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Chapter 2: Q7E (page 57)

An academic department has just completed voting by secret ballot for a department head. The ballot box contains four slips with votes for candidate Aand three slips with votes for candidate B.Suppose these slips are removed from the box one by one.

a. List all possible outcomes.

b. Suppose a running tally is kept as slips are removed. For what outcomes does Aremain ahead of B throughout the tally?

Short Answer

Expert verified

a. The possible outcomes are,

\(S = \left\{ \begin{aligned}{l}BBBAAAA,BBABAAA,BBAABAA,BBAAABA,BBAAAB,BABBAAA,BABABAA,BABAABA,BABAAAB,\\BAABBAA,BAABABA,BAABAAB,BAAABBA,BAAABAB,BAAAABB,ABBBAAA,ABBABAA,ABBAABA,\\ABBAAAB,ABABBAA,ABABABA,ABABAAB,ABAABBA,ABAABAB,ABAABB,AABBBAA,AABBABA,\\AABBAAB,AABABBA,AABABAB,AABAABB,AAABBBA,AAABBAB,AAABABB,AAAABBB\end{aligned} \right\}\)

b. The possible outcomes are,

\(C = \left\{ {AABABAB,AABAABB,AAABBAB,AAABABB,AAAABBB} \right\}\)

Step by step solution

01

Given information

The number of slips the ballot box contains with votes for candidate A is 4.

The number of slips the ballot box contains with votes for candidate B is 3.

Theslips are removed from the box one by one.

02

List the possible outcomes in event S

a.

Let S represents the event of all possible outcomes.

The outcomes are represented as,

\(S = \left\{ \begin{aligned}BBBAAAA,BBABAAA,BBAABAA,BBAAABA,BBAAAB,BABBAAA,BABABAA,BABAABA,BABAAAB,\\BAABBAA,BAABABA,BAABAAB,BAAABBA,BAAABAB,BAAAABB,ABBBAAA,ABBABAA,ABBAABA,\\ABBAAAB,ABABBAA,ABABABA,ABABAAB,ABAABBA,ABAABAB,ABAABB,AABBBAA,AABBABA,\\AABBAAB,AABABBA,AABABAB,AABAABB,AAABBBA,AAABBAB,AAABABB,AAAABBB\end{aligned} \right\}\)

03

List of the possible outcomes

b.

A running tally is kept as slips are removed.

Let C be the event that A remains ahead of B throughout the tally.

The possible outcomes are,

\(C = \left\{ {AABABAB,AABAABB,AAABBAB,AAABABB,AAAABBB} \right\}\)

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Most popular questions from this chapter

An electronics store is offering a special price on a complete set of components (receiver, compact disc player, speakers, turntable). A purchaser is offered a choice of manufacturer for each component:

A switchboard display in the store allows a customer to hook together any selection of components (consisting of one of each type). Use the product rules to answer the following questions:

a. In how many ways can one component of each type be selected?

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c. In how many ways can components be selected if none is to be Sony?

d. In how many ways can a selection be made if at least one Sony component is to be included?

e. If someone flips switches on the selection in a completely random fashion, what is the probability that the system selected contains at least one Sony component? Exactly one Sony component?

Reconsider the credit card scenario of, and show that A and B are dependent first by using the definition of independence and then by verifying that the multiplication property does not hold.

The accompanying table gives information on the type of coffee selected by someone purchasing a single cup at a particular airport kiosk.

Small

Medium

Large

Regular

\(14\% \)

\(20\% \)

\(26\% \)

Decaf

\(20\% \)

\(10\% \)

\(10\% \)

Consider randomly selecting such a coffee purchaser.

a. What is the probability that the individual purchased a small cup? A cup of decaf coffee?

b. If we learn that the selected individual purchased a small cup, what now is the probability that he/she chose decaf coffee, and how would you interpret this probability?

c. If we learn that the selected individual purchased decaf, what now is the probability that small size was selected, and how does this compare to the corresponding unconditional probability of (a)?

Reconsider the system defect situation described in Exercise.

a. Given that the system has a type \(1\) defect, what is the probability that it has a type \({\bf{2}}\) defect?

b. Given that the system has a type \(1\) defect, what is the probability that it has all three types of defects?

c. Given that the system has at least one type of defect, what is the probability that it has exactly one type of defect?

d. Given that the system has both of the first two types of defects, what is the probability that it does not have the third type of defect?

A certain shop repairs both audio and video components. Let A denote the event that the next component brought in for repair is an audio component, and let B be the event that the next component is a compact disc player (so the event B is contained in A). Suppose that \(P\left( A \right) = 0.6\)and\(P\left( B \right) = 0.05\). What is\(P\left( {B|A} \right)\)?
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