Course Outline
Unit 1: Foundations of Probability
Students begin by exploring the fundamental concepts of probability and uncertainty. They investigate sample spaces, events, probability rules, conditional probability, and graphical methods such as tree diagrams and probability tables. Emphasis is placed on developing a solid understanding of how probabilities are calculated and interpreted while building the logical reasoning required for more advanced probability models.
Subtopics
- Probability Concepts
- Sample Spaces and Events
- Probability Rules
- Conditional Probability
- Tree Diagrams and Tables
Unit 2: Counting Principles and Combinatorics
This unit introduces the mathematical techniques used to count large numbers of possible outcomes efficiently. Students investigate the Fundamental Counting Principle, permutations, combinations, binomial coefficients, and advanced counting strategies involving restrictions and repeated objects. These techniques form the foundation for solving many complex probability problems and have wide-ranging applications in mathematics and computer science.
Subtopics
- The Fundamental Counting Principle
- Permutations
- Combinations
- Binomial Coefficients
- Advanced Counting Techniques
Unit 3: Probability Distributions
Students explore random variables and probability distributions as mathematical models of uncertainty. They investigate expected value, variance, and standard deviation before studying important discrete and continuous probability distributions, including the binomial, geometric, Poisson, and continuous distributions. Students learn how these models describe random processes and support informed decision making in a variety of contexts.
Subtopics
- Random Variables
- Expected Value and Variance
- Binomial Distribution
- Geometric and Poisson Distributions
- Continuous Probability Distributions
Unit 4: Advanced Probability
In this unit, students examine more sophisticated probability concepts and mathematical models. They explore Bayes' Theorem, independent and dependent events, simple Markov processes, Monte Carlo simulations, and the construction and evaluation of probability models. Through these topics, students develop a deeper understanding of uncertainty, prediction, and the role of probability in modern scientific and technological applications.
Subtopics
- Bayes' Theorem
- Independent and Dependent Events
- Markov Processes
- Simulation and Monte Carlo Methods
- Probability Models
Unit 5: Applications of Probability and Combinatorics
The final unit demonstrates how probability and combinatorics are applied across numerous disciplines. Students investigate decision making under uncertainty, games of chance, cryptography, coding theory, engineering reliability, and scientific modelling before integrating concepts from the entire course to solve challenging multi-step problems. The course concludes by emphasising the importance of probabilistic reasoning in understanding and solving real-world problems.
Subtopics
- Decision Making Under Uncertainty
- Games of Chance
- Networks, Coding, and Cryptography
- Probability in Science and Engineering
- Comprehensive Problem Solving
By the end of this course, students will be able to calculate probabilities using a variety of mathematical techniques, solve complex counting problems using combinatorial methods, analyse random variables and probability distributions, apply advanced probability models to authentic situations, and communicate rigorous mathematical reasoning with confidence. They will understand how probability and combinatorics underpin fields such as statistics, computer science, artificial intelligence, engineering, finance, biology, economics, and operations research, providing a strong foundation for university-level mathematics and STEM studies.