted the previous year (i.e., in year $t$ ) and $P_{2}=$ the probability that a student is successfully promoted to the next grade level in year $t+1$ given that the student failed in the previous year. Compensatory advantage theory states that the difference between the two probabilities, $P_{1}-P_{2},$ is greater for disadvantaged families (say, the lower-class $L$ ) than for privileged families (say, the upper-class $U$ ).
a. Write $P_{1}$ as a conditional probability using symbols.
b. Write $P_{2}$ as a conditional probability using symbols.
c. The tables on p. 185 give the grade promotion results (number of students) for both upper- and lower-class students at a particular school. Does the compensatory advantage theory hold at this school?

Sheryl Ezze

Numerade Educator

01:26

Problem 93

Lucas Finney

Numerade Educator

01:57

Problem 100

Random mutation of cells. Chance (Spring 2010) presented an article on the random mutation hypothesis developed by microbiologists. Under this hypothesis, when a wild-type organic cell (e.g., a bacteria cell) divides, there is a chance that at least one of the two "daughter" cells is a mutant. When a mutant cell divides, both offspring will be mutant. The schematic in the next column shows a possible pedigree from a single cell that has divided. Note that one "daughter" cell is mutant (o) and one is a normal cell (o).
a. Consider a single, normal cell that divides into two offspring. List the different possible pedigrees.
b. Assume that a "daughter" cell is equally likely to be mutant or normal. What is the probability that a single, normal cell that divides into two offspring will result in at least one mutant cell?
c. Now assume that the probability of a mutant "daughter" cell is .2. What is the probability that a single, normal cell that divides into two offspring will result in at least one mutant cell?
d. The schematic below shows a possible secondgeneration pedigree from a single cell that has divided. Note that the first-generation mutant cell automatically produces two mutant cells in the second generation. List the different possible second-generation pedigrees. (Hint: Use your answer to part a.)
e. Assume that a "daughter" cell is equally likely to be mutant or normal. What is the probability that a single, normal cell that divides into two offspring will result in at least one mutant cell after the second
generation?

James Kiss

Numerade Educator

03:18

Problem 101

Testing a psychic's ability. Consider an experiment in which 10 identical small boxes are placed side by side on a table. A crystal is placed at random inside one of the boxes. A self-professed "psychic" is asked to pick the box that contains the crystal.
a. If the "psychic" simply guesses, what is the probability that she picks the box with the crystal?
b. If the experiment is repeated seven times, what is the probability that the "psychic" guesses correctly at least once?
c. A group called the Tampa Bay Skeptics tested a selfproclaimed "psychic" by administering the preceding experiment seven times. The "psychic" failed to pick the correct box all seven times (Tampa Tribune, Sept. 20,1998 ). What would you infer about this person's psychic ability?

Sarah X

Numerade Educator

01:26

Problem 102

Risk of a natural-gas pipeline accident. Process Safety Progress (Dec. 2004) published a risk analysis for a natural-gas pipeline between Bolivia and Brazil. The most likely scenario for an accident would be natural-gas leakage from a hole in the pipeline. The probability that the leak ignites immediately (causing a jet fire) is .01. If the leak does not immediately ignite, it may result in the delayed ignition of a gas cloud. Given no immediate ignition, the probability of delayed ignition (causing a flash fire) is .01. If there is no delayed ignition, the gas cloud will disperse harmlessly. Suppose a leak occurs in the natural-gas pipeline. Find the probability that either a jet fire or a flash fire will occur. Illustrate with a tree diagram.

Manik Pulyani

Numerade Educator

06:05

Problem 103

Encryption systems with erroneous ciphertexts. In cryptography, ciphertext is encrypted or encoded text that is unreadable by a human or computer without the proper algorithm to decrypt it into plaintext. The impact of erroneous ciphertexts on the performance of an encryption system was investigated in IEEE Transactions on Information Forensics and Security (Apr. 2013). For one data encryption system, the probability of receiving an erroneous ciphertext is assumed to be $\beta,$ where $0<\beta<1$. The researchers showed that if an erroneous ciphertext occurs, the probability of an error in restoring plaintext using the decryption system is .5. When no error occurs in the received ciphertext, the probability of an error in restoring plaintext using the decryption system is $\alpha \beta,$ where $0<\alpha<1$. Use this information to give an expression for the probability of an error in restoring plaintext using the decryption system.

Choosing portable grill displays. Refer to the Journal of Consumer Research (Mar. 2003) study of how people attempt to influence the choices of others, Exercise 3.29 (p. 159). Recall that students selected three portable grill displays to be compared from an offering of five different grill displays.
a. Use a counting rule to count the number of ways the three displays can be selected from the five available displays to form a three-grill-display combination.
b. The researchers informed students to select the three displays in order to convince people to choose Grill #2. Consequently, Grill #2 was a required selection. Use a counting rule to count the number of different ways the three grill displays can be selected from the five displays if Grill #2 must be selected. (Your answer should agree with the answer in 3.29 a. )
c. Now suppose the three selected grills will be set up in a specific order for viewing by a customer. (The customer views one grill first, then the second, and finally the third grill.) Again, Grill #2 must be one of the three selected. How many different ways can the three grill displays be selected if customers view the grills in order?

