Definition
The likelihood of something happening, defined as a number between 0 and 1
If something will never happen, it has a probability of 0; if it is certain to happen, it has a probability of 1.
Terms
 Experiment: observation of some activity or the act of taking some measurement.
 Outcome: result of an experiment
 Sample space: The set of all possible outcomes. Also known as Sample space. The notation is omega(Ω).
 Partition: set of mutually exclusive events, part of the Sample space.
 Probability mass: The sample space has a probability mass of 1.
 Uniform distribution: every event has the same likelihood of occurring.
Event
Collection of one or more outcomes of an experiment is known as an event. It is also described as a subset of the Sample space. The notation is Greek letter sigma (Σ).
Events can be impossible, certain or something in between.
Two or more events can be:
 mutually exclusive or disjoint: P (A ∩ B) = 0. The "∩" symbol means: union, is referred to as "cap" and may be read as "and".
 joint. The probability of A OR B occurring: P (A ∪ B). The "∪" symbol means: not the union, is referred to as "cup" and may be read as "or".
 independent : X ⊥ W
 conditional: P (A  B). The “” symbol means “given that”. In the table below it is written as "┃" for technical reasons.
 complement: P (A') or (¬ A)
Notation
 random variables written as upper case: X, Y,
 expected value: E[X]
Probability rules
Event

Key words

Rule

Formula

Formula

Formula

Formula

Mutually exclusive or disjoint 
The probability of A occurring or the probability of B occurring 
Special Rule of Addition 
P(A or B) = P(A)+ P(B) 
P(A + B) = P(A)+ P(B) 
P(A ∪ B) = P(A) + P(B) 
P(A ∩ B) = 0

Joint 
The probability that either A may occur or B may occur followed by the possibility that both A and B may occur 
General Rule of Addition 
P (A or B) = P(A)+ P(B) – P(AB) 
P (A or B) = P(A)+ P(B) – P(A and B) 
P(A ∪ B) = P(A) + P(B)  P (A ∩ B)

Independent 
The probability that A and B will occur 
Special Rule of Multiplication 
P(A and B) = P(A)P(B) 
P (A ∩ B) = P(A) P(B) 
P(A_{1}A_{2}...A_{n})=P(A_{1})P(A_{2})...P(A_{n})

Conditional 
P(B┃A) – probability that event B will occur given that event A has already occurred 
General Rule of Multiplication 
P(A and B) = P(A)P(B┃A) = P(B) P(A┃B) 
P(A ∩ B) = P(A) P(B┃A) = P(B) P(A┃B)

Complement 
Event Not occurring – or Neither/nor will happen 
Complement Rule 
P(A) = 1 – P (¬ A)

Probability distributions
Probability Mass Function

Probability Density Function



Bayes' Theorem