Chapter 3: Basic Ideas in Probability

Chapter 3: Basic Ideas in Probability#

Overview#

Probability theory provides the foundation for statistical inference, machine learning, and understanding uncertainty in computer science.

Learning Objectives#

  • Understand experiments, outcomes, and probability

  • Calculate probabilities using set theory

  • Apply independence rules

  • Master conditional probability

  • Avoid common probability fallacies

Why Probability?#

Computer scientists use probability for:

  • Randomized algorithms: Quicksort, hashing

  • Machine learning: Probabilistic models

  • Cryptography: Secure random number generation

  • Networks: Packet loss and reliability

  • AI: Decision making under uncertainty

Chapter Structure#

3.1 Experiments and Outcomes#

  • Sample spaces

  • Outcomes and events

  • Probability axioms

3.2 Events#

  • Computing event probabilities

  • Set operations: union, intersection, complement

  • Counting techniques

3.3 Independence#

  • Independent events

  • Multiplication rule

  • Applications

3.4 Conditional Probability#

  • Definition and interpretation

  • Bayes’ theorem

  • The Monty Hall problem

  • Prosecutor’s fallacy

Key Concepts#

Experiment: Process with uncertain outcome

Sample Space (\(\Omega\)): Set of all possible outcomes

Event: Subset of sample space

Probability: \(P(A)\) where \(0 \leq P(A) \leq 1\)

Axioms of Probability#

  1. \(P(A) \geq 0\) for any event \(A\)

  2. \(P(\Omega) = 1\)

  3. For disjoint events: \(P(A \cup B) = P(A) + P(B)\)

Getting Started#

→ Begin with 3.1 Experiments and Outcomes

Contents