User:Saul/probability

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Revision as of 23:52, 6 March 2020 by Saul (talk | contribs) (Notations)

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.
Ac - 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.
Possible outcome - may have a probability of 0.

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(Ac) = 1 - ℙ(A) Since A ∪ Ac = Ω
If A⊆B Then ℙ(A) ≤ ℙ(B) Since A∪(Ac∩B) = B
ℙ(A) ≤ 1 Since A ⊆ Ω
ℙ(A∪B) = ℙ(A) + ℙ(B) - ℙ(A∩B)