Lucas Finney

Numerade Educator

05:02

Problem 114

Monitoring impedance to leg movements. In an experiment designed to monitor impedance to leg movement, Korean engineers attached electrodes to the ankles and knees of volunteers. Of interest were the voltage readings between pairs of electrodes (IEICE Transactions on Information \& Systems, Jan. 2005). These readings were used to determine the signal-to-noise ratio (SNR) of impedance changes such as knee flexes and hip extensions.
a. Six voltage electrodes were attached to key parts of the ankle. How many electrode pairs on the ankle are possible?
b. Ten voltage electrodes were attached to key parts of the knee. How many electrode pairs on the knee are possible?
c. Determine the number of possible electrode pairs such that one electrode is attached to the knee and one is attached to the ankle.

Randomization in a study of TV commercials. Gonzaga University professors conducted a study of over 1,500 television commercials and published their results in the Journal of Sociology, Social Work and Social Welfare (Vol. 2,2008 ). Commercials from eight networks $-\mathrm{ABC}$, FAM, $\mathrm{FOX}, \mathrm{MTV}, \mathrm{ESPN}, \mathrm{CBS}, \mathrm{CNN},$ and $\mathrm{NBC}-$ were sampled
during an eight-day period, with one network randomly selected each day. The table below shows the order determined by random draw:
$$
\begin{array}{l}
\text { ABC-July } 6 \text { (Wed) } \\
\text { FAM - July 7 (Thr) } \\
\text { FOX - July 9 (Sat) } \\
\text { MTV - July 10 (Sun) } \\
\text { ESPN - July 11 (Mon) } \\
\text { CBS-July 12 (Tue) } \\
\text { CNN-July 16 (Sat) } \\
\text { NBC-July 17 (Sun) }
\end{array}
$$
a. Determine the number of possible orderings of the networks over the eight days.
b. What is the probability that ESPN is selected on Monday, July 11 th?
c. What is the probability that MTV is selected on a Sunday?

Lucas Finney

Numerade Educator

01:20

Problem 123

Multilevel marketing schemes. Successful companies employ multilevel marketing (MLM) schemes to distribute their products. In one company, each distributor recruits several additional distributors to work at the first level. Each of these first-level distributors, in turn, recruits distributors to work at the second level. Distributors at any level continue to recruit distributors, forming additional levels. A distributor at any particular level makes a $5 \%$ commission on the sales of all distributors at lower levels in his or her "group."
a. Elaine is a distributor for an MLM company. Elaine recruits seven distributors to work for her at the first level. Each of these distributors recruits an additional six distributors to work at the second level. How many second-level distributors are under Elaine?
b. Refer to part a. Suppose each distributor at Elaine's second level recruits six third-level distributors, and each of these third-level distributors recruits four level-four distributors. How many fourth-level distributors are under Elaine?

Hossam Mohamed

Numerade Educator

05:07

Problem 124

Mathematical theory of partitions. Mathematicians at the University of Florida solved a 30-year-old mathematics problem with the use of the theory of partitions (Explore, Fall 2000). In mathematical terminology, a partition is a representation of an integer as a sum of positive integers. (For example, the number 3 has three possible partitions:
$3,2+1,$ and $1+1+1 .$ ) The researchers solved the problem by using "colored partitions" of a number, where the colors correspond to the four suits $-$ red hearts, red diamonds, black spades, and black clubs-in a standard 52 -card bridge deck. Consider forming colored partitions of an integer.
a. How many colored partitions of the number 3 are possible? [Hint: One partition is $3 \mathbf{\varphi}$; another is $2 \bullet+1$.. .
b. How many colored partitions of the number 5 are possible?

Victoria Dollar

Numerade Educator

03:39

Problem 125

The "marriage" problem. A mathematics assignment problem in which each element from one set is matched with one and only one element from another set is often referred to as a "marriage" problem. Mathematical Social Sciences (May
2014) considered the following marriage problem. After passing their preliminary exams, doctoral students in a Ph.D. program are each assigned to a professor for research assistance. Each professor can accept only one additional student per year. Suppose there are three students, $a, b,$ and $c,$ who have passed their preliminary exams and three professors, $A, B$, and $C$. Also suppose that professor $A$ prefers student $a$, professor $B$ prefers student $b$, and professor $C$ prefers student $c$.
a. If the assignment is made at random, what are the chances that the preferred matches occur?
b. If the assignment is made at random, what are the chances that professor $A$ is matched with student $a$ ?
c. If the assignment is made at random, what are the chances that none of the professors are matched with their preferred student?

Nick Johnson

Numerade Educator

01:36

Problem 126

Florida license plates. In the mid-1980s, the state of Florida ran out of combinations of letters and numbers for its license plates. Then, each license plate contained three letters of the alphabet, followed by three digits selected from the 10 digits $0,1,2, \ldots, 9$
a. How many different license plates did this system allow?
b. New Florida tags were obtained by reversing the procedure, starting with three digits followed by three letters. How many new tags did the new system provide?
c. Since the new tag numbers were added to the old numbers, what is the total number of licenses available for registration in Florida?

