ST12010 Introduction to Probability
Introductory description
This module runs in Term 1 and is a core or listed optional module for some degree courses (primarily in Mathematics and Computer Science) and is also available as an unusual option to students on nonlisted degrees. You may be interested in this module if you wish to take further statistics modules.
Corequisites: MA132 Foundations and MA141 Analysis 1 (or equivalents)
Postrequisites: ST121 Statistical Laboratory, ST220 Introduction to Mathematical Statistics.
This module is not available to students who have their home department in Statistics, who take equivalent modules. Students who are considering transferring to a course in Data Science, Mathematics & Statistics or MORSE at the end of their first year should take this module.
Module aims
To lay the foundation for all subsequent modules in probability and statistics, by introducing the key notions of mathematical probability and developing the techniques for calculating with probabilities and expectations.
Outline syllabus
This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.
This module provides interdisciplinary coverage of mathematical techniques applying them to probabilistic methods. The module covers:
 Experiments with random outcomes, including the notions of events and their probability, operations with sets and their interpretations.
 Probability spaces, including the addition law and axiomatic definition of a probability space.
 Discrete probability spaces, including combinatorial probability methods such as the inclusionexclusion formula and multinomial coefficients.
 Continuous probability spaces, for example, points chosen uniformly at random in space.
 Conditioning and independence, including independence of events, conditional probabilities, the law of total probability and Bayes' theorem.
 Theory of Random variables, including discrete and continuous random variables, joint distributions, common families of distributions, expectation, variance and covariance.
 Moment generating functions, inequality theorems, central limit theorem.
Learning outcomes
By the end of the module, students should be able to:
 know and apply appropriate techniques to solve probabilistic problems;
 describe and interpret experiments with random outcomes using mathematical probability;
 know and interpret the concepts of conditional probability and independence;
 know and apply the theory of probability distributions, expectation, variance, and covariance associated with random variables;
 interpret problems and select appropriate distributions to create probability models.
Indicative reading list
Richard Durrett, (2009), Elementary Probability for Applications, Cambridge University Press, New York
Geoffrey Grimmett; D. J. A. Welsh, (2014), Probability  An Introduction, Oxford University, Oxford.
Geoffrey Grimmett, (2020) One Thousand Exercises in Probability. Third Edition, Oxford University Press.
View reading list on Talis Aspire
Subject specific skills
Demonstrate facility with advanced mathematical and probabilistic methods.
Select and apply appropriate mathematical and/or statistical techniques.
Demonstrate knowledge of key mathematical and statistical concepts, both explicitly and by applying them to the solution of mathematical problems.
Create structured and coherent arguments communicating them in written form.
Reason critically, carefully, and logically and derive (prove) mathematical results.
Transferable skills
Problem solving skills: The module requires students to solve problems presenting their conclusions as logical and coherent arguments.
Written communication: Written work requires precise and unambiguous communication in the manner and style expected in mathematical sciences.
Verbal communication: Dialogue with class tutors around problems prepared for each class.
Teaming working and working effectively with others: Students are encouraged to discuss and debate formative assessment and lecture material within smallgroup tutorials sessions.
Professionalism: Students work autonomously by developing and sustain effective approaches to learning, including time management, organisation, flexibility, creativity, collaboratively and intellectual integrity.
Study time
Type  Required  Optional 

Lectures  30 sessions of 1 hour (30%)  2 sessions of 1 hour 
Seminars  4 sessions of 1 hour (4%)  
Private study  54 hours (54%)  
Assessment  12 hours (12%)  
Total  100 hours 
Private study description
Weekly revision of lecture notes and materials, wider reading and practice exercises working on problem sets and preparing for the examination.
Costs
No further costs have been identified for this module.
You do not need to pass all assessment components to pass the module.
Assessment group D
Weighting  Study time  

Problem set 1  2%  2 hours 
A problem sheet that include problem solving and calculations. Problem sheets will be set at fortnightly intervals. The problem sheets will contain a number of questions for which solutions and / or written responses will be required. The preparation and completion time noted below refers to the amount of time in hours that a wellprepared student who has attended lectures and carried out an appropriate amount of independent study on the material could expect to spend on this assignment. 

Problem set 2  3%  3 hours 
A problem sheet that include problem solving and calculations. Problem sheets will be set at fortnightly intervals. The problem sheets will contain a number of questions for which solutions and / or written responses will be required. The preparation and completion time noted below refers to the amount of time in hours that a wellprepared student who has attended lectures and carried out an appropriate amount of independent study on the material could expect to spend on this assignment. 

Problem set 3  2%  2 hours 
A problem sheet that include problem solving and calculations. Problem sheets will be set at fortnightly intervals. The problem sheets will contain a number of questions for which solutions and / or written responses will be required. The preparation and completion time noted below refers to the amount of time in hours that a wellprepared student who has attended lectures and carried out an appropriate amount of independent study on the material could expect to spend on this assignment. 

Problem set 4  3%  3 hours 
A problem sheet that include problem solving and calculations. Problem sheets will be set at fortnightly intervals. The problem sheets will contain a number of questions for which solutions and / or written responses will be required. The preparation and completion time noted below refers to the amount of time in hours that a wellprepared student who has attended lectures and carried out an appropriate amount of independent study on the material could expect to spend on this assignment. 

Inperson Examination  90%  2 hours 
You will be required to answer all questions on this examination paper.

Assessment group R
Weighting  Study time  

Inperson Examination  Resit  100%  
You will be required to answer all questions on this examination paper.

Feedback on assessment
Individual feedback will be provided on problem sheets by class tutors. A cohortlevel feedback will be available for the examination. Students are actively encouraged to make use of office hours to build up their understanding, and to view all their interactions with lecturers and class tutors as feedback.
Courses
This module is Core for:
 Year 1 of UCSAG4G1 Undergraduate Discrete Mathematics
 Year 1 of UCSAG4G3 Undergraduate Discrete Mathematics
 Year 1 of UMAAG105 Undergraduate Master of Mathematics (with Intercalated Year)

UMAAG100 Undergraduate Mathematics (BSc)
 Year 1 of G100 Mathematics
 Year 1 of G100 Mathematics
 Year 1 of G100 Mathematics

UMAAG103 Undergraduate Mathematics (MMath)
 Year 1 of G100 Mathematics
 Year 1 of G103 Mathematics (MMath)
 Year 1 of G103 Mathematics (MMath)
 Year 1 of UMAAG106 Undergraduate Mathematics (MMath) with Study in Europe
 Year 1 of UMAAG1NC Undergraduate Mathematics and Business Studies
 Year 1 of UMAAG1N2 Undergraduate Mathematics and Business Studies (with Intercalated Year)
 Year 1 of UMAAGL11 Undergraduate Mathematics and Economics
 Year 1 of UECAGL12 Undergraduate Mathematics and Economics (with Intercalated Year)

UMAAGV17 Undergraduate Mathematics and Philosophy
 Year 1 of GV17 Mathematics and Philosophy
 Year 1 of GV17 Mathematics and Philosophy
 Year 1 of GV17 Mathematics and Philosophy
 Year 1 of UMAAG101 Undergraduate Mathematics with Intercalated Year