d) Chapter 5
5.1
An electrical goods retailer sold video recorders to 8200 customers last year and extended
warranties to 3500 of these customers. When the retailer sells a video recorder, what is
the probability that:
(a)
The customer will buy an extended warranty?
(b)
The customer will not buy an extended warranty?
5.2
Since it was set up 73825 people have visited the website of a fashion designer and 6301
of them purchased goods online. When someone visits the site what is the probability
that:
(a)
they do not purchase goods on-line?
(b)
they do purchase goods on-line?
5.3
A direct marketing company produces leaflets offering membership of a book club.
These leaflets are then inserted into the magazine sections of two Sunday newspapers, the
Citizen and the Despatch. 360 000 leaflets are put in copies of the Citizen and 2 130 000
are put in copies of the Despatch. The company receives 19447 completed leaflets from
Citizen readers and 58193 completed leaflets from Despatch readers. What is the
probability that:
(a)
A Citizen reader returns a leaflet?
(b)
A Despatch reader returns a leaflet?
5.4
A garage offers a breakdown recovery service for motorists that is available every day of
the year. According to their records the number of call-outs they received per day last
year were:
Number of call-outs
0
1
2
3
4
Number of days
68
103
145
37
12
What is the probability that:
(a)
They receive two call-outs in a day?
(b)
They receive two or fewer call-outs in a day?
(c)
They receive one or more call-outs in a day?
(d)
They receive less than four call-outs in a day?
(e)
They receive more than two call-outs in a day?
5.5
Last year 12966 people opened new accounts at a building society. Of these 5314 were
branch-based accounts, 4056 were postal accounts, and 3596 were Internet accounts.
What is the probability that when a customer opens a new account:
(a)
It is a postal account.
(b)
It is an Internet account.
(c)
It is either branch-based or postal.
5.6
The 120 employees at a factory were asked which would best improve their working life:
better promotion prospects, higher pay or more respect from other staff. The results are
tabulated below.
Response
Job type
Manual
Clerical
Managerial
Better promotion prospects
12
12
3
Higher pay
53
19
2
More respect
7
6
6
(a)
What is the probability that an employee selected more respect?
(b)
What is the probability that an employee is a clerical worker or selected better
promotion prospects?
(c)
What is the probability that a manual employee selected higher pay?
(d)
What is the probability that an employee selected more respect and is a manager?
(e)
What is the probability that a managerial employee selected higher pay?
5.7
A safety agency analysed road traffic accidents involving injury to pedestrians by private
cars and produced the following table:
Degree of injury
Type of car
4×4
Sports
Other
Fatal
8
5
14
Serious
21
9
38
Non-serious
13
7
95
What is the probability that:
(a)
An injury to a pedestrian proves fatal?
(b)
An accident involved a sports car?
(c)
An accident involved a 4×4 car or resulted in a non-serious injury?
(d)
An accident resulted in serious injury and involved an ‘other’ type of car?
(e)
An accident that involved a 4×4 car resulted in a fatal injury? Compare this figure
and your answer to (a), and comment on whether the type of car and degree of
injury are independent.
5.8
The following survey results show the social class and type of main holiday destination
of 250 adult holidaymakers.
Destination
Social class
AB
C1C2
DE
UK
13
25
26
The rest of Europe
29
60
23
Other
55
14
5
Use these figures to estimate
(a)
The probability that a holidaymaker belongs to social class DE.
(b)
The probability that a holidaymaker takes a main holiday in the UK.
(c)
The probability that a holidaymaker is in social class AB or takes a main holiday
outside Europe.
(d)
The probability that a holidaymaker takes a main holiday in the rest of Europe or
is in social class DE.
(e)
The probability that a holidaymaker is in social class C1C2 and takes a main
holiday in the UK.
(f)
The probability that a holidaymaker takes a main holiday in the rest of Europe and
is in social class AB.
(g)
(h)
(i)
The probability that a holidaymaker in social class DE takes a main holiday
outside Europe.
The probability that a holidaymaker in social class DE takes a main holiday in the
UK.
Compare your answers to (b) and (h). Are social class and main holiday
destination independent?
5.9
A final-year undergraduate applies for a well-paid job with a large and reputable
organization in a highly desirable location. Competition for the job is expected to be
fierce. The probability that a candidate is selected for first interview is 0.10. The
probability that a candidate is then selected for second interview is 0.4. The probability
that a candidate then passes the psychometric test is 0.8. The probability that a candidate
then passes the selection procedure at the assessment centre and is offered the job is 0.25.
What is the probability that the undergraduate will get the job?
5.10
A commuter’s journey to work consists of driving her car to her local train station, taking
a train to the city, and catching a bus to her place of work. If her car trip is delayed she
misses the train and is late for work. If the train is delayed, she misses the bus and is late
for work. If the bus is delayed she is late for work. The probability that her car journey is
delayed is 0.05, the probability that the train is delayed is 0.1, and the probability that the
bus is delayed is 0.07.
