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Introduction to Probability

Code: 104846 ECTS Credits: 6
Degree Type Year Semester
2503852 Applied Statistics FB 1 1


Rosario Delgado de la Torre

Use of Languages

Principal working language:
catalan (cat)
Some groups entirely in English:
Some groups entirely in Catalan:
Some groups entirely in Spanish:


As a subject of the first semester of the first year, it has no prerequisites except to take the subject Calculus 1 simultaneously.

To a lesser degree, it may also be convenient to take the Computer Tools for Statistics course at the same time.

Objectives and Contextualisation

What does a lottery draw have in common, a clinical trial to experimentally evaluate the efficacy and / or safety of a new medical treatment, the weather forecast of rain in a certain place, the management of a company's inventory, the transmission of genes from parents to children, the estimation of the size of the population of whales, an epidemiological study on the incidence of a certain disease, the inspection of the batches of products manufactured by a company to verify its quality, an experiment to study the effect of pressure and temperature in the result of a certain chemical reaction, or the effect of the use of different fertilizers in the agricultural production of a farm, ...?

They are real situations in which randomness intervenes.

To study them and be able to draw reliable conclusions, we have to use an appropriate mathematical model. This model is provided by Probability, which is the mathematical theory that allows the modeling of random phenomena, that is, situations where chance intervenes.

The objective of this subject is to introduce Probability, which studies the models that allow dealing with chance, and is fundamental in Statistics. The topics that will be introduced and developed in this subject include basic contents of Probability, which will be expanded and on which will be deepened in the subject "Probability" of the second semester, putting the emphasis on applications, among which the Statistics stand out. In the applications one should try to find the best possible probabilistic model in a given real situation and, using it in an appropriate way, with the tools that we will learn throughout the course, extract valuable information, knowledge, and reach useful conclusions.


  • Calculate and reproduce certain mathematical routines and processes with agility.
  • Critically and rigorously assess one's own work as well as that of others.
  • Make efficient use of the literature and digital resources to obtain information.
  • Select and apply the most suitable procedures for statistical modelling and analysis of complex data.
  • Students must be capable of applying their knowledge to their work or vocation in a professional way and they should have building arguments and problem resolution skills within their area of study.
  • Students must be capable of communicating information, ideas, problems and solutions to both specialised and non-specialised audiences.
  • Use quality criteria to critically assess the work done.

Learning Outcomes

  1. Critically assess the work done on the basis of quality criteria.
  2. Distinguish deterministic models from probabilistic-statistical models.
  3. Make effective use of references and electronic resources to obtain information.
  4. Reappraise one's own ideas and those of others through rigorous, critical reflection.
  5. Recognise the usefulness of mathematical methods (calculus, algebra, numerical methods) for probabilistic modelling.
  6. Students must be capable of applying their knowledge to their work or vocation in a professional way and they should have building arguments and problem resolution skills within their area of study.
  7. Students must be capable of communicating information, ideas, problems and solutions to both specialised and non-specialised audiences.
  8. Use probabilistic models to describe data in contexts of uncertainty and deduce behaviour patterns.


1. Probabilistic models.

  • Introduction.
  • Random phenomena.
  • Probability spaces.
  • Properties of probability.
  • Counting elements of a set: a little combinatorics.

2. Conditioned probability.

  • Definition of conditioned probability.
  • Independence of events.
  • Properties of the independence of events.
  • The Formula of the Total Probability.
  • The Bayes' Formula.

2. Random variables.

  • What is a random variable (r.v.)?
  • Distribution function of a random variable.
  • Discrete random variables.
  • (Absolutely) continuous random variables. 
  • Independence of random variables.

3. Mathematical Expectation and Variance.

  • Expectation of a random variable.
  • Variance of a random variable.
  • Covariance of two random variables.

*Unless the requirements enforced by the health authorities demand a prioritization or reduction of these contents.

IMPORTANT: To include the gender perspective in the teaching of this subject, we have reviewed the
possible androcentric biases and questioned hidden gender assumptions and stereotypes.
This revision involves including the contents of the subject, as far as possible, knowledge
produced by women scientists, often forgotten, seeking the recognition of
his contributions, as well as that of his works in the bibliographical references.


In this subject, the classical distinction is not made in the classroom activities of: theoretical classes, problems and practices with a computer, but will be combined according to the educational needs at each moment, thanks to the ease faced by the fact that students bring their computer to class.

In this way, the teacher will introduce the concepts and examples, while when appropriate the problems will be worked on in class or the statistical software and programming language R will be used to carry out some practice related to the subject that is " be working in class. It involves using an integral system that incorporates the three classical aspects of the classroom activities in an optimal way to facilitate the student's learning and achieve the goals set, while making the class as participatory as possible, following the The beginning of which you only learn what you try to do.

*The proposed teaching methodology may experience some modifications depending on the restrictions to face-to-face activities enforced by health authorities.

The students can communicate with the teacher through the email, always sent from the institutional address @e-campus.uab.cat.

IMPORTANT: To work more comfortably with R, it is recommended to use the RStudio interface: it's free, "open source" and works with Windows, Mac and Linux.

