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92 Introduction to probability
(5) Given below are the results of a survey of 100 cars:
Condition of tires Faulty
Condition of brakes
Not faulty Total
Faulty 25 5 30
Not faulty 15 55 70
Total 40 60 100
(a) Assuming that the survey is representative of all the cars on
the road, what is the probability that a car selected at random
will have:
(i) Faulty brakes;
(ii) Faulty tires;
(iii) Either faulty brakes or faulty tires;
(iv) Faulty brakes given that it has faulty tires;
(v) Faulty tires given that it has faulty brakes?
(b) What conclusion would you draw about the relationship
between the events ‘faulty tires’ and ‘faulty brakes’?
(6) Three machines, A, B and C, operate independently in a factory.
Machine A is out of action for 10% of the time, while B is out of
action for 5% of the time and C for 20% of the time. A rush order
has to be commenced at midday tomorrow. What is the probability
that at this time:
(a) All three machines will be out of action?
(b) None of the machines will be out of action?
(7) A speculator purchases three stocks on the London Stock Exchange.
He estimates that the probabilities that each of these stocks will
have risen in value by the end of the week are, respectively, 0.6, 0.8
and 0.4.
(a) Assuming that the price changes in the three stocks are independent,
determine the probability that all three stocks will
have risen in value by the end of the week.
(b) Do you think that it is reasonable to assume that movements in
the prices of individual stocks are independent?
(8) The managers of a company are considering the launch of a new
product and they are currently awaiting the results of a market
research study. It is thought that there is a 0.6 probability that
the market research will indicate that sales of the product in its
first three years will be high. If this indication is received then the
probability that sales will be high is thought to be 0.8. What is the