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Chapter 2: Q28E (page 66)

In Exercise 5, suppose that any incoming individual is equally likely to be assigned to any of the three stations irrespective of where other individuals have been assigned. What is the probability that

a. All three family members are assigned to the same station?

b. At most two family members are assigned to the same station?

c. Every family member is assigned to a different station?

Short Answer

Expert verified
  1. \({\rm{1/9}}\)
  2. \({\rm{8/10}}\)
  3. \({\rm{2/9}}\)

Step by step solution

01

definition of Aromatic hydrocarbon

We record the station number \(1,2,3\) for each family member. Member A visited station \({\rm{1}}{\rm{.}}\), member B visited station \({\rm{2,}}\), and member C visited station \({\rm{1}}{\rm{.}}\) (outcome 121).

We may make a table in which each row represents a sample and each column represents one of the three family members A, B, or C, as shown in the book. The table below shows all of the \({3^3} = 27\) tables (you will learn more about counting in the next chapters) \({\rm{S}}\) (Oc. stands for outcome and FM for Family Member)

If you're requested to list all outcomes, this is the way to go because it's easier to work with and you'll be able to see what you need more clearly. You may always start with all the same numbers and work your way down like we did when making the table; this is a classic strategy for doing so.

02

the probability if all three family members are assigned to the same station

Probability of an occurrence "All three members of the family are assigned to the same station," says A. On the benzene ring, each carbon atom possesses\({\rm{1}}\)atom.

Many known groups can also be replaced for that \({\rm{H}}\) atom.

\(\begin{aligned}\rm P(A) &= P((1,1,1) + (2,2,2) + (3,3,3))\\\rm &= \frac{{\rm{3}}}{{{\rm{27}}}}{\rm{ = }}\frac{{\rm{1}}}{{\rm{9}}}{\rm{,}}\end{aligned}\)

since all events have the same chance of occurring (probability \({\rm{1/27}}\)) and stands for the union of disjoint events

03

the probability if at most two family members are assigned to the same station

Probability of an occurrence \({\rm{B - }}\) "At most two family members are allocated to the same station" is the complement of \({\rm{B - }}\) "All three family members are assigned to the same station." are allocated to the same station", thus we get \({\rm{P(A) + P}}\left( {{\rm{A'}}} \right){\rm{ = 1}}\)

\({\rm{P(B) = 1 - P}}\left( {{\rm{B'}}} \right){\rm{ = 1 - }}\frac{{\rm{1}}}{{\rm{9}}}{\rm{ = }}\frac{{\rm{8}}}{{\rm{9}}}{\rm{,}}\)

where we used

04

the probability if Every family member is assigned to a different station

The occurrence C - "Every family member gets assigned to a separate station" has a probability of 1%.

\(\begin{aligned}\rm P(C) &= P(\{ (123),(132),(2,1,3),(2,3,1),(3,1,2),(3,2,1)\} )\\\rm &= \frac{{\rm{6}}}{{{\rm{27}}}}{\rm{ = }}\frac{{\rm{2}}}{{\rm{9}}}\end{aligned}\).

Because there are six outcomes that are all equally likely.

In the sample space, the probability of an event is #favourable outcomes.

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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)?

Consider purchasing a system of audio components consisting of a receiver, a pair of speakers, and a \({\rm{CD}}\) player. Let \({{\rm{A}}_{\rm{1}}}\) be the event that the receiver functions properly throughout the warranty period, \({{\rm{A}}_{\rm{2}}}\) be the event that the speakers function properly throughout the warranty period, and \({{\rm{A}}_{\rm{3}}}\) be the event that the \({\rm{CD}}\) player functions properly throughout the warranty period. Suppose that these events are (mutually) independent with \({\rm{P}}\left( {{{\rm{A}}_{\rm{1}}}} \right){\rm{ = }}{\rm{.95}}\), \({\rm{P}}\left( {{{\rm{A}}_{\rm{2}}}} \right){\rm{ = }}{\rm{.98}}\), and \({\rm{P}}\left( {{{\rm{A}}_{\rm{3}}}} \right){\rm{ = }}{\rm{.80}}\).

a. What is the probability that all three components function properly throughout the warranty period?

b. What is the probability that at least one component needs service during the warranty period?

c. What is the probability that all three components need service during the warranty period?

d. What is the probability that only the receiver needs service during the warranty period?

e. What is the probability that exactly one of the three components needs service during the warranty period?

f. What is the probability that all three components function properly throughout the warranty period but that at least one fails within a month after the warranty expires?

A computer consulting firm presently has bids out on three projects. Let \({A_i}\) = {awarded project i}, for i=1, 2, 3, and suppose that\(P\left( {{A_1}} \right) = .22,\;P\left( {{A_2}} \right) = .25,\;P\left( {{A_3}} \right) = .28,\;P\left( {{A_1} \cap {A_2}} \right) = .11,\;P\left( {{A_1} \cap {A_3}} \right) = .05,\)\(P\left( {{A_2} \cap {A_3}} \right) = .07\),\(P\left( {{A_1} \cap {A_2} \cap {A_3}} \right) = .01\). Express in words each of the following events, and compute the probability of each event:

a. {\({A_1} \cup {A_2}\)}

b. \(A_1' \cap A_2'\)(Hint: \(\left( {{A_1} \cup {A_2}} \right)' = A_1' \cap A_2'\) )

c.\({A_1} \cup {A_2} \cup {A_3}\)

d. \(A_1' \cap A_2' \cap A_3'\)

e. \(A_1' \cap A_2' \cap {A_3}\)

f. \(\left( {A_1' \cap A_2'} \right) \cup {A_3}\)

A department store sells sports shirts in three sizes (small, medium, and large), three patterns (plaid, print, and stripe), and two sleeve lengths (long and short). The accompanying tables give the proportions of shirts sold in the various category combinations.

a. What is the probability that the next shirt sold is a medium, long-sleeved, print shirt?

b. What is the probability that the next shirt sold is a medium print shirt?

c. What is the probability that the next shirt sold is a short-sleeved shirt? A long-sleeved shirt?

d. What is the probability that the size of the next shirt sold is the medium? That the pattern of the next shirt sold as a print?

e. Given that the shirt just sold was a short-sleeved plaid, what is the probability that its size was medium?

f. Given that the shirt just sold was a medium plaid, what is the probability that it was short-sleeved? Long-sleeved?

Consider the following information about travellers on vacation (based partly on a recent Travelocity poll): \({\rm{40\% }}\) check work email, \({\rm{30\% }}\) use a cell phone to stay connected to work, \({\rm{25\% }}\) bring a laptop with them, \({\rm{23\% }}\) both check work email and use a cell phone to stay connected, and \({\rm{51\% }}\) neither check work email nor use a cell phone to stay connected nor bring a laptop. In addition, \({\rm{88}}\) out of every \({\rm{100}}\) who bring a laptop also check work email, and \({\rm{70}}\) out of every \({\rm{100}}\) who use a cell phone to stay connected also bring a laptop. a. What is the probability that a randomly selected traveller who checks work email also uses a cell phone to stay connected? b. What is the probability that someone who brings a laptop on vacation also uses a cell phone to stay connected? c. If the randomly selected traveller checked work email and brought a laptop, what is the probability that he/ she uses a cell phone to stay connected?
See all solutions

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