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FV3102 Probabilistic Risk Analysis (PRA)
Assignment Brief (2024 – 2025)
The work shall be typed or word-processed in your own words. The deadline for submission is 11:59 p.m. (HKT) on 11 Nov 2024 (Mon).
Learning Outcomes
This piece of assessment will test your ability to meet learning outcomes as described hereunder:
Apply Transforms to solve differential equations for engineering problems (Learning Outcome 1)
Apply Linear programming and Markov modelling techniques to the solution of complex engineering problems (Learning Outcome 2)
Critically evaluate probabilistic analysis techniques (Learning Outcome 3)
Assignment Details
This assignment contains 8 questions. Answer all questions with words not exceeding 2,500. The assignment will carry 40% weighting of the total mark of this module.
Submission Details
(1) The deadline for submission is 11:59 p.m. (HKT) on 11 Nov 2024 (Mon). Late submission will be dealt with strictly in accordance with UCLan Regulations.
(2) All the assumptions and analysis steps in your answers should be clearly presented due to be awarded. The calculated results should be presented as fractions to maintain its precision. In cases where decimal values are involved, round the results to the nearest FOUR digits as possible.
(3) No hard copy is required to be submitted to the SCOPE counter. This assignment should be submitted through Turnitinto CityU SCOPE CANVAS assignment submission folder.
(4) In-text citations and referenced publications shall be added to the answer to each question.
(5) Using AI-generated text to complete your assignment is prohibited. Citation of Harvard style shall be used for all quoted references. UCLan regards any use of unfair means in an attempt to enhance performance or to influence the standard of award obtained as a serious academic and/or disciplinary offence.
(6) Submission of written assignment shall be type-written in .pdfor .docx format. The file name of your submission shall follow the format as the example below:
FV3102 CHAN Tai Man_G12345678
(7) Students should do whatever means to make sure the files are duly submitted via the CANVAS system and check whether the work is successfully uploaded (by downloading the file from CANVAS again). All claims on technical problem without strong evidence for unsuccessful uploading shall not be accepted.
(8) It should be the students’ responsibility to double-check the readability (pdf or docx format) of the submitted files.
(9) Administration team will not remove or replace student’s submitted assignment in CANVAS or help students to upload the soft copy of his/her assignments to CANVAS.
Q1. (Learning Outcome 3) Abox contains 4 red, 5 blue and 2 white balls. Four balls are selected from the box one by one without replacement.
a) What is the probability that exactly 3 balls are red?
b) What is the probability that the first and the last ball are white?
c) Suppose that we win $3 for each red ball selected, lose $1 for each blue ball selected and lose $2 for each white ball selected. What is the probability that we will win the money?
Q2. (Learning Outcome 3) For an electronics plant, the maximum number of fire incidents occurring every quarter was reported to be 5. Assume that the fire incidents occur independently every quarter. The probability of quartly fire inccidents is listed as below:
|
No. of Fire Incidents per Quarter |
1 |
2 |
3 |
4 |
5 |
|
Probability |
0.2 |
0.05 |
0.004 |
0.001 |
0.0001 |
a) What is the expected mean and standard deviation of quartly fire inccidents?
b) What is the probability that at least 2 fire incidents will occur in next quarter given that no fire incidents occur in this quarter?
c) To control the fire incident rate, a penalty of $50,000 per fire incident will be incurred when 3 or less fire incident occur in a quarter and a severe penalty of $300,000 in total will be incurred when 4 accidents occur and the highest penalty of $800,000 in total will be incurred when 5 accidents occur in a quarter. How much will be expected to pay for the penalty annually?
Q3. (Learning Outcome 3) Assume the average number of false alarms occurring at a certain commercial building is 2 per week. It is assumed that the Poisson process applies to the random variable “number of false alarms” .
a) What is the probability that there will be no false alarms in any particular week?
b) What is the probability of finding not more than 2 false alarms in a week?
c) What is the probability that totally 10 false alarms reported in a particular month and the building has at least 1 false alarm in the 1st week.
