MT2504 Combinatorics and Probability
Academic year
2025 to 2026 Semester 1
Curricular information may be subject to change
Further information on which modules are specific to your programme.
Key module information
SCOTCAT credits
15
SCQF level
SCQF level 8
Planned timetable
11am Mondays (weeks 1, 3, 5, 8, 10) and Wednesdays and Fridays
Module Staff
Prof Colva Roney-Dougal; Dr Michail Papathomas
Module description
This module provides an introduction to the study of combinatorics and finite sets and also the study of probability. It will describe the links between these two areas of study. It provides a foundation both for further study of combinatorics within pure mathematics and for the various statistics modules that are available. It is recommended that students in the Faculties of Arts and Divinity take an even number of the 15-credit 2000-level MT modules.
Relationship to other modules
Pre-requisites
BEFORE TAKING THIS MODULE YOU MUST PASS MT1002,IF MT1002 HAS NOT BEEN PASSED, A AT ADVANCED HIGHER MATHEMATICS OR A AT A-LEVEL FURTHER MATHEMATICS
Assessment pattern
2-hour Written Examination = 70%, Coursework = 30%
Re-assessment
2-hour Written Examination = 100%
Learning and teaching methods and delivery
Weekly contact
2.5 hours of lectures (x 10 weeks), 1-hour tutorial (x 4 weeks), 1-hour examples class (x 5 weeks)
Scheduled learning hours
34
Guided independent study hours
116
Intended learning outcomes
- Identify, prove, and apply relevant formulae from lectures to solve problems involving counting sets, functions, permutations, tuples and multisets, and problems involving recursively-defined sequences
- State the axioms of probability. Calculate elementary probabilities, including conditional probabilities, appropriately use rules of probability, and be able to work with the concept of independence
- Define a random variable and associated distribution functions. Understand and work with discrete and continuous distributions to calculate probabilities, expectations and variances. State and apply the uniqueness theorem for probability and moment generating functions
- Demonstrate an understanding of multivariate distributions and associated distribution functions. Define and calculate expectations, variance, covariance and correlation for multiple random variables
- Demonstrate computational skills in Python through programming basic combinatorial procedures, and be able to apply these to a range of combinatorial and probabilistic problems
Additional information from school
For guidance on module choice at 2000-level in Mathematics and Statistics please consult the School Handbook, at /mathematics-statistics/students/ug/module-choices-2000/
MT2504 Combinatorics and Probability
Academic year
2026 to 2027 Semester 1
Curricular information may be subject to change
Further information on which modules are specific to your programme.
Key module information
SCOTCAT credits
15
SCQF level
SCQF level 8
Planned timetable
11am Mondays (weeks 1, 3, 5, 8, 10) and Wednesdays and Fridays
Module Staff
TBD
Module description
This module provides an introduction to the study of combinatorics and finite sets and also the study of probability. It will describe the links between these two areas of study. It provides a foundation both for further study of combinatorics within pure mathematics and for the various statistics modules that are available. It is recommended that students in the Faculties of Arts and Divinity take an even number of the 15-credit 2000-level MT modules.
Relationship to other modules
Pre-requisites
BEFORE TAKING THIS MODULE YOU MUST PASS MT1002,IF MT1002 HAS NOT BEEN PASSED, A AT ADVANCED HIGHER MATHEMATICS OR A AT A-LEVEL FURTHER MATHEMATICS
Assessment pattern
2-hour Written Examination = 70%, Coursework = 30%
Re-assessment
2-hour Written Examination = 100%
Learning and teaching methods and delivery
Weekly contact
2.5 hours of lectures (x 10 weeks), 1-hour tutorial (x 4 weeks), 1-hour examples class (x 5 weeks)
Scheduled learning hours
34
Guided independent study hours
116
Intended learning outcomes
- Identify, prove, and apply relevant formulae from lectures to solve problems involving counting sets, functions, permutations, tuples and multisets, and problems involving recursively-defined sequences
- State the axioms of probability. Calculate elementary probabilities, including conditional probabilities, appropriately use rules of probability, and be able to work with the concept of independence
- Define a random variable and associated distribution functions. Understand and work with discrete and continuous distributions to calculate probabilities, expectations and variances. State and apply the uniqueness theorem for probability and moment generating functions
- Demonstrate an understanding of multivariate distributions and associated distribution functions. Define and calculate expectations, variance, covariance and correlation for multiple random variables
- Demonstrate computational skills in Python through programming basic combinatorial procedures, and be able to apply these to a range of combinatorial and probabilistic problems
Additional information from school
For guidance on module choice at 2000-level in Mathematics and Statistics see our Module choices at 1000 and 2000 level page.
Syllabus
- Counting & elementary probability: definition of sets, unions of disjoint sets, pigeonhole principle, notation needed for probabilities (e.g., events, complement); axioms of probability and concept of probability using counting argument; Inclusion-Exclusion.
- Basic rules of probability (building on elementary counting), conditional probability, multiplication rule, Bayes Theorem, independence.
- Ordered pairs, double-counting, size of Cartesian products of sets, choosing with repetition; functions, permutations.
- Recursion and generating functions: Binomial numbers, recursively and via generating functions; Fibonacci numbers including some recursive formulae; Catalan numbers, including some recursive formulae.
- Random variables and distributions: definition of a discrete random variable (r.v.), probability mass functions, Bernoulli distribution, Binomial distribution, Poisson distribution, geometric distribution (including lack of memory property).
- Continuous r.v.s, probability density functions, uniform distribution; exponential distribution, normal distribution; cumulative distribution function (c.d.f., discrete & continuous cases), inverse c.d.f.
- Expectation; variance, introduction to probability generating functions; moment generating functions.
- Bivariate distributions: discrete/continuous distributions, joint, marginal and conditional probability mass/density functions; expectation, covariance and correlation, independence.