Module: Probability

Module Specifications

Academic Year 2026 - 2027

Module Title	Probability
Module Code	MTH1098
Faculty	Science & Health	School	Mathematical Sciences
NFQ level	8	Credit Rating	7.5

Description

Probability 1 aims to introduce the basic concepts of probability theory through lectures and problem solving based tutorials. The module will give students a working knowledge of the main techniques of elementary probability and build a solid foundation for learning more advanced topics in probability and statistics.

Learning Outcomes

1. Define elementary concepts of probability and state the main theorems.
2. Use counting techniques to assign probabilities to events.
3. Compute and apply conditional probabilities.
4. Derive the basic properties of common discrete and continuous distributions.
5. Express probabilities in terms of multiple integrals and be able to evaluate such integrals.
6. Understand and apply limit theorems relating to sequences of independent random variables.

*Type*	*Hours*	*Description*
Workload	Full time hours per semester
Lecture	36	Presentation of course material.
Tutorial	24	2-hour tutorial per week.
Independent Study	65	Revising coursework, solving tutorials and exam preparation.
Total Workload: 125

Section Breakdown
CRN	21500	Part of Term	Semester 2
Coursework	20%	Examination Weight	80%
Grade Scale	40PASS	Pass Both Elements	N
Resit Category	RC3	Best Mark	Y
Module Co-ordinator	Martin Venker	Module Teacher

Type	Description	% of total	Assessment Date
Assessment Breakdown
In Class Test	n/a	20%	As required
Formal Examination	End-of-Semester Final Examination	80%	End-of-Semester

Reassessment Requirement Type
Resit arrangements are explained by the following categories; RC1: A resit is available for both^* components of the module. RC2: No resit is available for a 100% coursework module. RC3: No resit is available for the coursework component where there is a coursework and summative examination element. ^* ‘Both’ is used in the context of the module having a coursework/summative examination split; where the module is 100% coursework, there will also be a resit of the assessment

Pre-requisite	None
Co-requisite	None
Compatibles	None
Incompatibles	None

All module information is indicative and subject to change. For further information,students are advised to refer to the University's Marks and Standards and Programme Specific Regulations at: http://www.dcu.ie/registry/examinations/index.shtml

Indicative Content and Learning Activities

Modelling Chance
Probability spaces, discrete, continuous and mixed probability measures, distribution functions, random variables.

Conditional Probabilities and Independence
Tree diagrams, Law of Total Probability, Bayes' Theorem, independence of two event, independents of families of events, independence of random variables.

Combinatorics
Fundamental principles, permutations and combinations with and without repetitions, urn and box models.

Discrete and Continuous Distributions
Uniform, Dirac, product, Bernoulli, Binomial, Multinomial, Hypergeometric, Multivariate Hypergeometric, Negative Binomial, Negative Hypergeometric, Poisson, Gamma (including Exponential), Pareto, Normal.

Characteristics of Random Variables
Expectation, median, variance, standard deviation, moments, moment-generating-function.

Random Vectors
Joint densities, marginal densities, conditional densities, transformations, covariance matrix, sums of independent random variables, the multivariate normal distribution.

Limit Theorems
Poisson and normal approximations for the binomial distribution, approximation of Gamma distributions by Negative Binomial Distributions, Laws of Large Numbers, Central Limit Theorem.

Indicative Reading List

Books:

Hans-Otto Georgii: 2008, Stochastics: Introduction to Probability and Statistics., de Gruyter,
Geoffrey Grimmett and David Stirzaker: 2001, Probability and Random Processes, 3rd edition, Oxford University Press, Oxford,
Kai Lai Chung: 2003, Elementary Probability Theory with Stochastic Processes and an Introduction to Mathematical Finance, 4th edition, Springer, New York,
Richard Durrett: 1994, The Essentials of Probability, Duxbury Press, Belmont,
William Feller: 1971, An Introduction to Probability and its Applications, 3rd edition, Wiley, New York,
A. N. Shiryaev: 1996, Probability, 2nd edition, 1. chapter, Springer, New York,
Henk Tijms: 2007, Understanding Probability: Chance Rules in Everyday Life, 2nd edition, Cambridge University Press, Cambridge,
David Williams: 2001, Weighing the Odds: A Course in Probability and Statistics, Cambridge University Press, Cambridge,
Sheldon M. Ross: 2010, A first course in probability, 8th edition, Pearson, Englewood Cliffs,
Peter L. Bernstein: 1996, Against the Gods: The Remarkable Story of Risk, John Wiley & Sons, New York,

Articles:
None

Other Resources

None