Probability

Probability
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Probability is the branch of mathematics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an event is to occur.^{[note 1][1][2]} A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes ("heads" and "tails") are both equally probable; the probability of "heads" equals the probability of "tails"; and since no other outcomes are possible, the probability of either "heads" or "tails" is 1/2 (which could also be written as 0.5 or 50%).

These concepts have been given an

When dealing with random experiments – i.e., experiments that are random and well-defined – in a purely theoretical setting (like tossing a coin), probabilities can be numerically described by the number of desired outcomes, divided by the total number of all outcomes. This is referred to as theoretical probability (in contrast to empirical probability, dealing with probabilities in the context of real experiments). For example, tossing a coin twice will yield "head-head", "head-tail", "tail-head", and "tail-tail" outcomes. The probability of getting an outcome of "head-head" is 1 out of 4 outcomes, or, in numerical terms, 1/4, 0.25 or 25%. However, when it comes to practical application, there are two major competing categories of probability interpretations, whose adherents hold different views about the fundamental nature of probability:

• Objectivists assign numbers to describe some objective or physical state of affairs. The most popular version of objective probability is frequentist probability, which claims that the probability of a random event denotes the relative frequency of occurrence of an experiment's outcome when the experiment is repeated indefinitely. This interpretation considers probability to be the relative frequency "in the long run" of outcomes.^[4] A modification of this is propensity probability
  , which interprets probability as the tendency of some experiment to yield a certain outcome, even if it is performed only once.
• posterior probability distribution that incorporates all the information known to date.^[8] By Aumann's agreement theorem, Bayesian agents whose prior beliefs are similar will end up with similar posterior beliefs. However, sufficiently different priors can lead to different conclusions, regardless of how much information the agents share.^[9]

The word probability derives from the Latin probabilitas, which can also mean "probity", a measure of the authority of a witness in a legal case in Europe, and often correlated with the witness's nobility. In a sense, this differs much from the modern meaning of probability, which in contrast is a measure of the weight of empirical evidence, and is arrived at from inductive reasoning and statistical inference.^[10]

History

The scientific study of probability is a modern development of mathematics. Gambling shows that there has been an interest in quantifying the ideas of probability throughout history, but exact mathematical descriptions arose much later. There are reasons for the slow development of the mathematics of probability. Whereas games of chance provided the impetus for the mathematical study of probability, fundamental issues ^{[note 2]} are still obscured by superstitions.^[11]

According to Richard Jeffrey, "Before the middle of the seventeenth century, the term 'probable' (Latin probabilis) meant approvable, and was applied in that sense, univocally, to opinion and to action. A probable action or opinion was one such as sensible people would undertake or hold, in the circumstances."^[12] However, in legal contexts especially, 'probable' could also apply to propositions for which there was good evidence.^[13]

The sixteenth-century Italian polymath Gerolamo Cardano demonstrated the efficacy of defining odds as the ratio of favourable to unfavourable outcomes (which implies that the probability of an event is given by the ratio of favourable outcomes to the total number of possible outcomes^[14]). Aside from the elementary work by Cardano, the doctrine of probabilities dates to the correspondence of

The first two laws of error that were proposed both originated with Pierre-Simon Laplace. The first law was published in 1774, and stated that the frequency of an error could be expressed as an exponential function of the numerical magnitude of the error – disregarding sign. The second law of error was proposed in 1778 by Laplace, and stated that the frequency of the error is an exponential function of the square of the error.^[19] The second law of error is called the normal distribution or the Gauss law. "It is difficult historically to attribute that law to Gauss, who in spite of his well-known precocity had probably not made this discovery before he was two years old."^[19]

Daniel Bernoulli (1778) introduced the principle of the maximum product of the probabilities of a system of concurrent errors.

, editor of "The Analyst" (1808), first deduced the law of facility of error,

$${\displaystyle \phi (x)=ce^{-h^{2}x^{2}},}$$

where ${\displaystyle h}$ is a constant depending on precision of observation, and ${\displaystyle c}$ is a scale factor ensuring that the area under the curve equals 1. He gave two proofs, the second being essentially the same as

On the geometric side, contributors to The Educational Times included Miller, Crofton, McColl, Wolstenholme, Watson, and Artemas Martin.^[23] See integral geometry for more information.

Theory

Like other theories, the theory of probability is a representation of its concepts in formal terms – that is, in terms that can be considered separately from their meaning. These formal terms are manipulated by the rules of mathematics and logic, and any results are interpreted or translated back into the problem domain.

There have been at least two successful attempts to formalize probability, namely the

There are other methods for quantifying uncertainty, such as the Dempster–Shafer theory or possibility theory, but those are essentially different and not compatible with the usually-understood laws of probability.

Applications

Probability theory is applied in everyday life in

Consider an experiment that can produce a number of results. The collection of all possible results is called the sample space of the experiment, sometimes denoted as ${\displaystyle \Omega }$. The power set of the sample space is formed by considering all different collections of possible results. For example, rolling a die can produce six possible results. One collection of possible results gives an odd number on the die. Thus, the subset {1,3,5} is an element of the power set of the sample space of dice rolls. These collections are called "events". In this case, {1,3,5} is the event that the die falls on some odd number. If the results that actually occur fall in a given event, the event is said to have occurred.

