Unit 1 of 13
Probability Foundations
- 8 lessons
- Videos 1 to 8
A probability describes uncertainty under a stated model. This unit builds sample spaces and events, states the probability axioms and derives the complement and addition rules.
It starts by separating three ideas that are easy to mix up, a model probability, an observed frequency and a personal assessment. Everyday conditions such as at least, neither and not both are translated into sets before any probability is assigned. The axioms then give the complement rule and the addition rule. Equally likely outcomes turn probability into counting only when that assumption is justified, and a reproducible simulation is checked against an exact answer.
Every lesson is one video. Each lesson page has the definitions and an example for every key term.
Probability Models and Events
Probability Models and Uncertainty
How a probability model describes uncertainty, what experiments, trials, outcomes and events are, and how a theoretical probability differs from an empirical relative frequency and a subjective probability.
Sample Spaces and Events
How to build a finite sample space of recorded outcomes, describe events with union, intersection, complement and set difference, use De Morgan’s laws, and recognize disjoint events and partitions.
Video 2
Translating Event Language
How to translate everyday conditions such as inclusive or, neither, not both, exactly one, at least, at most and only if into sets of outcomes, and how one counterexample disproves a general claim.
Video 3
The Axioms and Basic Rules
Probability Axioms and Basic Rules
How the three probability axioms, nonnegativity, normalization and countable additivity over pairwise disjoint events, check a probability model and imply zero probability for the empty event and monotonicity.
Video 4
The Complement Rule
Why the complement rule follows from the axioms through a complementary partition, and how to name the correct opposite event for at least one, not all and at least k without assuming independence.
Video 5
The Addition Rule
Why the addition rule subtracts the overlap of two events, how the four disjoint regions of a Venn diagram give exactly one and neither, and how the union bound and monotonicity bracket a union when the overlap is unknown.
Video 6
Equally Likely Outcomes and Simulation
Equally Likely Outcomes
When a probability can be found by counting favorable outcomes, why counting needs the uniform finite model of equally likely outcomes, and how cardinality, the level of detail and ordered sequences decide what is counted.
Video 7
Probability Simulation
How a simulation estimates a probability from repeated simulated trials, how a random seed makes a pseudorandom run reproducible, and how a running estimate shows sampling variability around an exact answer.
Video 8