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Probability derives from the Latin probare (to prove, or to test). The word probable means roughly "likely to occur" in the case of possible future occurrences, or "likely to be true" in the case of inferences from evidence.

The idea is most often broken into two concepts:

1. aleatory probability, which represents the likelihood of future events whose occurrence is governed by some random physical phenomenon like tossing dice or spinning a wheel; and
2. epistemic probability, which represents our uncertainty of belief about propositions of which one does not know whether they are true. Such propositions may be about past or future events, but need not be.

Examples of epistemic probability:

• Assign a probability to the proposition that a proposed law of physics is true.
• Determine how "probable" it is that a suspect committed a crime, based on the evidence presented.

It is an open question whether aleatory probability is reducible to epistemic probability based on our inability to precisely predict every force that might affect the roll of a die, or whether such uncertainties exist in the nature of reality itself, particularly in quantum phenomena governed by Heisenberg's uncertainty principle. The same mathematical rules apply regardless of what interpretation you favor.

Probability in mathematics

In probability theory, an event is a "measurable" subset of a "sample space". "Events" are the things to which probabilities are assigned. A probability is a number in the closed interval from 0 to 1. Probabilities must be assigned to events in such a way that for pairwise disjoint (i.e., no two intersect each other) events A₁, A₂, A₃, ..., the probability of their union is the sum of their probabilities, or, in mathematical notation,

${\displaystyle P\left(A_{1}\cup A_{2}\cup A_{3}\cup \cdots \right)=P(A_{1})+P(A_{2})+P(A_{3})+\cdots .}$

In the special case of a "discrete probability distribution" the sample space is a set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \left\{\,x_1,x_2,x_3,...\,\}} of outcomes to each of which a positive number has been assigned as its probability. The one-members sets ${\displaystyle \{\,x_{i}\,\}}$ are "elementary events". One of the simplest of discrete sample spaces is a finite set ${\displaystyle \{\,x_{1},x_{2},x_{3},...x_{n}\,\}}$ to each of whose members the same probability 1/n is assigned. An example of a sample space that is not discrete is the closed interval to which the length of any subinterval (a, b) is assigned as the probability of that subinterval. The probability assigned to any one-member subset is 0.

Representation and interpretation of probability values

The value 0 is generally understood to represent impossible events, while the number 1 is understood to represent certain events (though there are more advanced interpretations of probability that use more precise definitions). Values between 0 and 1 quantify the probability of the occurrence of some event. In common language, these numbers are often expressed as fractions or percentages, and must be converted to real number form to perform calculations with them.

For example, if two events are equally likely, such as a flipped coin landing heads-up or tails-up, we express the probability of each event as "1 in 2" or "50%" or "1/2", where the numerator of the fraction is the relative likelihood of the target event and the denominator is the total of relative likelihoods for all events. To use the probability in math we must perform the division and express it as "0.5".

Another way probabilities are expressed is "odds", where the two numbers used represent the relative likelihood of the target event and the likelihood of all events other than the target event. Expressed as odds, tossing a coin will give heads odds of "1 to 1"or "1:1". To convert odds to probability, use the sum of the numbers given as the denominator of a fraction: "1:1" odds make a "1/2" probability; "3:2" odds make a "3/5" probability (or 0.6).

Distributions

The histogram of events versus occurrence is called a probability distribution. There are several important, discrete distributions, such as the discrete uniform distribution, the Poisson distribution, the binomial distribution, the negative binomial distribution and the Maxwell-Boltzmann distribution.

Remarks on probability calculations

The difficulty of probability calculations lie in determining the number of possible events, counting the occurrences of each event, counting the total number of possible events. Especially difficult is drawing meaningful conclusions from the probabilities calculated. An amusing probability riddle, the Monty Hall problem demonstrates the pitfalls nicely.

To learn more about the basics of probability theory, see the article on probability axioms and the article on Bayes' theorem that explains the use of conditional probabilities in case where the occurrence of two events is related.

See also

• Bayesian probability
• Bernoulli process
• Game theory
• Cox's theorem
• Information theory
• Law of averages
• Law of large numbers (weak version)
• Law of large numbers (strong version)
• Normal distribution
• Random variable
• Probability and Statistics
• Probability applications
• Statistical probability
• Stochastic process
• Wiener process

Buckley's chance means a very small chance, see http://www.anu.edu.au/ANDC/Ozwords/Oct%202000/Buckley%27s.html

A major impact of probability theory on everyday life is in risk assessment and in trade on commodity markets. Governments typically apply probability methods in environment regulation where it is called "pathway analysis", and are often measuring well-being using methods that are stochastic in nature, and choosing projects to undertake based on their perceived probable impact on the population as a whole, statistically. It is not correct to say that statistics are involved in the modelling itself, as typically the assessments of risk are one-time and thus require more fundamental probability models, e.g. "the probability of another 9/11". A law of small numbers tends to apply to all such choices and perception of the impact of such choices, which makes probability measures a political matter.

A good example is the impact of the perceived probability of any widespread Middle East conflict on oil prices - which have ripple effects in the economy as a whole. An assessment by a commodity trade that a war is more likely vs. less likely sends prices up or down, and signals other traders of that opinion. Accordingly, the probabilities are not assessed independently nor necessarily very rationally. The theory of behavioral finance emerged to describe the impact of such groupthink on pricing, on policy, and on peace and conflict.

It can reasonably be said that the discovery of rigorous methods to assess and combine probability assessments has had a profound impact on modern society. A good example is the application of game theory, itself based strictly on probability, to the Cold War and the mutual assured destruction doctrine. Accordingly, it may be of some importance to most citizens to understand how odds and probability assessments are made, and how they contribute to reputations and to decisions, especially in a democracy.