Christopher Stanley

Numerade Educator

01:39

Problem 127

Selecting a maintenance support system. In the Journal of Quality in Maintenance Engineering (Vol. 9,2003 ), researchers used an analytic hierarchy process to help build a preferred maintenance organization for ARTHUR, the Norwegian Army's high-tech radar system. The process requires the builder to select alternatives in three different stages (called echelons). In the first echelon, the builder must choose one of two mobile units (regular soldiers or soldiers with engineering training). In the second echelon, the builder chooses one of three heavy mobile units (units in the Norwegian Army, units from Supplier $2,$ or shared units). Finally, in the third echelon, the builder chooses one of three maintenance workshops (Norwegian Army, Supplier 1, or Supplier 2 ).
a. How many maintenance organization alternatives exist when choices are made in the three echelons?
b. The researchers determined that only four of the alternatives in part a are feasible alternatives for ARTHUR. If one of the alternatives is randomly selected, what is the probability that it is a feasible alternative?

Dominador Tan

Numerade Educator

01:33

Problem 128

Volleyball positions. Intercollegiate volleyball rules require that after the opposing team has lost its serve, each of the six members of the serving team must rotate into new positions on the court. Hence, each player must be able to play all six different positions. How many different team combinations, by player and position, are possible during a volleyball game? If players are initially assigned to the positions in a random manner, find the probability that the best server on the team is in the serving position.

Christopher Stanley

Numerade Educator

02:38

Problem 129

Studying exam questions. A college professor hands out a list of 10 questions, 5 of which will appear on the final examination for the course. One of the students taking the course is pressed for time and can prepare for only 6 of the 10 questions on the list. Suppose the professor chooses the 5 questions at random from the 10 .
a. What is the probability that the student will be prepared for all 5 questions that appear on the final examination?
b. What is the probability that the student will be prepared for fewer than 2 questions?
c. What is the probability that the student will be prepared for exactly 4 questions?

Christopher Stanley

Numerade Educator

03:17

Problem 130

Modeling the behavior of granular media. Granular media are substances made up of many distinct grains - including sand, rice, ball bearings, and flour. The properties of these materials were theoretically modeled in Engineering Computations: International Journal for Computer-Aided Engineering and Software (Vol. 30, No. 2, 2013). The model assumes there is a system of $N$ non-interacting granular particles. The particles are grouped according to energy level. Assume there are $r$ energy levels, with $N_{i}$ particles at energy level $i, i=1,2,3, \ldots, r .$ Consequently, $N=N_{1}+N_{2}+\cdots+N_{r} .$ A microset is defined as a possible grouping of the particles among the energy levels. For example, suppose $N=7$ and $r=3$. Then one possible microset is $N_{1}=1, N_{2}=2,$ and $N_{3}=4 .$ That is, there is one particle at energy level $1,$ two particles at energy level $2,$ and four particles at energy level 3. Determine the number of different microsets possible when $N=7$ and $r=3$.

Salamat Ali

Numerade Educator

02:37

Problem 131

A straight flush in poker. Consider 5 -card poker hands dealt from a standard 52-card bridge deck. Two important events are $A:\{$ You draw a flush. $\}$
$B:\{$ You draw a straight. $\}$
[Note: A flush consists of any 5 cards of the same suit. A straight consists of any 5 cards with values in sequence. In a straight, the cards may be of any suit, and an ace may be considered as having a value of 1 or, alternatively, a value higher than a king.]
a. How many different 5 -card hands can be dealt from a 52-card bridge deck?
b. Find $P(A)$.
c. Find $P(B)$.
d. The event that both $A$ and $B$ occur-that is, $A \cap B-$ is called a straight flush. Find $P(A \cap B)$.

Bob Bob

Numerade Educator

01:06

Problem 132

Explain the difference between the two probabilities $P(A \mid B)$ and $P(B \mid A)$

Reverse-engineering gene identification. In Molecular Systems Biology (Vol. 3, 2007), geneticists at the University of Naples (Italy) used reverse engineering to identify genes. They calculated $P(G \mid D),$ where $D$ is a gene expression data set of interest and $G$ is a graphical identifier. Several graphical identifiers were investigated. Suppose that, for two different graphical identifiers $G_{1}$ and $G_{2}, P\left(D \mid G_{1}\right)=.5$ and $P\left(D \mid G_{2}\right)=.3$. Also, $P(D)=.34, P\left(G_{1}\right)=.2,$ and $P\left(G_{2}\right)=.8$
a. Use Bayes's rule to find $P\left(G_{1} \mid D\right)$.
b. Use Bayes's rule to find $P\left(G_{2} \mid D\right)$.

