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Chapter 2: Q27E (page 66)

An academic department with five faculty members鈥� Anderson, Box, Cox, Cramer, and Fisher鈥攎ust select two of its members to serve on a personnel review committee. Because the work will be time-consuming, no one is anxious to serve, so it is decided that the representatives will be selected by putting the names on identical pieces of paper and then randomly selecting two.

a. What is the probability that both Anderson and Box will be selected? (Hint: List the equally likely outcomes.)

b. What is the probability that at least one of the two members whose name begins with C is selected?

c. If the five faculty members have taught for \({\rm{3, 6, 7, 10,}}\)and \({\rm{14}}\) years, respectively, at the university, what is the probability that the two chosen representatives have a total of at least \({\rm{15}}\) years鈥� teaching experience there?

Short Answer

Expert verified
  1. \({\rm{P(\{ A,B\} ) = }}\frac{{\rm{1}}}{{{\rm{10}}}}\)
  2. \({\rm{P(\{ }}\)starts with at least \({\rm{C\} ) = }}\frac{{\rm{7}}}{{{\rm{10}}}}{\rm{ }}\)
  3. \({\rm{P(}}\) has at least 15 years of experience \(\left. {\rm{\} }} \right){\rm{ = }}\frac{{\rm{6}}}{{{\rm{10}}}}{\rm{C\} )}}\)

Step by step solution

01

Because we are only choosing two papers, the results are limited.

We're going to have ten equally likely possibilities because we're picking at random. Before we go any further, suppose that instead of names, the initial letter of each name is written on a sheet of paper, with Cox having Cox and Cramer having Cr.

\(\begin{aligned}{{\rm{\{ A,B\} ,\{ A,Cox\} ,\{ A,Cr\} ,\{ A,F\} ,\{ B,Cox\} }}}\\{{\rm{\{ B,Cr\} ,\{ B,F\} ,\{ Cox,Cr\} ,\{ Cox,F\} ,\{ Cr,F\} }}}\end{aligned}\).

02

Determining is the probability that both Anderson and Box will be selected (Hint: List the equally likely outcomes 

The probability of both Anderson and Box being chosen is represented by letters \({\rm{A}}\)and \({\rm{B}}\)

\({\rm{P(\{ A,B\} ) = }}\frac{{\rm{1}}}{{{\rm{10}}}}{\rm{,}}\)

because the outcomes are equally likely to occur (probability is \({\rm{1/10}}\) for each simple occurrence out of ten).

03

Determining the probability that at least one of the two members whose name begins with C is selected 

The likelihood of at least one of the two members whose names begin with \({\rm{C}}\) being chosen equals the probability of one of the seven outcomes including at least \({\rm{Cox}}\) or \({\rm{Cr}}\) (disjoint union of the seven events)

\(\begin{aligned}P(\{~starts\text{ }with\text{ }at\text{ }least\text{ }one\text{ }C~\})&=P(\{A,Cox\}+\{A,Cr\}+\{B,Cox\}+\{B,Cr\} \\+\{Cox,Cr\}+\{Cox,F\}+\{Cr,F\}) \\&=\frac{7}{10}, \\\end{aligned}\)

  1. denotes the joining of discontinuous occurrences.
  2. equally likely events.
04

Determining the probability that the two chosen representatives have a total of at least fifteen years’ teaching experience there

We still obtain \({\rm{10}}\) outcomes if we replace the name with the number of years of teaching experience. There are six outcomes with a sum more than or equal to \({\rm{15}}\) years of teaching experience, and we want to find the likelihood that two chosen representatives have a total of at least \({\rm{15}}\) years of teaching experience.

\({\rm{\{ 3,14\} ,\{ 6,10\} ,\{ 6,14\} ,\{ 7,10\} ,\{ 7,14\} ,\{ 10,14\} }}\)

As a result, the likelihood is

\(\begin{aligned}P(~at\text{ }least~15~years\text{ }\exp erience~\})&=P(\{3,14\}+\{6,10\}+\{6,14\}+ \\+\{7,10\}+\{7,14\}+\{10,14\}) \\&=\frac{6}{10}, \\\end{aligned}\)

where the same explanations as before are used.

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Most popular questions from this chapter

An academic department with five faculty members narrowed its choice for department head to either candidate A or candidate B. Each member then voted on a slip of paper for one of the candidates. Suppose there are actually three votes for A and two for B. If the slips are selected for tallying in random order, what is the probability that A remains ahead of B throughout the vote count (e.g., this event occurs if the selected ordering is AABAB, but not for ABBAA)?

The probability that an individual randomly selected from a particular population has a certain disease is \({\rm{.05}}\). A diagnostic test correctly detects the presence of the disease \({\rm{98\% }}\)of the time and correctly detects the absence of the disease \({\rm{99\% }}\)of the time. If the test is applied twice, the two test results are independent, and both are positive, what is the (posterior) probability that the selected individual has the disease? (Hint: Tree diagram with first-generation branches corresponding to Disease and No Disease, and second- and third-generation branches corresponding to results of the two tests.)

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An experimenter is studying the effects of temperature, pressure, and type of catalyst on yield from a certain chemical reaction. Three different temperatures, four different pressures, and five different catalysts are under consideration.

a. If any particular experimental run involves the use of a single temperature, pressure, and catalyst, how many experimental runs are possible?

b. How many experimental runs are there that involve use of the lowest temperature and two lowest pressures?

c. Suppose that five different experimental runs are to be made on the first day of experimentation. If the five are randomly selected from among all the possibilities, so that any group of five has the same probability of selection, what is the probability that a different catalyst is used on each run?

The composer Beethoven wrote \({\rm{9}}\) symphonies, \({\rm{9}}\) piano concertos (music for piano and orchestra), and \({\rm{32}}\) piano sonatas (music for solo piano).

a. How many ways are there to play first a Beethoven symphony and then a Beethoven piano concerto?

b. The manager of a radio station decides that on each successive evening (\({\rm{7}}\) days per week), a Beethoven symphony will be played followed by a Beethoven piano concerto followed by a Beethoven piano sonata. For how many years could this policy be continued before exactly the same program would have to be repeated?
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