By: Neil E. Cotter	Probability
		Basic Probability
		Notation and definitions

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Def: Experiment ≡ process that generates data

Ex: Flip a fair coin twice.

Def: Observation ≡ recording of information

Ex: We flip the coin two times and get Heads Tails (or HT).

Def: Outcome ≡ result of a hypothetical experiment; an element or member or sample point of sample space.

Ex: When we flip the coin twice, there are four possible outcomes: HH, HT, TH, TT.

Def: Sample Space ≡ S ≡ the set of all possible outcomes of a statistical experiment

Ex: When we flip the coin twice, S is the set of possible outcomes:

S = {HH, HT, TH, TT}

Def: Event ≡ subset of sample space S

Ex: When we flip the coin twice, we may define an event A to be "we get Heads on the first flip".

A = {HH, HT}

Def: Probability (of event A) ≡ weight (of event A) ≡ likelihood of obtaining outcome in A as the result of a hypothetical experiment

Ex: For event A defined above, P(A) = 1/2 (meaning we have a 50-50 chance of getting Heads on the first flip).

Def: Complement (of event A) ≡ A' ≡ subset of all elements of sample space S that are not in A

Ex: For event A defined above, A' = {TH, TT} since the outcomes that start with Tails are the possible outcomes that are not in A.

Def: Intersection (of events A and B) ≡ A∩B ≡ event consisting of all elements of sample space S that are common to A and B

Ex: Suppose we define another event B to be "the outcome has one Heads and one Tails".

B = {HT, TH}

For event A defined earlier,

A∩B = {HT} since HT is in both A and B

Note: In general, we can have multiple outcomes in the intersection.

Def: Mutually exclusive (events B and C) ≡ disjoint (events B and C) ≡ B and C have no elements in common ≡ B∩C = Æ (empty set)

Ex: If we define event C = {HH}, then B (from earlier) and C are mutually exclusive since they have no elements (i.e., outcomes) in common.

B∩C = Æ

Def: Union (of events A and B) ≡ A∪B ≡ event consisting of all elements of sample space S that are in either A or B or both

Ex: For A and B defined above, the union of A and B has three elements:

A∪B = {HH, HT, TH}

Note: If an outcome is in both events A and B, it appears only once in the union.

Def: Partition (A1, A2, A3, ..., An of sample space S) ≡ events A1, A2, A3, ..., An are mutually exclusive and the union of A1, A2, A3, ..., An is S ≡ Ai∩Aj = Æ when i ≠ j and A1∪A2∪A3∪ ... ∪An = S

Ex: If we define a new event, D = {TT}, then B, C, and D form a partition.

B = {HT, TH}

C = {HH}

D = {TT}

Each outcome in sample space, S, appears in B, C, or D once and only once.