Brian Ketelobeter

Numerade Educator

03:48

Problem 139

Drug testing in athletes. When Olympic athletes are tested for illegal drug use (i.e., doping), the results of a single test are used to ban the athlete from competition. In Chance (Spring 2004), University of Texas biostatisticians demonstrated the application of Bayes's rule to making inferences about testosterone abuse among Olympic athletes. They used the following example: In a population of 1,000 athletes, suppose 100 are illegally using testosterone. Of the users, suppose 50 would test positive for testosterone. Of the nonusers, suppose 9 would test positive.
a. Given that the athlete is a user, find the probability that a drug test for testosterone will yield a positive result. (This probability represents the sensitivity of the drug test.)
b. Given that the athlete is a nonuser, find the probability that a drug test for testosterone will yield a negative result. (This probability represents the specificity of the drug test.)
c. If an athlete tests positive for testosterone, use Bayes's rule to find the probability that the athlete is really doping. (This probability represents the positive predictive value of the drug test.)

Patricia Berchiolli

Numerade Educator

01:53

Problem 140

Fingerprint expertise. A study published in $P$ sychological Science (Aug. 2011) tested the accuracy of experts and novices in identifying fingerprints. Participants were presented pairs of fingerprints and asked to judge whether the prints in each pair matched. The pairs were presented under three different conditions: prints from the same individual (match condition), non-matching but similar prints (similar distracter condition), and non-matching and very dissimilar prints (non-similar distracter condition). The percentages of correct decisions made by the two groups under each of the three conditions are listed in the table.
a. Given a pair of matched prints, what is the probability that an expert will fail to identify the match?
b. Given a pair of matched prints, what is the probability that a novice will fail to identify the match?
c. Assume the study included 10 participants -5 experts, and 5 novices. Suppose that a pair of matched prints are presented to a randomly selected study participant and the participant fails to identify the match. Is the participant more likely to be an expert or a novice?

Lucas Finney

Numerade Educator

02:54

Problem 141

Tests for Down syndrome. Currently, there are three diagnostic tests available for chromosome abnormalities in a developing fetus: triple serum marker screening, ultrasound, and amniocentesis. The safest (to both the mother and fetus) and least expensive of the three is the ultrasound test. Two San Diego State University statisticians investigated the accuracy of using ultrasound to test for Down syndrome (Chance, Summer 2007). Let $D$ denote that the fetus has a genetic marker for Down syndrome and $N$ denote that the ultrasound test is normal (i.e., no indication of chromosome abnormalities). Then, the statisticians desire the probability $P(D \mid N)$. Use Bayes's rule and the following probabilities (provided in the article) to find the desired probability: $P(D)=1 / 80, P\left(D^{C}\right)=$ $79 / 80, P(N \mid D)=1 / 2, P\left(N^{C} \mid D\right)=1 / 2, \quad P\left(N \mid D^{C}\right)=1$
and $P\left(N^{C} \mid D^{C}\right)=0$

Sheryl Ezze

Numerade Educator

04:15

Problem 142

HIV testing and false positives. Bayes's rule was applied to the problem of HIV testing in The American Statistician (Aug. 2008). In North America, the probability of a person having HIV is .008. A test for HIV yields either a positive or negative result. Given that a person has HIV, the probability of a positive test result is .99 . (This probability is called the sensitivity of the test.) Given that a person does not have HIV, the probability of a negative test result is also .99 . (This probability is called the specificity of the test.) The authors of the article are interested in the probability that a person actually has HIV given that the test is positive.
a. Find the probability of interest for a North American by using Bayes's rule.
b. In East Asia, the probability of a person having HIV is only .001. Find the probability of interest for an East Asian by using Bayes's rule. (Assume that both the sensitivity and specificity of the test are .99.)
c. Typically, if one tests positive for HIV, a follow-up test is administered. What is the probability that a North American has HIV given that both tests are positive? (Assume that the tests are independent.)
d. Repeat part $\mathbf{c}$ for an East Asian.

Sheryl Ezze

Numerade Educator

05:09

Problem 143

Mining for dolomite. Dolomite is a valuable mineral that is found in sedimentary rock. During mining operations, dolomite is often confused with shale. The radioactivity features of rock can aid miners in distinguishing between dolomite and shale rock zones. For example, if the gamma ray reading of a rock zone exceeds 60 API units, the area is considered to be mostly shale (and is not mined); if the gamma ray reading of a rock zone is less than 60 API units, the area is considered to be abundant in dolomite (and is mined). Data on 771 core samples in a rock quarry collected by the Kansas Geological Survey revealed that 476 of the samples are dolomite and 295 of the samples are shale. Of the 476 dolomite core samples, 34 had a gamma ray reading greater than $60 .$ Of the 295 shale core samples, 280 had a gamma ray reading greater than 60 . Suppose you obtain a gamma ray reading greater than 60 at a certain depth of the rock quarry. Should this area be mined?

