MATH 1103 · Probability and Statistics: Probability

Led by Andrey Kolmogorov Simulacrum

Probability from the axioms — the addition rule, conditional probability, independence, and Bayes' theorem, taught by the mathematician who gave probability its foundations.

1. Module 1

   The Foundations of Probability

   Led by Andrey Kolmogorov Simulacrum

   The question

   Three axioms. Everything else follows. What does it mean for intuition about probability to "contradict the axioms" — and what is the complement rule?

   Outcome

   The student can state the three axioms and compute probabilities using the classical definition and the complement rule.

   Sub-units

   1. ○ 1.1 Probability from First Principles Open this unit → ✓ ↻
   Begin the course → Print certificate of completion
2. Module 2

   The Addition Rule: Union and Intersection

   Led by Andrey Kolmogorov Simulacrum

   The question

   60 study maths, 40 study physics, 25 study both. P(maths or physics) ≠ 0.60 + 0.40. Why not — and why are mutually exclusive events and independent events completely different things?

   Outcome

   The student can apply the addition rule and inclusion-exclusion principle.

   Sub-units

   1. ○ 2.1 Addition Rule Problems Open this unit → ✓ ↻
   Open this module → Print certificate of completion
3. Module 3

   Conditional Probability and Independence

   Led by Andrey Kolmogorov Simulacrum

   The question

   A quality test is 95% accurate. 2% of components are defective. P(defective | flagged) turns out to be surprisingly low. Why — and what does this reveal about the difference between test accuracy and predictive value?

   Outcome

   The student can compute conditional probabilities, test for independence, and draw tree diagrams.

   Sub-units

   1. ○ 3.1 Conditional Probability and Independence Open this unit → ✓ ↻
   Open this module → Print certificate of completion
4. Module 4

   Bayes' Theorem

   Led by Andrey Kolmogorov Simulacrum

   The question

   Disease prevalence: 1%. Test sensitivity: 99%. Test specificity: 95%. A patient tests positive. What is the actual probability they have the disease — and what should the doctor say instead of "the test is 99% accurate"?

   Outcome

   The student can apply Bayes' theorem and explain the base rate problem.

   Sub-units

   1. ○ 4.1 Bayes' Theorem Applied Open this unit → ✓ ↻
   Open this module → Print certificate of completion
5. Module 5

   Probability in Context

   Led by Andrey Kolmogorov Simulacrum

   The question

   Why does switching doors in the Monty Hall problem double your probability of winning? Why does a group of 23 people have a >50% chance of a shared birthday? Both are consequences of the axioms. Show the calculation.

   Outcome

   The student can apply the law of total probability and analyse a counterintuitive probability problem.

   Sub-units

   1. ○ 5.1 Final Essay: The Axioms in Action Open this unit → ✓ ↻
   Open this module → Print certificate of completion