Chapter 8

In a past presidential election, it was estimated that the probability that the Republican candidate would be elected was \(\frac{3}{5}\), and therefore the probability that the Democratic candidate would be elected was \(\frac{2}{5}\) (the two Independent candidates were given no chance of being elected). It was also estimated that if the Republican candidate were elected, the probability that a conservative, moderate, or liberal judge would be appointed to the Supreme Court (one retirement was expected during the presidential term) was \(\frac{1}{2}, \frac{1}{3}\), and \(\frac{1}{6}\), respectively. If the Democratic candidate were elected, the probabilities that a conservative, moderate, or liberal judge would be appointed to the Supreme Court would be \(\frac{1}{8}, \frac{3}{8}\), and \(\frac{1}{2}\), respectively. A conservative judge was appointed to the Supreme Court during the presidential term. What is the probability that the Democratic candidate was elected?

A box contains two defective Christmas tree lights that have been inadvertently mixed with eight nondefective lights. If the lights are selected one at a time without replacement and tested until both defective lights are found, what is the probability that both defective lights will be found after exactly three trials?

If a 5-card poker hand is dealt from a well-shuffled deck of 52 cards, what is the probability of being dealt the given hand? A straight (but not a straight flush)

The deluxe model hair dryer produced by Roland Electric has a mean expected lifetime of 24 mo with a standard deviation of 3 mo. Find a bound on the probability that one of these hair dryers will last between 20 and 28 mo.

Bob, the proprietor of Midland Lumber, feels that the odds in favor of a business deal going through are 9 to 5\. What is the (subjective) probability that this deal will not materialize?

Applicants for temporary office work at Carter Temporary Help Agency who have successfully completed a typing test are then placed in suitable positions by Nancy Dwyer and Darla Newberg. Employers who hire temporary help through the agency return a card indicating satisfaction or dissatisfaction with the work performance of those hired. From past experience it is known that \(80 \%\) of the employees placed by Nancy are rated as satisfactory, and \(70 \%\) of those placed by Darla are rated as satisfactory. Darla places \(55 \%\) of the temporary office help at the agency and Nancy the remaining \(45 \%\). If a Carter office worker is rated unsatisfactory, what is the probability that he or she was placed by Darla?

It is estimated that \(0.80 \%\) of a large consignment of eggs in a certain supermarket is broken. a. What is the probability that a customer who randomly selects a dozen of these eggs receives at least one broken egg? b. What is the probability that a customer who selects these eggs at random will have to check three cartons before finding a carton without any broken eggs? (Each carton contains a dozen eggs.)

If a 5-card poker hand is dealt from a well-shuffled deck of 52 cards, what is the probability of being dealt the given hand? A flush (but not a straight flush)

A Christmas tree light has an expected life of \(200 \mathrm{hr}\) and a standard deviation of \(2 \mathrm{hr}\). a. Find a bound on the probability that one of these Christmas tree lights will last between \(190 \mathrm{hr}\) and \(210 \mathrm{hr}\). b. Suppose 150,000 of these Christmas tree lights are used by a large city as part of its Christmas decorations. Estimate the number of lights that are likely to require replacement between \(180 \mathrm{hr}\) and \(220 \mathrm{hr}\) of use.

a. Show that, for any number \(c\), $$E(c X)=c E(X)$$ b. Use this result to find the expected loss if a gambler bets \(\$ 300\) on \(\mathrm{red}\) in a single play in American roulette.

Based on data obtained from the National Institute of Dental Research, it has been determined that \(42 \%\) of 12 -yr-olds have never had a cavity, \(34 \%\) of 13 -yr-olds have never had a cavity, and \(28 \%\) of 14 -yrolds have never had a cavity. Suppose a child is selected at random from a group of 24 junior high school students that includes six 12 -yr-olds, eight 13 -yr-olds, and ten 14 -yrolds. If this child does not have a cavity, what is the probability that this child is 14 yrs old?

The accompanying data were obtained from the financial aid office of a certain uni- versity: $$\begin{array}{lccc}\hline & & {\text { Not }} & \\ & \begin{array}{c}\text { Receiving } \\\\\text { Financial } \\\\\text { Aid }\end{array} & \begin{array}{c}\text { Receiving } \\\\\text { Financial } \\\\\text { Aid }\end{array} & \text { Total} \\\\\hline \text { Undergraduates } & 4,222 & 3,898 & 8,120 \\\\\hline \text { Graduates } & 1,879 & 731 & 2,610 \\ \hline \text { Total } & 6,101 & 4,629 & 10,730 \\ \hline\end{array}$$ Let \(A\) be the event that a student selected at random from this university is an undergraduate student, and let \(B\) be the event that a student selected at random is receiving financial aid. a. Find each of the following probabilities: \(P(A), P(B)\), \(P(A \cap B), P(B \mid A)\), and \(P\left(B \mid A^{c}\right)\) b. Are the events \(A\) and \(B\) independent events?

