Probability and Statistics Questions

Probability and Statistics Questions

Show all your calculations and intermediate steps, with explanation, in your answers.
When giving your answers in decimal form, you should use 4 decimal places.
If you use RStudio then you should state and describe the commands and options used.
B1. (a) Let A, B and C be three arbitrary events. Using Venn diagrams, determine if the
following proposition is correct, or not:
(A ∩ B
c
) ∪ C = (A ∪ C) ∩ B
c
.
Now suppose that events B and C are mutually exclusive. Does this change your
conclusion regarding the above proposition? Justify your answer.
(b) In a study into the use of illegal drugs, the following two-stage procedure was applied:
(1) all participants were asked to toss a fair coin twice, (2) if the first toss was a head
they were instructed to answer the question “Have you ever taken illegal drugs?”;
otherwise they were instructed to answer the question “Was the second toss a head?”
For each question the possible answers were “Yes” and “No”. Probability and Statistics Questions
(i) Draw a tree diagram to describe this situation.
(ii) Suppose that in a large sample 35% of participants replied “Yes”. Estimate the
percentage of the population who have taken illegal drugs.
(iii) Now suppose that in a small sample 20% replied “Yes”. What now is your
estimate of the population percentage who have taken illegal drugs? Comment
on your answer.
(iv) Comment on the advantages of using this approach compared to an alternative
only using the question “Have you ever taken illegal drugs?”
(c) Suppose that the number of insurance claims per High Risk customer for a particular car insurance company can be well modelled by the following probability mass
function, where X denotes the number of claims.
x 0 1 2 3
pX(x) 0.8 0.1 0.08 0.02
(i) Calculate the mean and standard deviation of the number of claims.
(ii) Suppose that a particular High-Risk customer pays an annual premium of £900,
but that each claim will cost the company £2000. What is the expected profit
for the company and the standard deviation of the profit? Comment on your
answer.
Page 4 of 6 Turn the page over
Module Code: MATH171001
B2. (a) The Coombs blood test is 95% effective in detecting Iron-deficiency anaemia (IDA),
a condition which effects about 4% of the adult population, when it is in fact present;
however, the test also yields ‘false positive’ results for 7% of healthy people tested.
(i) What is the probability that a person randomly selected from the population has
a positive test result?
(ii) Suppose that a person does have a positive test, then what is the probability
that they actually have IDA? Comment.
(b) To investigate the number of defective items in an industrial production line a random
sample of size n was collected and the number of defective items counted.
It is believed that only 2% of items are defective.
(i) Suggest a suitable model for this situation. Justify your answer.
Supposing that the sample has n = 6 items, calculate the exact probability that
there are no defectives in the sample.
(ii) Now supposing that n = 100, and using a suitable Poisson approximation, evaluate the probability that there is at least 1 defective item in the sample. Comment
on the validity of the approximation in this case.
Suppose that the production line started to malfunction and that 45% of the items
were defective. Again a random sample of size n was collected.
(iii) Supposing that the sample has n = 200 items, and using a suitable normal approximation, evaluate the probability that more than half the sample is defective.
Comment on the validity of the approximation in this case.
When using the the normal distribution tables on page 6 to evaluate Φ(x), you
may round the corresponding x to 2 decimal places.
(c) Let T1, T2, . . . denote the times between trading events of a particular FTSE 100
company shares within a particular day, where T1 is the time from the start of the
day until the first trading event, T2 is the time between the first trade and the
second trade etc. Suppose that these times can be well modelled by an exponential
distribution with mean 6 minutes and that times are independent.
(i) Suppose that two times are randomly chosen. What is the probability that at
least one of these two times is longer than 12 minutes?
(ii) Suppose that a randomly chosen time is known to be shorter than 30 minutes,
then what is the probability that it is shorter then 12 minutes?
(iii) What is the distribution of the number of events occurring in an 8-hour day?
Comment on any necessary assumptions.

Probability and Statistics Questions

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