d) Chapter 6
6.1
The employees of the Brecht Bank decide to put on a five-a-side football tournament to
raise money for charity. If the teams are chosen at random and four tenths of the
employees are women, what is the probability that:
(a)
A team consists entirely of males?
(b)
There are three females in a team?
(c)
There are three or fewer females in a team?
(d)
The majority of a team are female?
6.2
A lingerie manufacturer produces ladies’ knickers in packs of five pairs. Quality control
has been erratic and they estimate that one in every ten pairs of knickers leaves the
factory without the trim sewn on. What is the probability that in a pack of five pairs of
knickers:
(a)
There are no pairs without the trim?
(b)
There are two or fewer pairs without the trim?
(c)
There is more than one pair without the trim?
(d)
The majority of pairs are without the trim?
6.3
The producers of a TV survival programme advertise for potential participants and then
select random samples of ten people to appear in the programme. If 30% of the people
who apply to take part in the programme are smokers, what is the probability that:
(a)
A team consists entirely of non-smokers?
(b)
A team includes only one smoker?
(c)
A team includes no more than three smokers?
(d)
A team includes more than four smokers?
(e)
The majority of a team are non-smokers?
6.4
An egg-packing company thinks that a tenth of all the eggs they buy from farms are
contaminated with harmful bacteria that could result in food poisoning. What is the
probability that in a pack of ten eggs:
(a)
None of the eggs contain harmful bacteria?
(b)
More than two eggs contain harmful bacteria?
(c)
Half the eggs contain harmful bacteria?
(d)
Less than two eggs contain harmful bacteria?
6.5
A large party of Russian managers is to visit a car plant in the UK. They will be shown
round the plant in small groups of five. Twenty per cent of the managers speak English.
If the groups are selected at random, what is the probability that:
(a)
There is no English speaker in a group?
(b)
There is at least one English speaker in a group?
(c)
All but one of the groups speaks English?
6.6
The mean number of accidents per day at a large building site is 1.0. What is the
probability that a day passes when there are:
(a)
No accidents?
(b)
Two accidents?
(c)
Four or fewer accidents?
(d)
More than three accidents?
6.7
An office worker receives an average of 22.5 e-mail messages per day. If his working
day lasts seven and a half hours, what is the probability that:
(a)
He receives no e-mails in an hour?
(b)
He receives one e-mail in an hour?
(c)
He receives two or fewer e-mails in an hour?
(d)
He receives more than four e-mails in an hour?
6.8
Everyone working in a large open-plan office takes cups of tea or coffee back to their
desk to drink. The mean number of spilled cups of hot drinks in an eight-hour working
day is 24. What is the probability that an hour passes in which:
(a)
There are no spillages?
(b)
There are two spillages?
(c)
There are less than four spillages?
(d)
There is at least one spillage?
6.9
A company produces mouse mats that are 20 cm long and 20 cm wide. These mats are
cut from sheets of material that are one metre long and one metre wide. Precise
machine setting means that 25 mouse mats can be cut from each sheet of material.
Problems that have arisen in the laminating process mean that the sheets of material
contain small holes. The mean number of holes per sheet is 25. What is the probability
that:
(a)
A mouse mat has no holes?
(b)
A mouse mat has two or fewer holes?
(c)
A mouse mat has more than one hole?
6.10
A pharmaceuticals company marketed a drug that proved to be an effective treatment
but unfortunately resulted in side effects for some patients. On the basis of initial
clinical research the probability that a patient who is treated with the drug suffers no
side effect is 0.85, the probability of a minor side effect is 0.11, and the probability of a
major side effect is 0.04. Under an agreement with the appropriate authorities the
company has agreed to pay £2500 in compensation to patients who suffer a minor side
effect and £20 000 in compensation to patients who suffer a major side effect. What is
the expected value of the compensation per patient?
