Theory 1
Axioms of probability
A probability measure is a function
satisfying: Kolmogorov Axioms:
Axiom 1:
for every event (probabilities are not negative!) Axiom 2:
(probability of “anything” happening is 1) Axiom 3: additivity for any countable collection of mutually exclusive events:
Notation: we write
instead of , even though is a function, to emphasize the fact that is a set.
Probability model
A probability model or probability space consists of a triple
:
the sample space
the set of valid events, where every satisfies
a probability measure satisfying the Kolmogorov Axioms
Finitely many exclusive events
It is a consequence of the Kolmogorov Axioms that additivity also works for finite collections of mutually exclusive events:
Inferences from Kolmogorov
A probability measure satisfies these rules. They can be deduced from the Kolmogorov Axioms.
- Negation: Can you find
but not ? Use negation:
- Monotonicity: Probabilities grow when outcomes are added:
- Inclusion-Exclusion: A trick for resolving unions:
(even when
and are not exclusive!)
Inclusion-Exclusion
The principle of inclusion-exclusion generalizes to three events:
The same pattern works for any number of events!
The pattern goes: “include singles” then “exclude doubles” then “include triples” then …
Include, exclude, include, exclude, include, …