INTRODUCTION TO PROBABILITY FOR ECONOMICS

This course is a foundational quantitative component of the degree programme. It provides the probabilistic and statistical concepts required for subsequent study in economics, econometrics and other empirically oriented disciplines. Students develop the ability to reason under uncertainty, work with random variables and probability distributions, and draw elementary conclusions about population parameters from sample data. The structure and selected materials draw on the Probability and Statistics component of the Oxford PPE prelims, adapted and streamlined for a 30-hour introductory course.

By the end of the course, students should:

1. Knowledge and understanding: understand the fundamental concepts and terminology of probability, random variables, probability distributions, sampling distributions, estimation, confidence intervals and hypothesis testing.

2. Applying knowledge and understanding: apply probability rules, calculate probabilities and moments for selected distributions, use normal approximations, and solve elementary problems involving confidence intervals and hypothesis tests for one population mean or proportion.

3. Making judgements: select and interpret appropriate elementary probabilistic and inferential procedures, recognising the role of uncertainty and sampling variation.

4. Communication skills: recognise, interpret and use appropriate statistical and probabilistic terminology and notation.

5. Learning skills: use textbooks, lecture notes and exercises to consolidate knowledge and solve new problems of a comparable level.

Students should be familiar with secondary-school algebra, including fractions, powers, roots, equations and inequalities, and with elementary functions, including linear, exponential and logarithmic functions. Basic knowledge of sets, summation notation, derivatives and definite integrals is also expected. No previous course in probability or statistics is required. There are no formal prerequisites.

PART I: PROBABILITY THEORY

1. Probability foundations: random experiments, sample spaces, events, probability axioms and basic rules; elementary counting.

2. Conditional probability: multiplication rule, Law of Total Probability, Bayes’ Rule and independence.

3. Random variables and probability distributions: discrete and continuous random variables; PMFs, PDFs and CDFs; Bernoulli, binomial and normal distributions; standardisation; expectation and variance.

PART II: STATISTICS

4. Sampling distributions and estimation: IID samples, sample mean and sample proportion, standard errors, Central Limit Theorem, point estimation and large-sample confidence intervals.

5. Hypothesis testing: null and alternative hypotheses, test statistics, significance levels, critical values, p-values, Type I and Type II errors, and large-sample tests for one population mean and one population proportion.

Students are not expected to study all the listed sources in full. Only the sections and exercises specified in the weekly programme are relevant.

Required textbooks:

J. K. Blitzstein and J. Hwang, Introduction to Probability, 2nd edition, CRC Press, 2019 (BH). Principal textbook for Weeks 1–3. A free online version is available at https://probabilitybook.net/ .

A. Agresti, C. Franklin and B. Klingenberg, Statistics: The Art and Science of Learning from Data, 4th edition, Pearson, 2017 (AFK). Principal textbook for Weeks 4–5 and an additional introductory resource for Weeks 1–3.

Supplementary materials:

J. A. Duffy, Review of Probability and Statistics, lecture notes, 2026. Concise review notes covering the main course topics, available at https://sites.google.com/site/jamesaduffy/tutorials/prelims-probability-and-statistics-2026 . A printed copy may be consulted during the written examination.

F. DiTraglia, LMH Prelims Probability and Statistics. Selected supplementary review questions and problem sets, available at https://ditraglia.com/prelims-prob-stats/ .

The examination consists of a 60-minute written test followed, where requested or required, by an oral examination.

The written test contains 15 multiple-choice questions. Each correct answer is worth 2 points; incorrect or unanswered questions receive 0 points, with no penalties. The pass mark is 18/30. The maximum final grade obtainable through the written examination alone is 26/30; therefore, raw scores of 26, 28 or 30 all correspond to a written grade of 26/30. Questions may be conceptual, interpretative or computational. Examples of written tests are available on Moodle.

Students who pass may accept the written grade or request an oral examination. Since a multiple-choice test alone is not sufficient to assess the deeper understanding, reasoning and problem-solving skills required for a very good or excellent grade, an oral examination is required for grades above 26/30. It may include theoretical questions, justification or correction of multiple-choice answers, short derivations, interpretation questions and/or open-ended exercises.

If an oral examination is taken, the final grade is based on the student’s overall performance in both examinations. It may be higher or lower than the written raw score, and an unsatisfactory oral performance may result in failure. Honours may be awarded only following an oral examination.

The instructor may require an oral examination when further assessment is necessary, including in cases of suspected irregularities.

During the written test, students may use a printed copy of Duffy’s Review of Probability and Statistics, a basic non-programmable calculator and the statistical tables provided by the instructor.

scritto e orale

Il/la docente ha il dovere di vigilare affinché siano rispettate le regole di autenticità e originalità delle prove d'esame. Di conseguenza, nei casi in cui vi sia il sospetto di un comportamento irregolare, l'esame può prevedere un ulteriore approfondimento, contestuale alla prova d'esame, che potrà essere realizzato anche in modalità differente rispetto alle modalità sopra riportate.

The final grade evaluates knowledge and understanding of the course contents, ability to apply probabilistic and statistical methods, reasoning and argumentation skills, and correct use of statistical and probabilistic terminology and notation.

18–20: minimally sufficient knowledge and application, with significant conceptual gaps.

21–23: adequate knowledge and ability to solve standard elementary problems.

24–26: good understanding and application of the methods covered in the course.

27–30: very good to excellent understanding, accurate application, autonomous reasoning and appropriate use of statistical and probabilistic language, as demonstrated in both the written and oral examinations.

Honours are awarded only for exceptional performance demonstrating excellent judgement and in-depth understanding of the course contents.

The course consists of 30 hours of lectures combining the presentation of concepts with worked examples and guided problem solving. The lectures develop conceptual understanding, while the examples and assigned exercises develop the ability to apply the methods and interpret the results. Weekly readings, review questions and exercises support individual study and preparation for the examination. Additional materials and practice tests are available through the University’s Moodle platform.

Students should register for the course on Moodle (moodle.unive.it), where they will find the weekly programme, supplementary materials, exercises, examination examples and announcements.

Students not properly enrolled for an examination through the University’s online system will not be admitted. Students should ensure that they have received the enrolment confirmation email.