6.11
A graduate wants to pursue a career in a profession. She will need to gain membership
of the professional institute by passing three stages of examinations. If she becomes a
full member of the institute she anticipates that she will be able to earn £60 000 per
year. If she fails the final stage of the examinations, but passes the first two stages she
can become an associate member of the institute. An associate member earns about £42
000 per year. If she fails at the second stage but passes the initial stage she can obtain a
certificate of competence from the institute and probably earn £34 000. If she fails the
initial stage of the examinations, she will have to consider other employment and
anticipates that she would be able to earn £28 000. The pass rates for the initial, second
and final stages of the institute’s examinations are 70%, 55% and 80%. What is the
expected value of her annual earnings?
6.12
An insurance company calculates that the probability that in a year a motor insurance
policyholder makes a claim arising from a major accident is 0.03, the probability that
he or she makes a claim as a result of a minor accident is 0.1, and the probability that he
or she makes a claim as a result of vehicle theft is 0.05. The typical payment for a
major accident claim is £4500, for a minor accident claim is £800, and for theft £4000.
The probability that a policyholder makes more than one claim in a year is zero. What
is the expected value of claims per policy?
6.13
A film production company is about to release a new movie. They estimate that there is
5% chance that the film will make a profit of $14 m, a 30% chance that it will make a
$1 m profit, a 25% chance that it will break even, and a 40% chance that it will make a
loss of $2 m. What is the expected return from the film?
6.14
A sportswear company markets a premium brand of trainers. At present the revenue
from sales of these trainers is $17m a year. They have been negotiating with
representatives of a top US sports star in order to obtain his endorsement of the product.
The cost to the company of the endorsement would be $3m a year, a cost that would be
met from the sales revenue from the trainers. If the star endorses the trainers the
company expects that the sales revenue will rise to $30m a year. However, just as the
deal is about to be signed a regional US news agency runs a story alleging that the star
has taken bribes. The sportswear company understands from their US representatives
that there is a 60% chance that the star will be able to refute the allegation, in which
case sales of the trainers will still be $30m. If the star is unable to refute the allegation
the negative publicity is likely to reduce sales revenue from the trainers to $10m a year.
If the deal is not completed sales revenue is expected to remain $17m a year.
(a)
Should the company cancel or complete the endorsement deal?
(b)
What should the company do if the chance of the star refuting the allegation is
40%?
6.15
A member of a successful girl band is considering leaving the band to pursue a solo
career. If she stays with the band she estimates that there is a 60% chance that the band
will continue to be successful and she would earn £1.7 m over the next three years. If
the band’s success does not continue she would still earn £0.6 m over the next three
years under her existing contract. If she embarks on a solo career she estimates that the
chance of success is 20%. A successful solo career would bring her £4 m in the next
three years. If her solo career is not successful she can expect to earn only £0.25 m in
the next three years. What should she do?
6.16
An arable farmer is thinking of sowing scientifically modified crops next year. She
believes that if she did so her profits would be £75 000, compared to £50 000 if she
sowed unmodified crops. A neighbouring farmer has made it clear that if his crops are
contaminated he will demand compensation. The arable farmer guesses that the
probability of contamination to be 40%. In the event that the neighbouring farmer
claims compensation there is a 30% chance that the arable farmer would have to pay
£25 000 in compensation and a 70% chance she would have to pay £50 000 in
compensation.
(a)
Construct a decision tree and use it to advise the arable farmer.
(b)
An expert puts the probability that there will be contamination of the crops of
the neighbouring farmer at 60%. Should the arable farmer change her strategy in
the light of this information?
6.17
Select the appropriate description for each term on the left hand side from the list on the
right-hand side.
(a)
cumulative probability
(i)
consists of probabilities of incidents over time
(b)
expected value
(ii)
the expected monetary value of a decision
(c)
binomial distribution
(iii)
the probability of one value or a lower one
(d)
Poisson distribution
(iv)
consists of probabilities of successes in trials
(e)
EMV
(v)
represent decisions and outcomes
(f)
decision trees
(vi)
a name for a probability distribution mean