Difference between revisions of "User:Saul/probability"
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'''ℙ(A∪B) = ℙ(A) + ℙ(B)''' - The sets '''A''' and '''B''' are disjoint.<br> | '''ℙ(A∪B) = ℙ(A) + ℙ(B)''' - The sets '''A''' and '''B''' are disjoint.<br> | ||
'''#A''' - The number of elements '''A''' contains if '''A''' is finite.<br> | '''#A''' - The number of elements '''A''' contains if '''A''' is finite.<br> | ||
| + | |||
| + | == Axioms == | ||
| + | '''ℙ(A) ≥ 0''' for all events '''A'''.<br> | ||
| + | '''ℙ(Ω) = 1'''<br> | ||
| + | ''(Countable additivity) For an infinite sequence of mutually exclusive events{A1,A2,A3,...}'' | ||
| + | |||
| + | === Deducible Properties === | ||
| + | '''ℙ(∅) = 0''' ''Since'' '''Ω ∪ ∅ ∪ ∅ ∪ ... = Ω'''<br> | ||
| + | ''Finite additivity.''<br> | ||
| + | '''ℙ(A<sup>c</sup>) = 1 - ℙ(A)''' ''Since'' '''A ∪ A<sup>c</sup> = Ω'''<br> | ||
| + | ''If'' '''A⊆B''' ''Then'' '''ℙ(A) ≤ ℙ(B)''' ''Since'' '''A∪(A<sup>c</sup>∩B) = B'''<br> | ||
| + | '''ℙ(A) ≤ 1''' ''Since'' '''A ⊆ Ω'''<br> | ||
| + | '''ℙ(A∪B) = ℙ(A) + ℙ(B) - ℙ(A∩B)'''<br> | ||
Revision as of 23:13, 6 March 2020
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.
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)
