Binominal Distributions Worksheet With Answer Key - Helm 2008 Section 372 Page 15
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Exercises continued
8. In a large school, 80% of the pupils like mathematics. A visitor to the school asks each of 4
pupils, chosen at random, whether they like mathematics.
(a) Calculate the probabilities of obtaining an answer yes from 0, 1, 2, 3, 4 of the pupils
(b) Find the probability that the visitor obtains the answer yes from at least 2 pupils:
(i) when the number of pupils questioned remains at 4
(ii) when the number of pupils questioned is increased to 8.
9. A machine has two drive belts, one on the left and one on the right. From time to time the
drive belts break. When one breaks the machine is stopped and both belts are replaced. Details of n
consecutive breakages are recorded. Assume that the left and right belts are equally likely to break
first. Let X be the number of times the break is on the left.
(a) How many possible different sequences of “left” and “right” are there?
(b) How many of these sequences contain exactly j “lefts”?
(c) Find an expression, in terms of n and j, for the probability that X = j.
(d) Let n = 6. Find the probability distribution of X.
10. A machine is built to make mass-produced items. Each item made by the machine has a
probability p of being defective. Given the value of p, the items are independent of each other.
Because of the way in which the machines are made, p could take one of several values. In fact
p = X/100 where X has a discrete uniform distribution on the interval [0, 5]. The machine is tested
by counting the number of items made before a defective is produced. Find the conditional probability
distribution of X given that the first defective item is the thirteenth to be made.
11. Seven batches of articles are manufactured. Each batch contains ten articles. Each article has,
independently, a probability of 0.1 of being defective. Find the probability that there is at least one
defective article
(a) in exactly four of the batches,
(b) in four or more of the batches.
12. A service engineer is can be called out for maintenance on the photocopiers in the offices of
four large companies, A, B, C and D. On any given week there is a probability of 0.1 that he will
be called to each of these companies. The event of being called to one company is independent of
whether or not he is called to any of the others.
(a) Find the probability that, on a particular day,
(i) he is called to all four companies,
(ii) he is called to at least three companies,
(iii) he is called to all four given that he is called to at least one,
(iv) he is called to all four given that he is called to Company A.
(b) Find the expected value and variance of the number of these companies which call the
engineer on a given day.
31
HELM (2008):
Section 37.2: The Binomial Distribution
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