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FV3102 Probabilistic Risk Analysis

FV3102 Probabilistic Risk Analysis (PRA)

Assignment Brief

Learning Outcomes

This piece of assessment will test your ability to meet learning outcomes 1, 2 and 3 as described in your module booklet: -

1. Apply Transforms to solve differential equations for engineering problems
2. Apply Linear programming and Markov modelling techniques to the solution of complex engineering problems 3. Critically evaluate probabilistic analysis techniques.

Assignment Details – Answer ALL Questions

Part A (64 Marks) (Learning Outcome 3)

A.1 There are 3 dice in a small box. The first die is unbiased; the second one is loaded and when tossed, the probability of obtaining 6 is 0.25, and the probability of obtaining each of the other faces is 0.15. The third die is also loaded, the probability of obtaining 6 is 0.4, and the probability of obtaining each of the other faces is 0.12.

1. A die is selected at random and tossed with the result of number 6, what is the probability that the selected die is the third die?
2. A die is selected at random and tossed twice, what is the probability that the product of the two rolls is NOT less than 24?
3. If 3 dice in the box are tossed simultaneously, what is the probability that the sum of the 3 dice is less than 16?

A.2 Urn A contains 4 Red and 4 Black balls, whereas urn B contains 3 Red and 5 White balls.

1. a ball is randomly selected from one of the two urns and found to be Red, what is the probability that the selected ball is from urn A?
2. Suppose we will select 4 balls one by one from any of the urns with replacement, what is the probability that the first 3 are Red and the last ball of different colour?
3. Suppose that we win $3 for each Red ball selected, lose $1 for each Black ball selected and lose $2 for each White ball selected. If two balls will be randomly selected from one of the urns without replacement, what is the probability that we will win the money?

A.3 Water pumps can be divided into high pressure, medium pressure and low pressure. Your company buys 30% of water pumps from supplier A, 50% from Supplier B and 20% from supplier C. Supplier A offers the three classes in the same amount. Supplier B offers the pumps of high, medium and low pressure in the ratio of 2:1:2 and Supplier C in the ratio of 1:2:3.

1. A water pump is chosen at random from your stock. What is the probability that the water pump was of high pressure?
2. Given that the pump chosen was of lower pressure, find the probability that it was made by supplier C.
3. Given that the pump chosen was of high pressure, what is the probability that the pump was NOT came from supplier A?

A.4 At a certain plant, the maximum number of accidents occurring every quarter was reported to 5 number. The past 10 years accidents were listed as below. Assume that the accidents occur independently every quarter.

1. List the probability distribution of the no. of accidents per quarter for the plant.
2. What is the probability that at least 2 accidents will occur in next quarter given that no accidents occur in this quarter?
3. What is the expected value and standard deviation of the no. of accidents per quarter?
4. To control the accidents rate, a penalty of $100,000 per accident will be incurred when 3 or less accidents occur in the quarter and a severe penalty of $600,000 in total will be incurred when 4 accidents occur and the highest penalty of $1,000,000 in total will be incurred when 5 accidents occur in the quarter. How much will be expected to pay for the penalty annually?

A.5 An engineering system consisting of 𝑛 components in parallel working is said to be a 𝑘-outof-𝑛 system, that is, the system functions well if and only if at least 𝑘 components from the total 𝑛 components operate normally without any failure. Suppose all the components works independently of each other.

1. For a 4-out-of-5 system, assume each component independently works with different The probability of 𝑖th component in normal working condition is assumed to be 𝑃_𝑖 = ^𝑖_𝑖⁺₊¹₂, 𝑖 = 1,2, … ,5, for example, the 1^st component (𝑖 = 1) has the probability of to operate normally. Compute the probability that the 4-out-of-5 system functions.
2. For a 2-out-of-4 system, assume each component independently works with the same probability 9, if we know the 2-out-of-4 system function well, what is the probability that at least 3 components could work in normal condition?

A.6 A manufacturer is concerned about a fault in a particular electrical device. The fault can, on rare occasions, cause a spike in voltage which can be harmful to the electrical device. The distribution of the number of electrical devices per year that will experience the fault is a Poisson random variable with average number of 3.