If a 5-card poker hand is dealt from a well-shuffled deck of 52 cards, what is the probability of being dealt the given hand? Four of a kind

The mean annual starting salary of a new graduate in a certain profession is $$\$ 52,000$$ with a standard deviation of $$\$ 500 .$$ Find a bound on the probability that the starting salary of a new graduate in this profession will be between $$\$ 50,000$$ and $$\$ 54,000 ?$$

In an examination given to a class of 20 students, the following test scores were obtained: $$\begin{array}{lllllllllr}40 & 45 & 50 & 50 & 55 & 60 & 60 & 75 & 75 & 80 \\ 80 & 85 & 85 & 85 & 85 & 90 & 90 & 95 & 95 & 100\end{array}$$ a. Find the mean (or average) score, the mode, and the median score. b. Which of these three measures of central tendency do you think is the least representative of the set of scores?

In a recent senatorial election, \(50 \%\) of the voters in a certain district were registered as Democrats, \(35 \%\) were registered as Republicans, and \(15 \%\) were registered as Independents. The incumbent Democratic senator was reelected over her Republican and Independent opponents. Exit polls indicated that she gained \(75 \%\) of the Democratic vote, \(25 \%\) of the Republican vote, and \(30 \%\) of the Independent vote. Assuming that the exit poll is accurate, what is the probability that a vote for the incumbent was cast by a registered Republican? -

The personnel department of Franklin National Life Insurance Company compiled the accompanying data regarding the income and education of its employees: $$\begin{array}{lcc}\hline & \text { Income } & \text { Income } \\ & \mathbf{\$ 5 0 , 0 0 0} \text { or Below } & \text { Above } \mathbf{\$ 5 0 , 0 0 0} \\ \hline \text { Noncollege Graduate } & 2040 & 840 \\ \hline \text { College Graduate } & 400 & 720 \\\\\hline\end{array}$$ Let \(A\) be the event that a randomly chosen employee has a college degree and \(B\) the event that the chosen employee's income is more than \(\$ 50,000\). a. Find each of the following probabilities: \(P(A), P(B)\), \(P(A \cap B), P(B \mid A)\), and \(P\left(B \mid A^{c}\right)\). b. Are the events \(A\) and \(B\) independent events?

If a 5-card poker hand is dealt from a well-shuffled deck of 52 cards, what is the probability of being dealt the given hand? A full house

Sugar packaged by a certain machine has a mean weight of \(5 \mathrm{lb}\) and a standard deviation of \(0.02 \mathrm{lb}\). For what values of \(c\) can the manufacturer of the machinery claim that the sugar packaged by this machine has a weight between \(5-c\) and \(5+c \mathrm{lb}\) with probability at least \(96 \%\) ?

The frequency distribution of the hourly wage rates (in dollars) among blue- collar workers in a certain factory is given in the following table. Find the mean (or average) wage rate, the mode, and the median wage rate of these workers. $$\begin{array}{lcccccc}\hline \text { Wage Rate } & 10.70 & 10.80 & 10.90 & 11.00 & 11.10 & 11.20 \\ \hline \text { Frequency } & 60 & 90 & 75 & 120 & 60 & 45 \\ \hline\end{array}$$

An insurance company has compiled the accompanying data relating the age of drivers and the accident rate (the probability of being involved in an accident during a 1 -yr period) for drivers within that group: $$\begin{array}{lcc}\hline & \begin{array}{c}\text { Percent of } \\\\\text { Insured Drivers }\end{array} & \begin{array}{c}\text { Accident } \\\\\text { Rate, \% }\end{array} \\\\\hline \text { Under 25 } & 16 & 5.5 \\\\\hline 25-44 & 40 & 2.5 \\ \hline 45-64 & 30 & 2 \\\\\hline 65 \text { and over } & 14 & 4 \\\\\hline\end{array}$$ What is the probability that an insured driver selected at random a. Will be involved in an accident during a particular 1 -yr period? b. Who is involved in an accident is under 25 ?

Two cards are drawn without replacement from a wellshuffled deck of 52 cards. Let \(A\) be the event that the first card drawn is a heart, and let \(B\) be the event that the second card drawn is a red card. Show that the events \(A\) and \(B\) are dependent events.

If a 5-card poker hand is dealt from a well-shuffled deck of 52 cards, what is the probability of being dealt the given hand? Two pairs

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. Both the variance and the standard deviation of a random variable measure the spread of a probability distribution.

