MAS3901 : Applied Probability

Semester 1 Credit Value: 10
ECTS Credits: 5.0


To develop skills in probabilistic reasoning and to gain familiarity with some of the main techniques involved in the analysis of random systems. To develop skills in the analysis of a variety of simple queues.

Module summary
How can we predict the chance of a gambler winning given a certain strategy? On average how much will they win? Uncertainty is a central feature of almost every real‐life problem, and questions of this nature arise naturally in many applications ranging from economics and finance through to engineering, sports betting and queues. In the first part of this course, we shall discover how certain probabilistic techniques can be used to model and analyse systems or phenomena that evolve randomly over time. In the second part of the course, we will look in detail at the analysis of queues. Queues occur in many situations. For example in telecommunications, traffic engineering, computing and in the design of factories, shops and hospitals all of which are subject to congestion. We will discuss how we construct and analyse models for systems of queues. Crucial questions are how congested any such system is likely to become over time and how we design such systems optimally.

Outline Of Syllabus

Review of probability ideas: conditioning arguments. Gambler's ruin problems, random walks. Markov chains: definition and examples, Chapman‐Kolmogorov equations, classification of states, notions of transience and recurrence, stationary distributions, simulation. Components of a queue; Poisson process; the M/M/1 queue; Birth-Death models; the M/M/c system; optimum number of servers; embedded Markov process; the M/G/1 queue; priority queues, acyclic queuing networks.

Teaching Methods

Teaching Activities
Category Activity Number Length Student Hours Comment
Scheduled Learning And Teaching ActivitiesLecture21:002:00Class tests
Guided Independent StudyAssessment preparation and completion113:0013:00Revision for unseen exam
Guided Independent StudyAssessment preparation and completion12:002:00Unseen exam
Scheduled Learning And Teaching ActivitiesLecture31:003:00Problem classes
Scheduled Learning And Teaching ActivitiesLecture21:002:00Revision lectures
Scheduled Learning And Teaching ActivitiesLecture241:0024:00Formal lectures
Scheduled Learning And Teaching ActivitiesDrop-in/surgery120:102:00Office hours
Guided Independent StudyIndependent study119:0019:00Studying, practising and gaining understanding of course material
Guided Independent StudyIndependent study33:009:00Review of tests and group project
Guided Independent StudyIndependent study112:0012:00Preparation for group project
Guided Independent StudyIndependent study26:0012:00Revision for class tests
Teaching Rationale And Relationship

Lectures are used for the delivery of theory and explanation of methods, illustrated with examples, and for giving general feedback on marked work. Problem Classes are used to help develop the students’ abilities at applying the theory to solving problems. Tutorials are used to identify and resolve specific queries raised by students and to allow students to receive individual feedback on marked work. In addition, office hours (two per week) will provide an opportunity for more direct contact between individual students and the lecturer: a typical student might spend a total of one or two hours over the course of the module, either individually or as part of a group.

Assessment Methods

The format of resits will be determined by the Board of Examiners

Description Length Semester When Set Percentage Comment
Written Examination1201A90N/A
Written Examination401M52 Class tests (test 1: 2%, test 2: 3%)
Other Assessment
Description Semester When Set Percentage Comment
Prob solv exercises1M5Group Project
Assessment Rationale And Relationship

A substantial formal unseen examination is appropriate for the assessment of the material in this module. The tests and the group project allow the students to develop their problem solving techniques, to practise the methods learnt in the module, to assess their progress and to receive feedback; these assessments have a secondary formative purpose as well as their primary summative purpose.

Reading Lists