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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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