(a)
What is the probability that she arrives at work on time?
(b)
What is the probability that she is late for work?
(c)
What is the probability that she is late for work because the bus is delayed?
5.11
Bella is taking her friend out for the evening but has forgotten to take some cash out of
her bank account. She reaches the cash machine, but can’t remember the exact sequence
of her PIN number. She knows that the digits in her PIN number are 2, 7, 8, and 9.
(a)
What is the probability that Bella enters the correct PIN number?
(b)
Bella’s friend Mel offers to draw some cash out of her account instead. She also
cannot remember her PIN number but recalls that there are two fours and two
sixes in it. What is the probability that Mel enters the correct PIN number?
5.12
Two friends have a favourite compilation CD that contains some of their all-time
favourite tracks. There are 20 tracks on the album. Sam likes five tracks and Chris likes
six other tracks. If they program their CD player to play three tracks selected at random
from the album, and assuming that the CD player can only select a track once, what is:
(a)
The probability that none of their favourites is selected?
(b)
The probability that at least one of Sam’s favourites is selected?
(c)
The probability that at least one of Chris’s favourites is selected?
5.13
Every morning Jack makes tea for the other four people in his section. Two of them take
sugar. Jack always forgets which teas he has sugared. One of the two senior members of
the section, Alicia, takes sugar in her tea; the other, Ben, does not. If Jack takes tea first
to Alicia and then to Ben, what is the probability that:
(a)
He gives them both the right tea?
(b)
(c)
He gets one wrong?
He gets both wrong?
5.14
As a result of flood damage a supermarket has a very large stock of tins without labels.
Forty per cent of the tins contain soup, 30% contain carrots, 25% contain raspberries, and
5% contain asparagus. The tins are to be offered for sale at three for 50 pence. When a
customer buys three tins what is the probability that:
(a)
None of the tins contain asparagus?
(b)
All three tins contain soup?
(c)
One tin contains raspberries?
(d)
Two tins contain carrots?
(e)
The contents of the three tins are different?
5.15
Declan, Emily and Farid each intend to start their own business after leaving college.
Declan wants to start a computer software company, Emily intends to open a recruitment
agency, and Farid would like to launch a graphic design business. According to available
evidence 60% of computer software companies, 75% of recruitment agencies, and 80%
of graphic design businesses fail within their first trading year. What is the probability
that:
(a)
The businesses that Declan, Emily and Farid start all stay in operation for at least
a year?
(b)
The businesses that two of them start are still trading after a year?
(c)
All three businesses fail within a year?
5.16
You win a prize in a charity raffle. The prize, donated by a small hotel chain, is a voucher
entitling you to a free double room for a weekend at each of the three hotels in the chain,
the Xerxes, the York and the Zetland. The room you stay in at each hotel will be picked
at random by the hotel manager. The Xerxes has 12 double rooms, 7 of which are en
suite. The York has 28 double rooms, 16 of which are en suite. The Zetland has 18
double rooms, of which 13 are en suite.
(a)
What is the probability that none of the rooms you get is en suite?
(b)
What is the probability that one of the rooms you get is en suite?
(c)
What is the probability that two or more of the rooms you get are en suite?
5.17
An insurance company offers quotations for motor insurance by telephone. The company
employs permanent staff to do this work but as a result of a dramatic rise in the number
of telephone inquiries 35% of quotations are provided by temporary staff. Unfortunately
22% of the quotations provided by temporary staff prove to be wrong, compared to the
8% of the quotations provided by full-time staff that turn out to be wrong. Under the
contract with the agency supplying the temporary staff, the agency will pay a proportion
of total costs arising from the mistakes based on the proportion of mistakes that are made
by the temporary staff. Use Bayes’ rule to determine the probability that if a mistake has
been made it has been made by one of the temporary staff and use it to suggest what
proportion of the total costs of mistakes the agency should pay.
5.18
Select the appropriate definition for each term on the left hand side from the list on the
right-hand side.
(a)
compound probability
(i) basing a probability on opinion
(b)
multiplication rule
(ii) outcomes that cannot occur together
(c)
collectively exhaustive
(iii) P (A and B) = P (A) * P (B | A)
(d)
dependency
(iv) a probability of a single outcome
(e)
judgmental
(v) basing a probability on deduction
(f)
simple probability
(vi) all possible outcomes
(g)
mutually exclusive
(vii) P (A or B) = P (A) + P (B) – P (A and B)
(h)
experimental
(viii) when P (B | A) is not equal to P (B)
(i)
addition rule
(ix) a probability of more than one outcome
(j)
theoretical
(x) basing a probability on evidence