OBSERVATION: Although we have already talked about the gender perspective in teaching in the section of the contents of the subject, we go further by doing a review of the teaching methodology and the interactions between studentsand teachers. In this sense, a participatory teaching methodology will be implemented, generating an egalitarian environment, less hierarchical in the classroom, avoiding stereotypical examples in gender and sexist vocabulary, with the objective of developing critical reasoning and respect for diversity and A plurality of ideas, people and situations, which will be more favorable to the integration and full participation of the students.

Annotation: Within the schedule set by the centre or degree programme, 15 minutes of one class will be reserved for students to evaluate their lecturers and their courses or modules through questionnaires.


Title Hours ECTS Learning Outcomes
Type: Directed      
Problems in the classroom 18 0.72 4, 7, 6, 8
Theory in the classroom 26 1.04 4, 1, 2, 7, 6, 5, 3, 8
Type: Supervised      
Practical sessions 8 0.32 1, 7, 6, 3
Type: Autonomous      
Personal work 89 3.56 4, 1, 2, 7, 6, 5, 3, 8


The evaluation of this subject will consist of:

Continued avaluation:
• Two continuous assessment tests (PAC), with a weight of 20% each (total = 40%)
• Problem examination (EProb), with 50% weight.
• Examination of practices with R (EPract), with a 10% of weight.

Therefore, Grade1 = 0.2 * PAC1 + 0.2 * PAC2 + 0.5 * Eprob + 0.1 * EPract.

If Grade1> = 5, the student passes the subject. Otherwise, you have the opportunity to take the grade recoveri exam, which does NOT serve to improve your grade nor to obtain "Matrícula d'Honor".

Recovery exam (ERec): worth 80% of the grade. The practice exam with R will be worth the remaining 20% of the grade.

Therefore, Grade2 = 0.8 * ERec + 0.2 * EPract

The FINAL GRADE will be Grade1 for students such that Grade1> = 5, and will be the maximum between Grade1 and Grade2 otherwise.

IMPORTANT: in order for any test (PAC or exam) to be taken into account in the calculation of the grade of the subject (Grade or Final Grade), at least 3.5 out of 10 in the 'evaluation of the test is required.
Otherwise, the test will score as a 0 on the calculation.
If a student presents at least one of the two PACs, or takes one of the exams, he/she will be considered as Presented; otherwise, their rating will be "Not evaluable". For a student who is considered Presented, the mark of any evaluable test to which it is not presented will be a 0.








The evaluation of this subject will consist of:

Continuous assessment:

control of practices with a computer (20% of the final grade). UNRECOVERABLE.
Two non-eliminatory partial exams of matter, with 30 and 50% respecius weights.

Final exam of recovery: it is worth 80% of the final grade and will allow to increase the joint grade of the two partial ones.

If a student presents at least one of the two partial exams, or in the final exam of recovery, it will be consideredas

presented; otherwise, your qualification will be "Not evaluable".

*Students assessment may experience some modifications depending on the restrictions to face-to-face activities enforced by health authorities.

Assessment Activities

Title Weighting Hours ECTS Learning Outcomes
Continuous Assessment Test, PAC1 0.20 2 0.08 4, 6, 3
Continuous Assessment Test, PAC2 0.20 2 0.08 4, 1, 2, 7, 6, 5, 8
Exam 0.60 5 0.2 4, 1, 2, 7, 6, 5, 8



Bardina, Xavier. Càlcul de Probabilitats. Servei de Publicacions UAB, 2004.

DeGroot, Morris H., Schervish, Mark J. Probability and statistics. Pearson, 2012, 4th ed., international ed.

Delgado, Rosario. Probabilidad y Estadística con aplicaciones.

Devore, Jay L. Probabilidad y Estadística para ingeniería y ciencias. Cengage Learning, 2016

Julià, Olga; Márquez, David; Rovira, Carles i Sarrà, Mónica. Probabilitats: Problemes i més problemes.

Publicacions i edicions de la Universitat de Barcelona, 2005.

Kai Lai, Chung. Teoría elemental de la probabilidad y los procesos estocásticos. Reverté, cop., 1983.

Sanz-Solé, Marta. Probabilitats. Edicions de la Universitat de Barcelona, 1999.


Ross, Sheldon M. Introduction to Probability Models, Academic Press, 12th Edition. Elsevier, 2019.
Rao, C. Radhakrishna. Estadística y verdad. Aprovechando el azar. Colección Universitas-73. Serie Estadística y Análisis de datos. PPU, S.A., 1994.


In this subject the R software will be used (https://cran.r-project.org/)
R is a programming environment consisting of a set of very flexible tools that can be easily expanded through packages, libraries, or by defining our own functions. It is also free and open source, an Open Source part of the GNU project, and this is one of its main advantages. Any user can download and create their code for free, without restrictions of use, the only rule is that the distribution is always free (GPL). Because it can freely access its code, R software has no limited functions, unlike other commercial statistical tools.
Preferably, we will use it using the RStudio platform (https://www.rstudio.com/)