Presentation and Context

Focus:

• The course introduces the fundamental concepts and properties of Probability Calculation, particularly for discrete probability models. It builds upon basic knowledge from high school or introductory statistics courses.

Objective

• To provide a deeper understanding of probability models through analyzing situations with finite or countable outcomes.

Continuation:

• The subject extends into Cálculo de Probabilidades II, which covers continuous and multidimensional probability models, and Procesos Estocásticos (Stochastic Processes), which deals with dynamic probability models.

Requirements and Recommendations

Prerequisites

• Basic knowledge of mathematical analysis, particularly related to sequences and series of real numbers, and familiarity with combinatorics is expected.

Content Review

• A review of combinatorial techniques is included for those needing reinforcement in this area.

Main Competencies Acquired

• Critical reasoning and the ability to evaluate both personal and others' work.
• Capability to solve mathematical problems using correct formulations and mathematical language.
• Developing logical arguments and identifying inconsistencies.
• Presenting mathematical reasoning clearly in both written and oral forms.

Learning Outcomes

• Master the fundamental properties of discrete probability models.
• Perform probability and expectation calculations.
• Model real-world situations using mathematical probability models.
• Approach probability problems intuitively.
• Understand discrete approximations to continuous distributions and laws of large numbers.

Key Topics (Temario)

• Random Experience

  • Introduction to randomness and chance.

• Mathematical Model of Probability

  • Formalizing probabilistic situations.

• Probability Assignment

  • How to allocate probabilities to events.

• Inclusion-Exclusion Formulas

  • Techniques for calculating probabilities of combined events.

• Extensions of the Probability Model

  • Broader applications and generalizations.

• Conditional Probability

  • Understanding dependent events.

• Event Independence

  • Recognizing and working with independent events.

• Random Variables

  • Defining and using random variables in models.

• Mathematical Expectation

  • Calculating expected values.

• Descriptive Analysis of Probability Distributions

  • Analyzing and interpreting distributions.

• Repeated Trials

  • Modeling repeated random experiments.

• Random Fluctuations

  • Exploring the variability in random processes.

Methodology

• Study is based on a textbook designed for autonomous learning, supplemented by exercises and examples.
• Demonstrations of results are included in the text but aren't necessary to memorize, only to understand the logic behind them.

Evaluation System

Exams

• A test with 10 multiple-choice questions, with both theoretical and practical content.
• Correct answers are worth 1 point, wrong answers deduct 0.4 points.

Continuous Assessment (PEC)

• An optional online questionnaire consisting of 5 questions.
• Correct answers earn 0.4 points, wrong answers deduct 0.15 points.

Recommended Textbooks

Basic: "Cálculo de Probabilidades I" by Víctor Hernández Morales and Ricardo Vélez Ibarrola.
Complementary: "An Introduction to Probability Theory and Its Applications" by William Feller.