6.3 Probability, tree diagrams and decision trees 245
4 Metco make components for small
electrical equipment. One production line
makes 500,000 switches each year. They
currently use a manual inspection system
with one quality control operative. 1% of
the production is faulty and the operative
finds and rejects 90% of these. Metco sell
the switches for $2 each but any faulty
ones which are delivered cost the company
$25 in replacement and compensation
costs. The quality control operative costs
$40,000 per year to employ.
Metco’s management are considering
installing an automatic quality control
system. This will mean the quality control
operative will be made redundant, for
which they will have to make a single
payment of $10,000. The manufacturers
of the new system claim that it will
pick up 99% of faulty switches, but the
production manager is sceptical about
this. He estimates that there is only a
20% chance of it being this good and an
80% chance that it will only pick up 95%
of the faulty switches. The new equipment
will cost $180,000 which will be written
off over four years (^14 each year). Other
manufacturing costs will not be affected by
the quality control system used.
By calculating the average income
and costs per year for a four-year period,
investigate the economics of the old and
new systems, considering which is most
likely to be the best to use.Answers and comments are on pages 336–39.1 My drawer contains eight blue socks and
six black socks. If I take four socks out at
random, what are the chances that they
will make up two matching pairs?
2 My wife has sent me to the bank with her
cash card. I do not know the four-digit
number I have to enter into the machine to
withdraw money. I know the first two digits
are the two digits of her month of birth, in
the right order. The last two digits are the
date in the month of her birthday. There
are no zeros and I have forgotten my wife’s
birthday.
What are the chances of my getting it
right first time? What are the chances of
my getting it right in the three attempts I
am allowed?
3 A fairground game involves taking three
throws to get a ring over two poles in the
ground at different distances from the
throwing position. Throws must be taken
alternately at the two poles, but you may
start with either one. You win a prize if your
ring lands over a pole in two successive
throws out of the three.Clearly, it is easier to throw the ring over
the nearer pole than the farther one. Is it
better to make your attempts in the order
‘near, far, near’ or ‘far, near, far’, or doesn’t
it matter?End-of-chapter assignments