Probability

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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(A1A2...An)=P(A1)P(A2)...P(An)
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
discrete variables
continuous variables

Bayes' Theorem