Ren Jie Tuieng

Numerade Educator

01:29

Problem 144

Nondestructive evaluation. Nondestructive evaluation (NDE) describes methods that quantitatively characterize materials, tissues, and structures by noninvasive means, such as X-ray computed tomography, ultrasonics, and acoustic emission. Recently, NDE was used to detect defects in steel castings (JOM, May 2005). Assume that the probability that NDE detects a "hit" (i.e., predicts a defect in a steel casting) when, in fact, a defect exists is .97 . (This is often called the probability of detection.) Assume also that the probability that NDE detects a "hit" when, in fact, no defect exists is $.005 .$ (This is called the probability of a false call.) Past experience has shown that a defect occurs once in every 100 steel castings. If NDE detects a "hit" for a particular steel casting, what is the probability that an actual defect exists?

Hast Aggarwal

Numerade Educator

04:14

Problem 145

Intrusion detection systems. Refer to the Journal of Research of the National Institute of Standards and Technology (Nov.-Dec. 2003) study of a double intrusion detection system with independent systems, presented in Exercise 3.98 (p. 186). Recall that if there is an intruder, system A sounds an alarm with probability .9 and system B sounds an alarm with probability .95. If there is no intruder, system A sounds an alarm with probability .2 and system $\mathrm{B}$ sounds an alarm with probability $.1 .$ Now, assume that the probability of an intruder is $.4 .$ If both systems sound an alarm, what is the probability that there is an intruder?

Lucas Finney

Numerade Educator

01:31

Problem 146

Confidence of feedback information for improving quality. In the semiconductor manufacturing industry, companies strive to improve product quality. One key to improved quality is having confidence in the feedback generated by production equipment. A study of the confidence level of feedback information was published in Engineering Applications of Artificial Intelligence (Vol. 26, 2013). At any point in time during the production process, a report can be generated. The report is classified as either "OK" or "not OK." As an example, consider the following probabilities: The probability an "OK" report is generated in any time period $(t)$ is .8. The probability of an "OK" report at one time period $(t+1)$, given an "OK" report in the previous time period $(t),$ is $.9 .$ Also, the probability of an "OK " report at one time period $(t+1)$, given a "not $\mathrm{OK}$ " report in the previous time period $(t),$ is $.5 .$ If an "OK " report is generated in one time period, what is the probability that an "OK" report was generated in the previous time period?

Sheryl Ezze

Numerade Educator

10:57

Problem 147

Forensic analysis of JFK assassination bullets. Following the assassination of President John F. Kennedy (JFK) in $1963,$ the House Select Committee on Assassinations (HSCA) conducted an official government investigation. The HSCA concluded that although there was a probable conspiracy, involving at least one additional shooter other than Lee Harvey Oswald, the additional shooter missed all limousine occupants. A recent analysis of assassination bullet fragments, reported in the Annals of Applied Statistics (Vol. 1,2007 ), contradicted these findings, concluding that evidence used to rule out a second assassin by the HSCA is fundamentally flawed. It is well documented that at least two different bullets were the source of bullet fragments used in the assassination. Let $E=\{$ bullet evidence used by the HSCA\},
$T=\{$ two bullets used in the assassination $\},$ and $T^{C}=$ $\{$ more than two bullets used in the assassination $\}$. Given the evidence $(E),$ which is more likely to have occurred $-$ two bullets used $(T)$ or more than two bullets used $\left(T^{C}\right) ?$
a. The researchers demonstrated that the ratio, $P(T \mid E) / P\left(T^{C} \mid E\right),$ is less than $1 .$ Explain why this result supports the theory of more than two bullets used in the assassination of JFK.
b. To obtain the result in part a, the researchers first showed that $P(T \mid E) / P\left(T^{C} \mid E\right)=[P(E \mid T) \cdot P(T)] /\left[P\left(E \mid T^{C}\right) \cdot P\left(T^{C}\right)\right]$
Demonstrate this equality using Bayes's theorem.

Given that $P(A \cap B)=.3$ and $P(B \mid A)=.9,$ find $P(A)$

Lucas Finney

Numerade Educator

04:36

Problem 154

Do you have a library card? According to a Harris poll, $68 \%$ of all American adults have a public library card. The percentages do differ by gender $-62 \%$ of males have library cards compared to $73 \%$ of females.
a. Consider the three probabilities: $P(A)=.68$, $P(A \mid B)=.62,$ and $P(A \mid C)=.73 .$ Define events $A, B$ and $C$.
b. Assuming that two-thirds of all American adults are males and one-thirds are females, what is the probability that an American adult is a male who owns a library card?

