User:Saul/probability

Notations

Ω - The outcome space.
ω - An outcome.
∅ - A non event.
∪ - Union.
∩ - Intersection.
∈ - Is in.
∉ - Is not in.
⊂ - Subset of.
⊆ - Subset or equal to.

Note: there is a small difference between ⊂ & ⊆ but sometimes they get used interchangeably.

⊄ - Is not a subset of.
⊃ - Superset of.
⊇ - Is not a superset of.
ℙ(A) - Probability of A.
A^c - A compliment - The event A does not occur.
ω∈A - The outcome ω is in the event A
A∪B - The union of A and B - The set containing all the elements from A and B without duplicates.
A∩B - The intersection of A and B - The set containing all the common elements from A and B.
A∩B = ∅ - The sets A and B are disjoint.
ℙ(A∪B) = ℙ(A) + ℙ(B) - The sets A and B are disjoint.
#A - The number of elements A contains if A is finite.
ℙ(S) = 1 - The set S is a exhaustive set as it contains all the outcomes.

Axioms

ℙ(A) ≥ 0 for all events A.
ℙ(Ω) = 1
(Countable additivity) For an infinite sequence of mutually exclusive events{A1,A2,A3,...}

Deducible Properties

ℙ(∅) = 0 Since Ω ∪ ∅ ∪ ∅ ∪ ... = Ω
Finite additivity.
ℙ(A^c) = 1 - ℙ(A) Since A ∪ A^c = Ω
If A⊆B Then ℙ(A) ≤ ℙ(B) Since A∪(A^c∩B) = B
ℙ(A) ≤ 1 Since A ⊆ Ω
ℙ(A∪B) = ℙ(A) + ℙ(B) - ℙ(A∩B)