Probability

Course objectives

General objectives: to acquire basic knowledge in probability theory. Specific objectives: Knowledge and understanding: at the end of the course, students will be able to use basic notions in combinatorics to solve math problems, derive laws for discrete random variables. Applying knowledge and understanding: at the end of the course, students will be able to solve simple problems in discrete probability, problems concerning discrete random vectors and random numbers represented by continuous random variables. They will understand the role of indipendence and conditioning in discrete models and understand the meaning of some limit theorems, like the law of large numbers. Critical and judgmental skills: students will have the bases to analize and to build simple probabilistic models for physics, biology and technology, simulate discrete probability distribution, as well as the Gaussian distribution and understand the use of some elementry tools in statistics, like inference, sampling and simulation. Communication skills: ability to expose the contents of the course in the oral part of the test and in any theoretical questions present in the written test. Learning skills: the acquired knowledge will allow a study, individual or given in a course related to more specialized aspects of probability theory.

Channel 1
GIOVANNI FRANZINA Lecturers' profile

Program - Frequency - Exams

Course program
[1] Historical notes on the birth of probability. [2] Mathematical models for random experiments using set theory. [3] Finite probability spaces and indicator functions. [4] De Morgan’s laws, inclusion–exclusion principle, cardinality of a union. [5] Classical model: finite spaces of equiprobable events. [6] Counting: elementary combinatorial calculus. [7] Combinatorics: ordered and unordered selections. [8] Probabilistic models of drawing from an urn (with or without replacement). [9] Axioms of probability. Classic problems (birthdays, matching, collector’s problem). [10] Conditional probability. Independence of events. [11] Total probability and Bayes’ theorems. [12] Partititions problems [13] Independent events and correlation between events. [14] Discrete random variables, discrete probability mass function and distribution function. [15] Models of random variables on finite spaces: Bernoulli, Binomial, Hypergeometric, Uniform. [16] Expected value of Bernoulli, Binomial, Hypergeometric, Uniform random variables. [17] Joint and marginal densities, marginalization, joint and marginal distribution functions. Independence of random variables and factorization properties. [18] Expected value of the sum; expected value of the product of independent random variables. Variance. Variance of the sum of independent random variables. Variance of Bernoulli, Binomial, Hypergeometric, Uniform random variables. [19] Random walks. [20] Conditional expectation. [21] Countable probability spaces: consequences of the axioms, law of total expectation. [22] Random variables on countable probability spaces: geometric random variables (expected value, variance, waiting times, memoryless property). [23] Random variables on countable probability spaces: Poisson random variables (expected value, variance, approximation of binomials, number of rare events in an interval). [24] Law of large numbers, concentration of probability (Chebyshev’s inequality).
Prerequisites
Previous Knowledge of basic Calculus, elementary linear algebra, set theory and logic.
Books
S. Ross, A First course in Probability
Exam mode
Learning is assessed through a final written exam consisting of 6 exercises covering the main topics of the syllabus. The exam awards up to 40 points (each point corresponds to 1/30 of the final grade) and is the sole formal assessment of the acquired competences.
  • Lesson code1020421
  • Academic year2025/2026
  • CourseComputer Science
  • CurriculumSingle curriculum
  • Year2nd year
  • Semester1st semester
  • SSDMAT/06
  • CFU9