MA32048: Discrete probability
[Page last updated: 22 May 2025]
| Academic Year: | 2025/26 |
| Owning Department/School: | Department of Mathematical Sciences |
| Credits: | 5 [equivalent to 10 CATS credits] |
| Notional Study Hours: | 100 |
| Level: | Honours (FHEQ level 6) |
| Period: |
- Semester 2
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| Assessment Summary: | EXCB 100% |
| Assessment Detail: |
- Closed-book written examination (EXCB 100%)
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| Supplementary Assessment: |
- Like-for-like reassessment (where allowed by programme regulations)
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| Requisites: |
Before taking this module you must take MA22039 OR take MA20225 OR take MA20301
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| Learning Outcomes: |
By the end of the unit, you will be able to:
- Describe the giant component phenomenon in random graphs and the phase transition phenomenon in percolation theory;
- Apply tools such as the first and second moment method and the branching process approximation;
- Perform simple calculations on discrete probabilistic structures;
- Compute percolation thresholds on graphs such as trees;
- Explain ideas behind theorems such as the infinite cluster's uniqueness.
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| Synopsis: | In this unit you will focus on some discrete probabilistic structures with relevance to networks, statistical physics and communications - in particular, random graphs and percolation. You will learn about some of the key results on these structures and the mathematical ideas and techniques behind these results. You will become acquainted with notions of phase transition, spatial and combinatorial methods in probability theory.
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| Content: | Random Graphs: definition and motivation, small subgraphs; clique number, threshold functions, connectivity, giant component and phase transition.
Percolation Theory: non-triviality of the phase transition, properties and interpretation of the percolation probability, percolation on trees. Optional topics in percolation: uniqueness of the infinite component, the critical value on the square lattice, lattice animals.
Further topics in discrete probability may be considered such as: random walks on graphs and connection to electrical networks; mixing times for Markov chains; introductory information theory.
For some theorems the proofs will be optional reading material'.
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| Course availability: |
MA32048 is Optional on the following courses:
Department of Computer Science
- USCM-AFB32 : BSc(Hons) Computer Science and Mathematics (Year 3)
- USCM-AAB20 : BSc(Hons) Computer Science and Mathematics with Study year abroad (Year 4)
- USCM-AKB20 : BSc(Hons) Computer Science and Mathematics with Year long work placement (Year 4)
Department of Economics
- UHES-AFB12 : BSc(Hons) Economics and Mathematics (Year 3)
- UHES-AAB04 : BSc(Hons) Economics and Mathematics with Study year abroad (Year 4)
- UHES-AKB04 : BSc(Hons) Economics and Mathematics with Year long work placement (Year 4)
- UHES-ACB04 : BSc(Hons) Economics and Mathematics with Combined Placement and Study Abroad (Year 4)
Department of Mathematical Sciences
- USMA-AAB16 : BSc(Hons) Mathematical Sciences with Study year abroad (Year 4)
- USMA-AKB16 : BSc(Hons) Mathematical Sciences with Year long work placement (Year 4)
- USMA-AFB30 : BSc(Hons) Mathematics (Year 3)
- USMA-AAB14 : BSc(Hons) Mathematics with Study year abroad (Year 4)
- USMA-AKB14 : BSc(Hons) Mathematics with Year long work placement (Year 4)
- USMA-AFB32 : BSc(Hons) Mathematics and Statistics (Year 3)
- USMA-AAB02 : BSc(Hons) Mathematics and Statistics with Study year abroad (Year 4)
- USMA-AKB02 : BSc(Hons) Mathematics and Statistics with Year long work placement (Year 4)
- USMA-AFB33 : BSc(Hons) Mathematics, Statistics and Data Science (Year 3)
- USMA-AAB20 : BSc(Hons) Mathematics, Statistics, and Data Science with Study year abroad (Year 4)
- USMA-AKB20 : BSc(Hons) Mathematics, Statistics, and Data Science with Industrial Placement (Year 4)
- USMA-AFM30 : MMath(Hons) Mathematics (Year 3)
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Notes: - This unit catalogue is applicable for the 2025/26 academic year only. Students continuing their studies into 2026/27 and beyond should not assume that this unit will be available in future years in the format displayed here for 2025/26.
- 好色tv and units are subject to change in accordance with normal University procedures.
- Availability of units will be subject to constraints such as staff availability, minimum and maximum group sizes, and timetabling factors as well as a student's ability to meet any pre-requisite rules.
- Find out more about these and other important University terms and conditions here.
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