A probability function is a real-valued unit set function P() that returns a probability value for every event E of sample space Ω such that: P(Ω)=1; P(E)≥0; and P(E1 ∪ E2) = P(E1) + P(E2) if distinct(E1, E2)
- AKA: P, Event Probability Function, Probability Measure, Probability, Prior Probability, Outcome Probability, Odds, Probability Distribution Function.
- Context:
- Example(s):
- Counter-Example(s):
- See: Sample Space, Event Space, Probability Theory, Experimental Probability, Certainty, Conditional Probability Function, Likelihood Function, Bayesians, Frequentists, Statistical Independence Relation, Identical Distribution Relation.
References
- WordNet
- probability - a measure of how likely it is that some event will occur; a number expressing the ratio of favorable cases to the whole number of cases possible ...
- probability - the quality of being probable; a probable event or the most probable event; "for a while mutiny seemed a probability"; "going by past experience there was a high probability that the visitors were lost"
- probable - likely but not certain to be or become true or real; "a likely result"; "he foresaw a probable loss"
- probable - an applicant likely to be chosen
- probable - apparently destined; "the probable consequences of going ahead with the scheme"
- http://en.wiktionary.org/wiki/Probability
- 1. the state of being probable; likelihood
- 2. an event that is likely to occur
- 3. the relative likelihood of an event happening
- 4. (mathematics) a number, between 0 and 1, expressing the precise likelihood of an event happening
- (Wikipedia, 2009) http://en.wikipedia.org/wiki/Probability
- Probability, or chance, is a way of expressing knowledge or belief that an event will occur or has occurred. In mathematics the concept has been given an exact meaning in probability theory, that is used extensively in such areas of study as mathematics, statistics, finance, gambling, science, and philosophy to draw conclusions about the likelihood of potential events and the underlying mechanics of complex systems.
- The word probability does not have a consistent direct definition. In fact, there are two broad categories of probability interpretations, whose adherents possess different (and sometimes conflicting) views about the fundamental nature of probability:
- 1. Frequentists talk about probabilities only when dealing with experiments that are random and well-defined. The probability of a random event denotes the relative frequency of occurrence of an experiment's outcome, when repeating the experiment. Frequentists consider probability to be the relative frequency "in the long run" of outcomes.[1]
- 2. Bayesians, however, assign probabilities to any statement whatsoever, even when no random process is involved. Probability, for a Bayesian, is a way to represent an individual's degree of belief in a statement, given the evidence.
- http://en.wiktionary.org/wiki/theoretical_probability
- 1. (mathematics) the probability that a certain outcome will occur, as determined through reasoning or calculation. Given a die which is a regular octahedron of uniform density, and given that one and only one of its faces is painted black, then if the die is cast, the theoretical probability that the outcome will be the black face is 1/8.
- http://www.econ.cam.ac.uk/faculty/weeks/Paper6/Exercise1.pdf
- A probability function is defined as a real-valued set function on the class of all subsets of the sample space Ω: the value associated with a subset A is denoted Pr(A).
- The assignment of probabilities must satisfy the following three axioms:
- i) Pr(Ω) = 1
- ii) Pr(A) ≥ 0 for all A ⊂ Ω
- iii) Pr(A ∪ B) = Pr(A) + Pr(B) if A ∩ B = Ø
2006
1991
- Yuri Gurevich. (1991). "Average Case Completeness." In: Journal of Computer and System Sciences, 42(3). [doi>0.1016/0022-0000(91)90007-R].
- We will consider only finite or infinite countable sample spaces, i.e., probability spaces. The function that assigns probabilities to sample points is the probability function. If μ is a probability function and X is a collection of sample points then the μ-probability of the event X will be denoted μ(X); in other words, μ(X) = Σx∈X μ(x). The letters μ and ν are reserved for probability functions. If μ(x) is a probability function on an ordered sample space then μ*(x) = Σ y<x μ(y) is the corresponding probability distribution. A probability function μ is positive if every value of μ is positive. The restriction μ|X of a probability function μ to a set X of sample points with μ(x) > 0 is the probability function proportional to μ on X and zero outside X.
1986
- (Larsen & Marx, 1986) => Richard J. Larsen, and Morris L. Marx. (1986). "An Introduction to Mathematical Statistics and Its Applications, 2nd edition. Prentice Hall
- Quote: Consider a sample space, S, and any event, A, defined on S. If our experiment were performed one time, either A or AC would be the outcome. If it were performed n times, the resulting set of sample outcomes would be members of A on m occasions, m being some integer between 0 and n, inclusive. Hypothetically, we could continue this process an infinite number of times. As n gets large, the ratio m/n will fluctuate less and less (we will make that statement more precise a little later). The number that m/n convert to is called the empirical probability of A : that is, P(A) = limn→∞(m/n). ... the very act of repeating an experiment under identical conditions an infinite number of times is physically impossible. And left unanswered is the question of how large n must be to give a good approximation for limn→∞(m/n).
- The next attempt at defining probability was entirely a product of the twentieth century. Modern mathematicians have shown a keen interest in developing subjects axiomatically. It was to be expected, then , that probability would come under such scrutiny ... The major breakthrough on this front came in 1933 when Andrei Kolmogorov published Grundbegriffe der Wahscheinlichkeitsrechnung (Foundations of the Theory of Probability.). Kolmogorov's work was a masterpiece of mathematical elegance - it reduced the behavior of the probability function to a set of just three or four simple postulates, three if the same space is limited to a finite number of outcomes and four if S is infinite.