Gus Steppen

Numerade Educator

04:50

Problem 167

Beach erosional hot spots. Beaches that exhibit high erosion rates relative to the surrounding beach are defined as erosional hot spots. The U.S. Army Corps of Engineers is conducting a study of beach hot spots. Through an online questionnaire, data are collected on six beach hot spots. The data are listed in the next table.
$$
\begin{array}{llll}
\begin{array}{l}
\text { Beach Hot } \\
\text { Spot }
\end{array} & \begin{array}{c}
\text { Beach } \\
\text { Condition }
\end{array} & \begin{array}{c}
\text { Nearshore } \\
\text { Bar Condition }
\end{array} & \begin{array}{c}
\text { Long-Term } \\
\text { Erosion Rate } \\
\text { (miles/year) }
\end{array} \\
\begin{array}{l}
\text { Miami } \\
\text { Beach, FL }
\end{array} & \text { No dunes/flat } & \begin{array}{l}
\text { Single,shore } \\
\text { coney } \\
\text { Island, NY }
\end{array} & \text { No dunes/flat } & \text { Other } & 13 \\
\text { Surfside, CA } & \text { Bluff/scarp } & \begin{array}{l}
\text { Single, shore } \\
\text { parallel }
\end{array} & 35 \\
\begin{array}{c}
\text { Monmouth } \\
\text { Beach, NJ }
\end{array} & \text { Single dune } & \text { Planar } & \text { Not estimated } \\
\text { Ocean City, NJ } & \text { Single dune } & \text { Other } & \text { Not estimated } \\
\text { Spring Lake, NJ } & \text { Not observed } & \text { Planar } & 14
\end{array}
$$
a. Suppose you record the nearshore bar condition of each beach hot spot. Give the sample space for this experiment.
b. Find the probabilities of the sample points in the sample space you defined in part a.
c. What is the probability that a beach hot spot has either a planar or single, shore-parallel nearshore bar condition?
d. Now suppose you record the beach condition of each beach hot spot. Give the sample space for this experiment.
e. Find the probabilities of the sample points in the sample space you defined in part $\mathbf{d}$.
fo What is the probability that the condition of the beach at a particular beach hot spot is not flat?

Christopher Stanley

Numerade Educator

01:19

Problem 168

Chemical insect attractant. An entomologist is studying the effect of a chemical sex attractant (pheromone) on insects. Several insects are released at a site equidistant from the pheromone under study and a control substance. If the pheromone has an effect, more insects will travel toward it rather than toward the control. Otherwise, the insects are equally likely to travel in either direction. Suppose the pheromone under study has no effect, so that it is equally likely that an insect will move toward the pheromone or toward the control. Suppose four insects are released.
a. List or count the number of different ways the insects can travel.
b. What is the chance that all four travel toward the pheromone?
c. What is the chance that exactly three travel toward the pheromone?
d. What inference would you make if the event in part $\mathbf{c}$ actually occurs? Explain.

Lottie Adams

Numerade Educator

04:00

Problem 169

Process Safety Progress (Sept. 2004) reported on an emergency response system for incidents involving toxic chemicals in Taiwan. The system has logged over 250 incidents since being implemented. The next table gives a breakdown of the locations where these incidents occurred. Consider the location of a toxic chemical incident in Taiwan.

Nicholas Majtenyi

Numerade Educator

13:23

Problem 170

Federal civil trial appeals. The Journal of the American Law and Economics Association (Vol. 3, 2001) published the results of a study of appeals of federal civil trials. The accompanying table, extracted from the article, gives a breakdown of 2,143 civil cases that were appealed by either the plaintiff or the defendant. The outcome of the appeal, as well as the type of trial (judge or jury), was determined for each civil case. Suppose one of the 2,143 cases is selected at random and both the outcome of the appeal and the type of trial are observed.
$$
\begin{array}{lccc}
\hline & \text { Jury } & \text { Judge } & \text { Totals } \\
\hline \begin{array}{l}
\text { Plaintiff trial win }- \\
\text { reversed }
\end{array} & 194 & 71 & 265 \\
\text { Plaintiff trial win }- & & & \\
\quad \text { affirmed/dismissed } & 429 & 240 & 669 \\
\begin{array}{c}
\text { Defendant trial win }- \\
\text { reversed }
\end{array} & 111 & 68 & 179 \\
\begin{array}{c}
\text { Defendant trial win }- \\
\text { affirmed/dismissed }
\end{array} & 731 & 299 & 1,030 \\
\hline \text { Totals } & 1,465 & 678 & 2,143
\end{array}
$$
a. Find $P(A),$ where $A=\{$ jury trial $\}$.
b. Find $P(B)$, where $B=\{$ plaintiff trial win is reversed $\}$.
c. Are $A$ and $B$ mutually exclusive events?
d. Find $P\left(A^{c}\right)$.
e. Find $P(A \cup B)$.
fo Find $P(A \cap B)$.

