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: 
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.
A.2 Urn A contains 4 Red and 4 Black balls, whereas urn B contains 3 Red and 5 White balls.
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.
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.
No. of accidents 

1st Quarter 
2nd Quarter 
3rd Quarter 
4th Quarter 

year 1 
 
1 
 
2 
year 2 
1 
2 
 
4 
year 3 
 
 
1 
3 
year 4 
 
2 
1 
 
year 5 
3 
 
 
1 
year 6 
 
1 
2 
3 
year 7 
 
1 
4 
 
year 8 
2 
 
1 
 
year 9 
 
2 
1 
5 
year 10 
1 
 
 
3 
A.5 An engineering system consisting of π components in parallel working is said to be a πoutofπ system, that is, the system functions well if and only if at least π components from the total π components operate normally without any failure. Suppose all the components works independently of each other.
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.
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,
A.8 The lifetime of a certain type of large machine is known to follow the exponential distribution with a mean of 10 years.
A.9 A warehouse develops a πoutofπ 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:
As the designer, you are required to design the πoutofπ detection and alarm system (i.e. find out the value of k and N). A welldesigned 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?
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?
B.1 Use Laplace Transform method to solve the following initialvalue problems:
π¦^{′ }= π₯ + π¦
π§^{′ }= −2π₯ − π§
B.2 Assume that an engineering system can either be in the three states: π = 0 (operating normally), π = 1 (failed due to hardware problem), or π = 2 (failed due to human errors). The following set of differential equations describes the probability ( π_{π}(π‘)) that the engineering system is in state π at time π‘ (in hours) where π = 0, 1,2:
ππ^{π}^{(}π‘^{) }+ 0.03π_{π}(π‘) = 0
ππ‘
ππ_{1}(π‘) − 0.02ππ(π‘) = 0
ππ‘
ππ_{2}(π‘)
_{{ }_{ππ‘ }− 0.01π_{π}(π‘) = 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?
C.1 Use graphical method to solve the following linear programming problem: minimize π subject to
π₯
C.2 Use simplex tableau method to solve the following linear programming problem:
maximize π subject to π₯ 2
π₯ π₯_{1}
π₯
Marking Criteria for Assignment
The submitted assignment will be marked according to the following criteria:
Questions 
Marking Allocation 
Marking Criteria 
Part A (A.1 A.9) 
64 
Demonstrate the application of probabilistic analysis techniques to solve engineering problems. 
Part B (B.1B.2) 
24 
Apply Laplace Transform and Markov modelling techniques to solve differential equations for engineering problems 
Part C (C.1C.2) 
12 
Demonstrate the application of Graphic methods and Simplex tableau method to solve the linear programming problems. 
Total 
100 
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