1. What is the probability that more than 2 electrical devices per year will experience the fault?
2. What is the probability that exactly 1 electrical device per year will experience the fault in successive 3 year?
3. Given that 5 electrical devices experience the fault in successive 4 years, what is the probability that at least 1 electrical device experiences the fault in the first year?

A.7 The lifetime of a laser printer is approximately normally distributed with a mean of 20,000 hours and standard deviation of 3,000 hours. Consider a batch of 1,000 laser printers,

1. How many printers in the batch would be expected to have operating life between 10,000 and 30,000 hours?
2. How many printers in the batch would be expected to survive longer than 25,000 hours?
3. Find the length of operating time in hours above which approximate 50 printers will survive?

A.8 The lifetime of a certain type of large machine is known to follow the exponential distribution with a mean of 10 years.

1. What is the probability that the machine will last at most 15 years?
2. What is the probability that the machine will NOT survive the first 5 years?
3. If 5 of these machines are installed in different systems, which is the probability that at least 2 are still functioning at the end of 12 years?

A.9 A warehouse develops a 𝑘-out-of-𝑁 detection and alarm system in which the fire alarm will be actuated once at least k detectors identify the fire occurrence and send the fire signals to the fire control panel for trigging the building fire alarm signal. All the detectors are identical and independently functioning of each other. For individual detector, it can not only miss the detection of real fire incidents in the area, but also falsely report fire incident while nothing happened in the warehouse. As a summary, two types of failures modes for each detector may be considered:

1. Fail dangerous: The detector fails to detect the real fire incidents and results in NO fire signal being sent to the fire control panel. Assume the probability that each detector being failed to send the fire signal when a fire breaks out is 0.01.
2. False alarm: The detector reports fire detection and sends out a fire signal to the fire control panel in absence of fire. Assume the probability that each detector gives a false alarm in absence of fire is 0.01.

As the designer, you are required to design the 𝑘-out-of-𝑁 detection and alarm system (i.e. find out the value of k and N). A well-designed system must be protected from both types of faulty operation. How will you propose the value of k and N to meet both of the following two requirements?

1. System fails to danger: The probability that the building alarm system fails to detect the real fire incidents is no more than 0.0005; and
2. System sends false alarm: The probability that the building alarm system reports the fire incident regardless the true state of the area is no more than 0.0005.

If the probability for both the two requirements will be changed as no more than 0.00001, how will you propose the value of k and N to meet the new requirements?

Part B (24 Marks) (Learning Outcome 1~2)

B.1 Use Laplace Transform method to solve the following initial-value problems:

• 𝑦^′′ − 2𝑦^′ − 3𝑦 = 𝑡𝑒^𝑡, 𝑦(0) = 0, 𝑦^′(0) = −1

• 𝑦^′′′ + 𝑦^′′ = 𝑒^𝑡 + 𝑡 + 1, 𝑦(0) = 0, 𝑦′(0) = 0, 𝑦′′(0) = 0

• 𝑥^′ = 𝑥 + 𝑧 𝑥(0) = 1, 𝑦(0) = 0, 𝑧(0) = 0

𝑦^′ = 𝑥 + 𝑦

𝑧^′ = −2𝑥 − 𝑧

B.2 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 ( 𝑋_𝑖(𝑡)) that the engineering system is in state 𝑖 at time 𝑡 (in hours) where 𝑖 = 0, 1,2:

𝑑𝑋^𝑜(𝑡⁾ + 0.03𝑋_𝑜(𝑡) = 0

𝑑𝑡

𝑑𝑋₁(𝑡) − 0.02𝑋𝑜(𝑡) = 0

𝑑𝑡

𝑑𝑋₂(𝑡)

_{{ 𝑑𝑡} − 0.01𝑋_𝑜(𝑡) = 0

At the initial time, we know that the engineering system is operating normally. Apply Laplace Transform to solve the engineering problem and find the probability that the engineering system will be out of services after 1,000 hours?

Part C (12 Marks) (Learning Outcome 2)

C.1 Use graphical method to solve the following linear programming problem: minimize 𝑓 subject to

𝑥

C.2 Use simplex tableau method to solve the following linear programming problem:

maximize 𝑓 subject to 𝑥 2

𝑥 -𝑥₁

𝑥

Marking Criteria for Assignment

The submitted assignment will be marked according to the following criteria: