## APP MTH 4101 - Applied Probability - Honours

### North Terrace Campus - Semester 1 - 2024

Many processes in the real world involve some random variation superimposed on a deterministic structure. For example, the experiment of flipping a coin is best studied by treating the outcome as a random one. Mathematical probability has its origins in games of chance with dice and cards, originating in the fifteenth and sixteenth centuries. This course aims to provide a basic tool kit for modelling and analysing discrete-time problems in which there is a significant probabilistic component. We will consider Markov chain examples in the course including population branching processes (with application to genetics), random walks (with application to games), and more general discrete time examples using Martingales. Topics covered are: basic probability and measure theory, discrete time Markov chains, hitting probabilities and hitting time theorems, population branching processes, homogeneous random walks on the line, solidarity properties and communicating classes, necessary and sufficient conditions for transience and positive recurrence, global balance, partial balance, reversibility, Martingales, stopping times and stopping theorems with a link to Brownian motion.

• General Course Information
##### Course Details
Course Code APP MTH 4101 Applied Probability - Honours Mathematical Sciences Semester 1 Undergraduate North Terrace Campus 3 Up to 3 hours per week Y (MATHS 1012 or MATHS 2103) or (MATHS 2201 and MATHS 2202) or (MATHS 2106 and MATHS 2107) APP MTH 3001 Knowledge of Markov Chains such as would be obtained from MATHS 2103 Honours students only Many processes in the real world involve some random variation superimposed on a deterministic structure. For example, the experiment of flipping a coin is best studied by treating the outcome as a random one. Mathematical probability has its origins in games of chance with dice and cards, originating in the fifteenth and sixteenth centuries. This course aims to provide a basic tool kit for modelling and analysing discrete-time problems in which there is a significant probabilistic component. We will consider Markov chain examples in the course including population branching processes (with application to genetics), random walks (with application to games), and more general discrete time examples using Martingales. Topics covered are: basic probability and measure theory, discrete time Markov chains, hitting probabilities and hitting time theorems, population branching processes, homogeneous random walks on the line, solidarity properties and communicating classes, necessary and sufficient conditions for transience and positive recurrence, global balance, partial balance, reversibility, Martingales, stopping times and stopping theorems with a link to Brownian motion.
##### Course Staff

Course Coordinator: Dr Andrew Black

##### Course Timetable

The full timetable of all activities for this course can be accessed from Course Planner.

• Learning Outcomes
##### Course Learning Outcomes
1. demonstrate understanding of the mathematical basis of discrete-time Markov chains and martingales
2. demonstrate the ability to formulate discrete-time Markov chain models for relevant practical systems
3. demonstrate the ability to apply the theory developed in the course to problems of an appropriate level of difficulty
4. demonstrate the ability to conduct a group project applying the theory developed in this course
5. develop an appreciation of the role of applied probability in mathematical modelling
6. demonstrate skills in communicating mathematics orally and in writing

This course will provide students with an opportunity to develop the Graduate Attribute(s) specified below:

University Graduate Attribute Course Learning Outcome(s)

Attribute 1: Deep discipline knowledge and intellectual breadth

Graduates have comprehensive knowledge and understanding of their subject area, the ability to engage with different traditions of thought, and the ability to apply their knowledge in practice including in multi-disciplinary or multi-professional contexts.

1,2,3,4

Attribute 2: Creative and critical thinking, and problem solving

Graduates are effective problems-solvers, able to apply critical, creative and evidence-based thinking to conceive innovative responses to future challenges.

1,2,3,4

Attribute 3: Teamwork and communication skills

Graduates convey ideas and information effectively to a range of audiences for a variety of purposes and contribute in a positive and collaborative manner to achieving common goals.

4,5,6

Graduates engage in professional behaviour and have the potential to be entrepreneurial and take leadership roles in their chosen occupations or careers and communities.

3,4,5,6

Attribute 5: Intercultural and ethical competency

Graduates are responsible and effective global citizens whose personal values and practices are consistent with their roles as responsible members of society.

4,6

Attribute 8: Self-awareness and emotional intelligence

Graduates are self-aware and reflective; they are flexible and resilient and have the capacity to accept and give constructive feedback; they act with integrity and take responsibility for their actions.

4
• Learning Resources
None.
##### Recommended Resources
There are many good books on probability and statistics in the Barr Smith Library, with the following texts being recommended for this course.

1. "Probability and Random Processes" (Oxford, 2001).
2. "Introduction to Probability Models" by Sheldon Ross (Academic Press, 2010).
3. "An introduction to Stochastic Modelling" by Taylor and Karlin (Academic Press, 1998).
4. "A First Course in Stochastic Processes" by Karlin and Taylor (Academic Press, 1975).
5. "Elementary Probability Theory with Stochastic Processes" by Kai Lai Chung (Springer-Verlag, 1975).
6. "An Introduction to Probability Theory and its Applications" by Feller (Wiley, 1968).
7. "Introduction to Stochastic Models" by Roe Goodman (2nd edition, Dover, 2006).
8. "Markov chains" by James Norris (Cambridge, 1997).

For other texts on probability and statistics, try browsing books with call numbers beginning with 519.2.
##### Online Learning
This course uses MyUni exclusively for providing electronic resources, such as notes, videos, quizzes, assignments and solutions et cetera.
• Learning & Teaching Activities
##### Learning & Teaching Modes
Each week, lecture notes will be provided, designed to be read in advance of viewing videos. Videos will consist of the lecturer explaining key material and examples from the lecture notes.

