Binominal Distributions Worksheet With Answer Key - Helm 2008 Section 372 Page 16
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20Exercises continued
13. There are five machines in a factory. Of these machines, three are working properly and two
are defective. Machines which are working properly produce articles each of which has independently
a probability of 0.1 of being imperfect. For the defective machines this probability is 0.2. A machine
is chosen at random and five articles produced by the machine are examined. What is the probability
that the machine chosen is defective given that, of the five articles examined, two are imperfect and
three are perfect?
14. A company buys mass-produced articles from a supplier. Each article has a probability p of being
defective, independently of other articles. If the articles are manufactured correctly then p = 0.05.
However, a cheaper method of manufacture can be used and this results in p = 0.1.
(a) Find the probability of observing exactly three defectives in a sample of twenty articles
(i) given that p = 0.05
(ii) given that p = 0.1.
(b) The articles are made in large batches. Unfortunately batches made by both methods
are stored together and are indistinguishable until tested, although all of the articles
in any one batch will be made by the same method. Suppose that a batch delivered
to the company has a probability of 0.7 of being made by the correct method. Find the
conditional probability that such a batch is correctly manufactured given that, in a sample
of twenty articles from the batch, there are exactly three defectives.
(c) The company can either accept or reject a batch. Rejecting a batch leads to a loss for
the company of 150. Accepting a batch which was manufactured by the cheap method
will lead to a loss for the company of
400. Accepting a batch which was correctly
manufactured leads to a profit of 500. Determine a rule for what the company should
do if a sample of twenty articles contains exactly three defectives, in order to maximise
the expected value of the profit (where loss is negative profit). Should such a batch be
accepted or rejected?
(d) Repeat the calculation for four defectives in a sample of twenty and hence, or otherwise,
determine a rule for how the company should decide whether to accept or reject a batch
according to the number of defectives.
32
HELM (2008):
Workbook 37: Discrete Probability Distributions
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