There were 80 male guests at a party. The number of men in each of four age categories is given in the following table. The table also gives the probability that a man in the respective age category will keep his paper money in order of denomination. $$\begin{array}{lll}\hline \text { Age } & \text { Men } & \text { Keep Paper Money in Order, \% } \\\\\hline 21-34 & 25& 9 \\\\\hline 35-44 & 30 & 61 \\\\\hline 45-54 & 15 & 80 \\\\\hline 55 \text { and over } & 10 & 80 \\\\\hline\end{array}$$ A man's wallet was retrieved and the paper money in it was kept in order of denomination. What is the probability that the wallet belonged to a male guest between the ages of 35 and 44 ?

A nationwide survey conducted by the National Cancer Society revealed the following information. Of 10,000 people surveyed, 3200 were "heavy coffee drinkers" and 160 had cancer of the pancreas. Of those who had cancer of the pancreas, 132 were heavy coffee drinkers. Using the data in this survey, determine whether the events "being a heavy coffee drinker" and "having cancer of the pancreas" are independent events.

There are 12 signs of the Zodiac: Aries, Taurus, Gemini, Cancer, Leo, Virgo, Libra, Scorpio, Sagittarius, Capricorn, Aquarius, and Pisces. Each sign corresponds to a different calendar period of approximately 1 month. Assuming that a person is just as likely to be born under one sign as another, what is the probability that in a group of five people at least two of them a. Have the same sign? b. Were born under the sign of Aries?

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, explain why or give an example to show why it is false. Let \(\sum_{n=1}^{\infty}(-1)^{n+1} a_{n}\) be an alternating series, where \(a_{n}>0\). If \(\lim _{n \rightarrow \infty} a_{n}=0\), then \(\sum_{n=1}^{\infty}(-1)^{n+1} a_{n}\) converges.

The normal daily minimum temperature in degrees Fahrenheit for the months of January through December in San Francisco follows: $$\begin{array}{l}\begin{array}{l} 0\end{array}\\\\\begin{array}{llllll} 46.2 & 48.4 & 48.6 & 49.2 & 50.7 & 52.5 \\ 53.1 & 54.2 & 55.8 & 54.8 & 51.5 & 47.2 \end{array}\end{array}$$ Find the average and the median daily minimum temperature in San Francisco for these months.

According to a survey conducted in 2004 of 1000 American adults with Internet access, one in four households plans to switch ISPs in the next 6 months. Of those who plan to switch, \(1 \%\) of the households are likely to switch to a satellite connection, \(27 \%\) to digital subscriber line (DSL), \(28 \%\) to cable modem, \(35 \%\) to dial-up modem, and \(9 \%\) don't know what kind of service provider they will switch to. a. What is the probability that a randomly selected survey participant who was planning to switch ISPs will switch to a dial-up modem connection? b. What is the probability that a randomly selected survey participant will upgrade to high-speed service (satellite, DSL, or cable)?

What is the probability that at least two of the nine justices of the U.S. Supreme Court have the same birthday?

The weights, in ounces, of ten packages of potato chips are \(\begin{array}{llllllllll}16.1 & 16 & 15.8 & 16 & 15.9 & 16.1 & 15.9 & 16 & 16 & 16.2\end{array}\) Find the average and the median of these weights.

Before being allowed to enter a maximum-security area at a military installation, a person must pass three independent identification tests: a voice-pattern test, a fingerprint test, and a handwriting test. If the reliability of the first test is \(97 \%\), the reliability of the second test is \(98.5 \%\), and that of the third is \(98.5 \%\), what is the probability that this security system will allow an improperly identified person to enter the maximumsecurity area?

Fifty people are selected at random. What is the probability that none of the people in this group have the same birthday?

The relative humidity, in percent, in the morning for the months of January through December in Boston follows: $$\begin{array}{llllll}68 & 67 & 69 & 69 & 71 & 73 \\ 74 & 76 & 79 & 77 & 74 & 70\end{array}$$ Find the average and the median of these humidity readings.

Researchers weighed 1976 3-yr-olds from low-income families in 20 U.S. cities. Each child is classified by race (white, black, or Hispanic) and by weight (normal weight, overweight, or obese). The results are tabulated as follows: $$\begin{array}{lcccc}\hline & &{\text { Weight, \% }} & \\ \text { Race } & \text { Children } & \text { Normal Weight } & \text { Overweight } & \text { Obese } \\\\\hline \text { White } & 406 & 68 & 18 & 14 \\ hline \text { Black } & 1081 & 68 & 15 & 17 \\ \hline \text { Hispanic } & 489 & 56 & 20 & 24 \\ \hline\end{array}$$ If a participant in the research is selected at random and is found to be obese, what is the probability that the 3 -yr-old is white? Hispanic?

In a home theater system, the probability that the video components need repair within \(1 \mathrm{yr}\) is \(.01\), the probability that the electronic components need repair within 1 yr is \(.005\), and the probability that the audio components need repair within 1 yr is .001. Assuming that the events are independent, find the probability that a. At least one of these components will need repair within 1 yr. b. Exactly one of these components will need repair within 1 yr.

The probabilities that the three patients who are scheduled to receive kidney transplants at General Hospital will suffer rejection are \(\frac{1}{2}, \frac{1}{3}\), and \(\frac{1}{10}\). Assuming that the events (kidney rejection) are independent, find the probability that a. At least one patient will suffer rejection. b. Exactly two patients will suffer rejection.

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If the odds in favor of an event \(E\) occurring are \(a\) to \(b\), then the probability of \(E\) occurring is \(b /(a+b)\).

The following table gives the percent of eligible voters grouped according to profession who responded with "voted" in the 2000 presidential election. The table also gives the percent of people in a survey categorized by their profession. $$\begin{array}{lcc}\hline \text { Profession } & \begin{array}{c}\text { Percent } \\ \text { Who Voted } \end{array} & \begin{array}{c}\text { Percent in } \\\\\text { Each Profession }\end{array} \\\\\hline \text { Professionals } &84 & 12 \\\\\hline \text { White collar } & 73 & 24 \\\\\hline \text { Blue collar } & 66 & 32 \\\\\hline \text { Unskilled } & 57 & 10 \\\\\hline \text { Farmers } & 68 & 8 \\ \hline \text { Housewives } & 66 & 14 \\\\\hline\end{array}$$ If an eligible voter who participated in the survey and voted in the election is selected at random, what is the probability that this person is a housewife?

Copykwik has four photocopy machines: \(A, B, C\), and \(D\). The probability that a given machine will break down on a particular day is \(P(A)=\frac{1}{50} \quad P(B)=\frac{1}{60} \quad P(C)=\frac{1}{75} \quad P(D)=\frac{1}{40}\) Assuming independence, what is the probability on a particular day that a. All four machines will break down? b. None of the machines will break down?

A study was conducted among a certain group of union members whose health insurance policies required second opinions prior to surgery. Of those members whose doctors advised them to have surgery, \(20 \%\) were informed by a second doctor that no surgery was needed. Of these, \(70 \%\) took the second doctor's opinion and did not go through with the surgery. Of the members who were advised to have surgery by both doctors, \(95 \%\) went through with the surgery. What is the probability that a union member who had surgery was advised to do so by a second doctor?

The proprietor of Cunningham's Hardware Store has decided to install floodlights on the premises as a measure against vandalism and theft. If the probability is \(.01\) that a certain brand of floodlight will burn out within a year, find the minimum number of floodlights that must be installed to ensure that the probability that at least one of them will remain functional for the whole year is at least .99999. (Assume that the floodlights operate independently.)

According to the Centers for Disease Control and Prevention, the percentage of adults 25 yr and older who smoke, by educational level, is as follows: $$\begin{array}{lcccccc}\hline & & \text { High } &{\text { Under- }} \\\\\begin{array}{l} \text { Fducational } \\\\\text { Level } \end{array} & \begin{array}{c} \text { No } \\ \text { diploma } \end{array} & \begin{array}{c} \text { GED } \\ \text { diploma }\end{array} & \begin{array}{c} \text { school } \\\\\text { graduate } \end{array} & \begin{array}{c} \text { Some } \\\\\text { college } \end{array} & \begin{array}{c}\text { graduate } \\\\\text { level }\end{array} & \begin{array}{c}\text { Graduate } \\\\\text { degree }\end{array} \\ \hline \text { Respondents, } \% & 26 & 43 & 25 & 23 & 10.7 & 7 \\ \hline\end{array}$$ In a group of 140 people, there were 8 with no diploma, 14 with GED diplomas, 40 high school graduates, 24 with some college, 42 with an undergraduate degree, and 12 with a graduate degree. (Assume that these categories are mutually exclusive.) If a person selected at random from this group was a smoker, what is the probability that he or she is a person with a graduate degree?

Let \(E\) be any event in a sample space \(S .\) a. Are \(E\) and \(S\) independent? Explain your answer. b. Are \(E\) and \(\varnothing\) independent? Explain your answer.

Suppose the probability that an event will occur in one trial is \(p .\) Show that the probability that the event will occur at least once in \(n\) independent trials is \(1-(1-p)^{n} .\)

Let \(E\) and \(F\) be events such that \(F \subset E\). Find \(P(E \mid F)\) and interpret your result.

Suppose that \(A\) and \(B\) are mutually exclusive events and that \(P(A \cup B) \neq 0 .\) What is \(P(A \mid A \cup B)\) ?

Let \(E\) and \(F\) be independent events; show that \(E\) and \(F^{c}\) are independent.

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If \(A\) and \(B\) are mutually exclusive and \(P(B) \neq 0\), then \(P(A \mid B)=0\)