The videos will be supported by two weekly classes, a tutorial and a workshop. In the workshop the lecturer will guide you through the week’s material, incorporating active learning exercises, whilst the tutorial is focused on practicing problems to reinforce this learning.

Five written assignments and five online quizzes provide the assessment opportunities for students to strengthen their understanding of the theory and their skills in applying it, and gauge and demonstrate their progress and understanding.

The mid-semester test allows students to guage their progress with mastering the course material.

Interaction with the lecturer is encouraged during contact hours.

The information below is provided as a guide to assist students in engaging appropriately with the course requirements.

 Activity Quantity Workload hours Weekly online materials 12 weeks 70 Tutorials 12 24 Workshops 12 12 Online quizzes 5 5 Assignments 5 30 Test revision 1 15 Total 156
##### Learning Activities Summary
 Week 1 Basic probability theory Sample space and events. Laws of large numbers and the central limit theorem and their interpretation. Week 2 Basic probability theory. Algebras and sigma-algebras of events and probability measure. Week 3 Discrete time Markov chains Definition of a discrete time Markov chain (DTMC). Random walks. Week 4 Discrete time Markov chains Hitting probabilities and hitting times. Classification of states. Week 5 Discrete time Markov chains Recurrence and transience. Week 6 Discrete time Markov chains Irreducible DTMCs. Branching processes. Periodicity. Week 7 Discrete time Markov chains Limiting behaviour. Long term behaviour and global balance. Week 8 Discrete time Markov chains. Partial balance. Time reversal and reversibility. Week 9 Martingales Definition of a martingale. Fair games, branching processes and random walks. Week 10 Martingales Stopping times and optional stopping theorem. Dominated martingales and Optional stopping times. Week 11 Martingales Two dimensional random walks. Identifying martingales. Week 12 Martingales and Brownian motion Sub-martingales, super-martingales and construction of martingales. Motivation and definition of Brownian motion with examples. Review.

Tutorials will cover the content of the previous week, while the first tutorial in Week 1 will focus on revision.
• Assessment

The University's policy on Assessment for Coursework Programs is based on the following four principles:

1. Assessment must encourage and reinforce learning.
2. Assessment must enable robust and fair judgements about student performance.
3. Assessment practices must be fair and equitable to students and give them the opportunity to demonstrate what they have learned.
4. Assessment must maintain academic standards.

##### Assessment Summary
 Assessment task When due Weighting Learning outcomes Examination Examination period 60% All Assignments Weeks 2, 4, 8, 10 and 12 20% All Mid-semester test Week 6 15% All Quizzes Weeks 3, 5, 7, 9 and 11 5% All

##### Assessment Related Requirements
An aggregate score of 50% is required to pass the course. Furthermore students must achieve at least 40% on the final examination to pass the course.
##### Assessment Detail

 Task Week released Week due Weight Assignment 1 1 2 4% Assignment 2 3 4 4% Mid-semester test 6 15% Assignment 3 7 8 4% Assignment 4 9 10 4% Assignment 5 11 12 4%
Quizzes are due in weeks 3, 5, 7, 9 and 11 and the combined marks contribute a total of 5% of the overall grade. The quizzes are primarily intended to be formative assessment and multiple attempts are allowed.
##### Submission
Assignments must be submitted on time and online via MyUni. Late assignments will not be accepted. Students may be excused from an assignment for medical or compassionate reasons. In such cases, documentation is required and the lecturer must be notified as soon as possible.

Grades for your performance in this course will be awarded in accordance with the following scheme:

M11 (Honours Mark Scheme)
Fail A mark between 1-49 F
Third Class A mark between 50-59 3
Second Class Div B A mark between 60-69 2B
Second Class Div A A mark between 70-79 2A
First Class A mark between 80-100 1
Result Pending An interim result RP
Continuing Continuing CN

Further details of the grades/results can be obtained from Examinations.

Grade Descriptors are available which provide a general guide to the standard of work that is expected at each grade level. More information at Assessment for Coursework Programs.

Final results for this course will be made available through Access Adelaide.

• Student Feedback

The University places a high priority on approaches to learning and teaching that enhance the student experience. Feedback is sought from students in a variety of ways including on-going engagement with staff, the use of online discussion boards and the use of Student Experience of Learning and Teaching (SELT) surveys as well as GOS surveys and Program reviews.

SELTs are an important source of information to inform individual teaching practice, decisions about teaching duties, and course and program curriculum design. They enable the University to assess how effectively its learning environments and teaching practices facilitate student engagement and learning outcomes. Under the current SELT Policy (http://www.adelaide.edu.au/policies/101/) course SELTs are mandated and must be conducted at the conclusion of each term/semester/trimester for every course offering. Feedback on issues raised through course SELT surveys is made available to enrolled students through various resources (e.g. MyUni). In addition aggregated course SELT data is available.

• Student Support
• Policies & Guidelines
• Fraud Awareness

Students are reminded that in order to maintain the academic integrity of all programs and courses, the university has a zero-tolerance approach to students offering money or significant value goods or services to any staff member who is involved in their teaching or assessment. Students offering lecturers or tutors or professional staff anything more than a small token of appreciation is totally unacceptable, in any circumstances. Staff members are obliged to report all such incidents to their supervisor/manager, who will refer them for action under the university's studentâ€™s disciplinary procedures.

The University of Adelaide is committed to regular reviews of the courses and programs it offers to students. The University of Adelaide therefore reserves the right to discontinue or vary programs and courses without notice. Please read the important information contained in the disclaimer.

```
```