Robin Corrigan

Numerade Educator

05:15

Problem 171

Winning at roulette. Roulette is a very popular game in many American casinos. In Roulette, a ball spins on a circular wheel that is divided into 38 arcs of equal length, bearing the numbers $00,0,1,2, \ldots, 35,36 .$ The number of the arc on which the ball stops is the outcome of one play of the game. The numbers are also colored in the manner shown in the following table.
$$
\begin{aligned}
&\text { Red: } 1,3,5,7,9,12,14,16,18,19,21,23,25,27,30,32,34,36\\
&\text { Black: } 2,4,6,8,10,11,13,15,17,20,22,24,26,28,29,31,33,35\\
&\text { Green: } 000
\end{aligned}
$$
Players may place bets on the table in a variety of ways, including bets on odd, even, red, black, high, low, etc. Consider the following events:
$A:\{$ The outcome is an odd number $(00$ and 0 are considered neither odd nor even.) $\}$
$B:\{$ The outcome is a black number. $\}$
$C:\{$ The outcome is a low number $(1-18) .\}$
a. Define the event $A \cap B$ as a specific set of sample points.
b. Define the event $A \cup B$ as a specific set of sample points.
c. Find $P(A), P(B), P(A \cap B), P(A \cup B),$ and $P(C)$ by summing the probabilities of the appropriate sample points.
d. Define the event $A \cap B \cap C$ as a specific set of sample points.
e. Use the additive rule to find $P(A \cup B)$. Are events $A$ and $B$ mutually exclusive? Why?
f. Find $P(A \cap B \cap C)$ by summing the probabilities of the sample points given in part $\mathbf{d}$.
g. Define the event $(A \cup B \cup C)$ as a specific set of sample points.
h. Find $P(A \cup B \cup C)$ by summing the probabilities of the sample points given in part $\mathrm{g}$.

Sheryl Ezze

Numerade Educator

03:09

Problem 172

Cigar smoking and cancer. The Journal of the National Cancer Institute (Feb. 16,2000 ) published the results of a study that investigated the association between cigar smoking and death from tobacco-related cancers. Data were obtained for a national sample of 137,243 American men. The results are summarized in the table below. Each male in the study was classified according to his cigar-smoking status and whether or not he died from a tobacco-related
cancer.
a. Find the probability that a randomly selected man never smoked cigars and died from cancer.
b. Find the probability that a randomly selected man was a former cigar smoker and died from cancer.
c. Find the probability that a randomly selected man was a current cigar smoker and died from cancer.
d. Given that a male was a current cigar smoker, find the probability that he died from cancer.
e. Given that a male never smoked cigars, find the probability that he died from cancer.

Trinity Steen

Numerade Educator

05:14

Problem 173

Errors in estimating job costs. A construction company employs three sales engineers. Engineers $1,2,$ and 3 estimate the costs of $30 \%, 20 \%,$ and $50 \%,$ respectively, of all jobs bid on by the company. For $i=1,2,3,$ define $E_{i}$ to be the event that a job is estimated by engineer $i$. The following probabilities describe the rates at which the engineers make serious errors in estimating costs:
$P\left(\right.$ error $\left.\mid E_{1}\right)=.01, P\left(\right.$ error $\left.\mid E_{2}\right)=.03,$ and
$P\left(\right.$ error $\left.\mid E_{3}\right)=.02$
a. If a particular bid results in a serious error in estimating job cost, what is the probability that the error was made by engineer $1 ?$
b. If a particular bid results in a serious error in estimating job cost, what is the probability that the error was made by engineer $2 ?$
c. If a particular bid results in a serious error in estimating job cost, what is the probability that the error was made by engineer $3 ?$
d. Based on the probabilities given in parts a-c, which engineer is most likely responsible for making the serious error?

Spencer Bahr

Numerade Educator

03:38

Problem 174

Elderly wheelchair user study. The American Journal of Public Health (Jan. 2002) reported on a study of elderly wheelchair users who live at home. A sample of 306 wheelchair users, age 65 or older, were surveyed about whether they had an injurious fall during the year and whether their home featured any one of five structural modifications:
bathroom modifications, widened doorways/hallways, kitchen modifications, installed railings, and easy-open doors. The responses are summarized in the accompanying table. Suppose we select, at random, one of the 306 wheelchair users surveyed.
a. Find the probability that the wheelchair user had an injurious fall.
b. Find the probability that the wheelchair user had all five features installed in the home.
c. Find the probability that the wheelchair user had no falls and none of the features installed in the home.
d. Given no features installed in the home, find the probability of an injurious fall.

Hossam Mohamed

Numerade Educator

View

Problem 175

Matching medical students with residencies. The National Resident Matching Program (NRMP) is a service provided to match graduating medical students with residency appointments at hospitals. After students and hospital officials have evaluated each other, they submit rank-order lists of their preferences to the NRMP. Using a matching algorithm, the NRMP then generates final, nonnegotiable assignments of students to the residency programs of hospitals (www.nrmp.org). Assume that three graduating medical students $(\# 1, \# 2,$ and $\# 3)$ have applied for positions at three different hospitals $(A, B,$ and $C),$ each of which has one and only one resident opening.
a. How many different assignments of medical students to hospitals are possible? List them.
b. Suppose student $\# 1$ prefers hospital $\mathrm{B}$. If the NRMP algorithm is entirely random, what is the probability that the student is assigned to hospital $\mathrm{B}$ ?

Rashmi Sinha

Numerade Educator

01:05

Problem 176

Rashmi Sinha

Numerade Educator

01:12

Problem 183

Sheryl Ezze

Numerade Educator

01:37

Problem 190

Chance of winning at blackjack. Blackjack, a favorite game of gamblers, is played by a dealer and at least one opponent. At the outset of the game, 2 cards of a 52-card bridge deck are dealt to the player and 2 cards to the dealer. Drawing an ace and a face card is called blackjack. If the dealer does not draw a blackjack and the player does, the player wins. If both the dealer and player draw blackjack, a "push" (i.e., a tie) occurs.
a. What is the probability that the dealer will draw a blackjack?
b. What is the probability that the player wins with a blackjack?