Q4. (Learning Outcome 3) The lifespan of fire extinguishers in a facility is normally distributed with a mean of 10 years and a standard deviation of 2 years. Assume that the facility has totally 100 fire extinguishers,
a) How many extinguishers will fail within 3 years?
b) How many extinguishers will be expected to operate longer than 12 years?
Q5. (Learning Outcome 3) Suppose the time between emergency calls to a fire station is exponentially distributed with a meantime between calls of 30 seconds.
a) What is the probability that the fire station will receive the next emergency call within 10 seconds?
b) What is the probability that the fire station will receive the next emergency call between 30 seconds to 1 min?
c) What is the probability that 2 calls arrived within first 30 seconds given that 4 calls arrived within 1 min?
Q6. (Learning Outcome 3) To minimize the possible false alarm, the building fire alarm system is designed as a k-out-of-N system, that is, at least k detectors identify the fire occurrence for trigging the building fire alarm signal. All the detectors are identical and independently functioning of each other. Two failures modes of detectors maybe considered:
i) Fail-to-danger: The detector is designed to detect the fire occurrence but fails to send the fire signal to the building fire alarm system and therefore results in “danger” situation. Assume the probability that each detector being failed to identify the fire occurrence is 0.02.
ii) False alarm: The detector sends out a fire signal in absence of fire. Assume the probability that each detector gives a false alarm in absence of fire is 0.02.
As the designer, you are required to design the building fire alarm system that meets the following requirement:
i) Fail-to-danger probability of the system: The probability of the system failing to detect a fire must be less than 0.0001.
ii) False alarm probability of the system: The probability of a false alarm in absence of fire must also be less than 0.0001.
You need to combine fail-to-danger and false alarm requirements and explore the minimum number of detectors and the design strategy (i.e. k-out-of-N) to ensure that both the probability are below the required threshold.
Q7. (Learning Outcome 1) Use Laplace Transform method to solve the following initial-value problem:
a) y ′′ + 2y′ + y = tet , y(0) = 0, y′(0) = 0
b) y ′′ + 1 = cos (t), y(0) = 0, y′(0) = 0
c) Assume that an engineering system can either be in the three states: i = 0 (operating normally), i = 1 (failed due to hardware problem), or i = 2 (failed due to human errors). The following set of differential equations describes the probability (Xi(t)) that the engineering system is in state i at time t (in hours) where i = 0, 1, 2:
At the initial time, we know that the engineering system is operating normally. Apply Laplace Transform to solve the engineering problem and and find out the mean time to system failure, MTTF = ∫∞0(t)dt
Q8. (Learning Outcome 2) Solve the below linear programming problems:
a) Use graphical method to solve the following linear programming problem:
minimize f = 5x1 + x2
subject to x1 + 2x2 ≤ 5
2x1 + 5x2 ≤ 18
x1 + x2 ≥ 3
x1 ≥ 0
x2 ≥ 1
b) Use simplex tableau method to solve the following linear programming problem:
maximize f = 3x1 + 2x2 + 4x3
subject to x1 + 2x2 + x3 ≤ 8
x1 + x3 ≤ 5
x1 + x2 + 2x3 ≤ 6
x1, x2, x3 ≥ 0
x1, x2, x3 ≥ 0
Marking Criteria
Marks will be allocated according to the following criteria:
|
Marking Criteria |
Marks allocation |
|
Knowledge of relevant material and grasp of themes: Students to use own words in demonstrating awareness and appreciation of key issues. |
20 |
|
Analysis, synthesis and depth of argument: Identification of key points and justified put forward clearly and succinctly. |
30 |
|
Engineering principle/calculation: Correct application of concept/formulae with complete accuracy and correct answer. |
40 |
|
Structure: Logical structure with introduction, background and executive summary. |
10 |
|
Total |
100 |