Your company insures recreational light aircraft. Over the last 10 years there have been an average of 17 crashes per year. Legislation this year has increased your exposure: a crash will now cost you Lognorm(50,22) thousand pounds. You have not been able to increase your premiums though, and have the same revenue as last year of 1.09 million pounds.
Assuming the same level of exposure for next year, what is the probability that you will make a loss next year?
You are faced with a number of risks (A to F) that may impact on your project. The size of impact is uncertain and specified by minimum, most likely and maximum values. The probability of each risk occurring is estimated below: some probabilities are conditional on whether other risks have occurred.
Determine the distribution of costs you might occur from these risks.
What is the probability that none of the risk events will occur?
Your company has been invited to bid for a project to design and construct a line of 16 electricity pylons. Estimates are based on PERT distributions (using absolute minimum, most likely and absolute maximum) unless indicated.
Design
The design work could be done in-house. However, if you are awarded an upcoming contract in France that your company has bid for, you will not have the resources to design this job and will have to sub-contract out. There is some disagreement about the probability of getting the French job, so it is also represented by a probability distribution.
Your designers estimate the following:
Minimum
Most Likely
Maximum
In-house design cost:
Ј150 000
Ј165 000
Ј190 000
Sub-contracted costs:
Ј180 000
Ј200 000
Ј235 000
Probability of French job:
40%
Foundation construction
Your foundation engineers estimate the following:
Minimum
Most Likely
Maximum
Material costs/pylon:
Ј6 200
Ј7 000
Ј7 900
Man hours/pylon:
Normal(600,80) dependent on
individual ground conditions at pylon sites
Cost/man hour:
Ј7.50
Total plant cost:
Ј250 000
Ј300 000
Ј380 000
Pylon construction
Your steel work engineers estimate the following:
Minimum
Most Likely
Maximum
Steel cost/pylon at construction time:
Ј52 000
Ј54 000
Ј58 500
Man hours/pylon (dependent on final pylon design):
200
240
270
Cost/man hour:
Ј13.00
Time constraints
The project's components are estimated to take the following time (weeks):
Minimum
Most Likely
Maximum
Design:
10
12
15
Foundations:
14
16
21
Time from foundation completion to last pylon completed:
4
4.5
5.5
The bid document has a clause specifying a 35 week completion time with a penalty of Ј25,000 per week or part thereof for overrunning.
Produce distributions of the total project cost and duration.
Use a scatter plot to illustrate the relationship between the project's cost and duration. Try plotting iterations above and below 35 weeks separately on the same graph and fit regression lines through the two data sets.
Determine a suitable bid price you would be happy to offer.
Which uncertain variable within the model is most influencing the maximum tail of the cost distribution?
Rerun the model, but make the design time and cost 90% correlated and the time to do the foundations 80% correlated with the associated plant cost and 90% correlated with the total foundation man hours. What is the effect of including these correlations?
20 people have randomly been selected off the streets of a city. They write down whether they are male or female on a piece of paper and put them in a hat. You pick out 10 pieces of paper and read them. 6 are female. Estimate how many females are there in total in the whole group? (Hint: The problem is one of uncertainty, and lends itself well to Bayesian analysis.)
Sum of two equal independent uniform distributions
What is the shape of the distribution of the sum of two independent, equal Uniform distributions?
Too easy to give a model answer to, but could you work it out mathematically? What would the answer be if the two Uniform distributions had different ranges?
100 people were exposed to some bacteria (eating food at a fair perhaps). The food was subsequently found to be contaminated and 10. They were all tested for the presence of infection by the bacteria.
Unfortunately, the testing method is not perfect. The test has a sensitivity of 0.65 (Sensitivity is the probability of testing positive given the patient is infected). The test has a specificity of 0.85 (Specificity is the probability of testing negative given the patient is not infected). 6 of the 10 people traced tested positive.
Estimate how many of the 100 people exposed are actually infected.
Two people agree to arrive at the town clock sometime tomorrow at some time between 1p.m. and 2p.m., and remain for 20 minutes. What is the probability that they will be there at the same moment? (Assume that there is equal probability that they will arrive at any particular time).
There are 20 virus particles in a bottle of water. The water is poured into glasses in equal amounts and one person drinks each glass of water. A single particle has a 30% probability of infecting a person. How many people would be infected if a) there were 5 glasses, b) there were 10 glasses?