Hunza Gilgit

Numerade Educator

01:29

Problem 191

Accuracy of pregnancy tests. Sixty percent of all women who submit to pregnancy tests are really pregnant. A certain pregnancy test gives a false positive result with probability .03 and a valid positive result with probability
98. If a particular woman's test is positive, what is the probability that she really is pregnant? [Hint: If $A$ is the event that a woman is pregnant and $B$ is the event that the pregnancy test is positive, then $B$ is the union of the two mutually exclusive events $A \cap B$ and $A^{c} \cap B .$ Also, the probability of a false positive result may be written as $\left.P\left(B \mid A^{c}\right)=.03 .\right]$

Maxime Rossetti

Numerade Educator

09:44

Problem 192

Chance of winning at "craps." A version of the dice game "craps" is played in the following manner. A player starts by rolling two balanced dice. If the roll (the sum of the two numbers showing on the dice) results in a 7 or 11 , the player wins. If the roll results in a 2 or a 3 (called craps), the player loses. For any other roll outcome, the player continues to throw the dice until the original roll outcome recurs (in which case the player wins) or until a 7 occurs (in which case the player loses).
a. What is the probability that a player wins the game on the first roll of the dice?
b. What is the probability that a player loses the game on the first roll of the dice?
c. If the player throws a total of 4 on the first roll, what is the probability that the game ends (win or lose) on the next roll?

Reagan Chuang

Numerade Educator

03:43

Problem 193

The perfect bridge hand. According to a morning news program, a very rare event recently occurred in Dubuque, Iowa. Each of four women playing bridge was astounded to note that she had been dealt a perfect bridge hand. That is, one woman was dealt all 13 spades, another all 13 hearts, another all the diamonds, and another all the clubs. What is the probability of this rare event?

Pratyush Raitan

Numerade Educator

01:17

Problem 194

Odd Man Out. Three people play a game called "Odd Man Out." In this game, each player flips a fair coin until the outcome (heads or tails) for one of the players is not the same as that for the other two players. This player is then "the odd man out" and loses the game. Find the probability that the game ends (i.e., either exactly one of the coins will fall heads or exactly one of the coins will fall tails) after only one toss by each player. Suppose one of the players, hoping to reduce his chances of being the odd man out, uses a two-headed coin. Will this ploy be successful? Solve by listing the sample points in the sample space.

Hunza Gilgit

Numerade Educator

03:01

Problem 195

"Let's Make a Deal." Marilyn vos Savant, who is listed in the Guinness Book of World Records Hall of Fame as having the "Highest IQ," writes a weekly column in the Sunday newspaper supplement Parade Magazine. Her column, "Ask Marilyn," is devoted to games of skill, puzzles, and mind-bending riddles. In one issue ( Parade Magazine, Feb. 24,1991 ), vos Savant posed the following question:

Suppose you're on a game show and you're given a choice of three doors. Behind one door is a car; behind the others, goats. You pick a door -say, #1-and the host, who knows what's behind the doors, opens another door-say $\# 3-$ which has a goat. He then says to you, "Do you want to pick door $\# 2 ?$ " Is it to your advantage to switch your choice?

Marilyn's answer: "Yes, you should switch. The first door has a $1 / 3$ chance of winning [the car], but the second has a $^{2} / 3$ chance [of winning the car]." Predictably, vos Savant's surprising answer elicited thousands of critical letters, many of them from Ph.D. mathematicians, that disagreed with her. Who is correct, the Ph.D.s or Marilyn?

Prabhakar Kumar

Numerade Educator

01:22

Problem 196

Most likely coin-toss sequence. In Parade Magazine's (Nov. 26, 2000) column "Ask Marilyn," the following question was posed: "I have just tossed a [balanced] coin 10 times, and I ask you to guess which of the following three sequences was the result. One (and only one) of the sequences is genuine."
1. $\mathrm{H} \mathrm{H} \mathrm{H} \mathrm{H} \mathrm{H} \mathrm{H} \mathrm{H} \mathrm{H} \mathrm{H} \mathrm{H}$
2. $\mathrm{H} \mathrm{H} \mathrm{T} \mathrm{T} \mathrm{H} \mathrm{T} \mathrm{T} \mathrm{H} \mathrm{H} \mathrm{H}$
3. $\mathrm{T} \mathrm{T} \mathrm{T} \mathrm{T} \mathrm{T} \mathrm{T} \mathrm{T} \mathrm{T} \mathrm{T} \mathrm{T}$
Marilyn's answer to the question posed was "Though the chances of the three specific sequences occurring randomly are equal ... it's reasonable for us to choose sequence
(2) as the most likely genuine result." Do you agree?

Christopher Stanley

Numerade Educator