A probability is a way of assigning every event a value between zero and one, with the requirement that the event made up of all possible results (in our example, the event {1,2,3,4,5,6}) is assigned a value of one. To qualify as a probability, the assignment of values must satisfy the requirement that for any collection of mutually exclusive events (events with no common results, such as the events {1,6}, {3}, and {2,4}), the probability that at least one of the events will occur is given by the sum of the probabilities of all the individual events.^[28]

The probability of an event A is written as ${\displaystyle P(A)}$,^[29] ${\displaystyle p(A)}$, or ${\displaystyle {\text{Pr}}(A)}$.^[30] This mathematical definition of probability can extend to infinite sample spaces, and even uncountable sample spaces, using the concept of a measure.

The opposite or complement of an event A is the event [not A] (that is, the event of A not occurring), often denoted as ${\displaystyle A',A^{c}}$, ${\displaystyle {\overline {A}},A^{\complement },\neg A}$, or ${\displaystyle {\sim }A}$; its probability is given by P(not A) = 1 − P(A).^[31] As an example, the chance of not rolling a six on a six-sided die is 1 – (chance of rolling a six) = 1 − 1/6 = 5/6. For a more comprehensive treatment, see Complementary event.

If two events A and B occur on a single performance of an experiment, this is called the intersection or

${\displaystyle P(A\cap B).}$

Independent events

If two events, A and B are independent then the joint probability is^[29]

$${\displaystyle P(A{\mbox{ and }}B)=P(A\cap B)=P(A)P(B).}$$

For example, if two coins are flipped, then the chance of both being heads is ${\displaystyle {\tfrac {1}{2}}\times {\tfrac {1}{2}}={\tfrac {1}{4}}.}$^[32]

Mutually exclusive events

If either event A or event B can occur but never both simultaneously, then they are called mutually exclusive events.

If two events are

${\displaystyle P(A\cap B)}$

$${\displaystyle P(A{\mbox{ and }}B)=P(A\cap B)=0}$$

${\displaystyle P(A\cup B)}$

$${\displaystyle P(A{\mbox{ or }}B)=P(A\cup B)=P(A)+P(B)-P(A\cap B)=P(A)+P(B)-0=P(A)+P(B)}$$

${\displaystyle P(1{\mbox{ or }}2)=P(1)+P(2)={\tfrac {1}{6}}+{\tfrac {1}{6}}={\tfrac {1}{3}}.}$

Not (necessarily) mutually exclusive events

If the events are not (necessarily) mutually exclusive then

$${\displaystyle P\left(A{\hbox{ or }}B\right)=P(A\cup B)=P\left(A\right)+P\left(B\right)-P\left(A{\mbox{ and }}B\right).}$$

$${\displaystyle P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)-P\left(A\cap B\right)}$$

${\displaystyle {\tfrac {13}{52}}+{\tfrac {12}{52}}-{\tfrac {3}{52}}={\tfrac {11}{26}},}$

This can be expanded further for multiple not (necessarily) mutually exclusive events. For three events, this proceeds as follows:

$${\displaystyle {\begin{aligned}P\left(A\cup B\cup C\right)=&P\left(\left(A\cup B\right)\cup C\right)\\=&P\left(A\cup B\right)+P\left(C\right)-P\left(\left(A\cup B\right)\cap C\right)\\=&P\left(A\right)+P\left(B\right)-P\left(A\cap B\right)+P\left(C\right)-P\left(\left(A\cap C\right)\cup \left(B\cap C\right)\right)\\=&P\left(A\right)+P\left(B\right)+P\left(C\right)-P\left(A\cap B\right)-\left(P\left(A\cap C\right)+P\left(B\cap C\right)-P\left(\left(A\cap C\right)\cap \left(B\cap C\right)\right)\right)\\P\left(A\cup B\cup C\right)=&P\left(A\right)+P\left(B\right)+P\left(C\right)-P\left(A\cap B\right)-P\left(A\cap C\right)-P\left(B\cap C\right)+P\left(A\cap B\cap C\right)\end{aligned}}}$$

Conditional probability

Conditional probability is the probability of some event A, given the occurrence of some other event B. Conditional probability is written ${\displaystyle P(A\mid B)}$, and is read "the probability of A, given B". It is defined by^[33]

$${\displaystyle P(A\mid B)={\frac {P(A\cap B)}{P(B)}}.\,}$$

${\displaystyle P(B)=0}$

${\displaystyle P(A\mid B)}$

${\displaystyle A}$

${\displaystyle B}$

${\displaystyle P(A\cap B)=P(A)P(B)=0.}$

In probability theory and applications, Bayes' rule relates the odds of event ${\displaystyle A_{1}}$ to event ${\displaystyle A_{2},}$ before (prior to) and after (posterior to) conditioning on another event ${\displaystyle B.}$ The odds on ${\displaystyle A_{1}}$ to event ${\displaystyle A_{2}}$ is simply the ratio of the probabilities of the two events. When arbitrarily many events ${\displaystyle A}$ are of interest, not just two, the rule can be rephrased as posterior is proportional to prior times likelihood, ${\displaystyle P(A|B)\propto P(A)P(B|A)}$ where the proportionality symbol means that the left hand side is proportional to (i.e., equals a constant times) the right hand side as ${\displaystyle A}$ varies, for fixed or given ${\displaystyle B}$ (Lee, 2012; Bertsch McGrayne, 2012). In this form it goes back to Laplace (1774) and to Cournot (1843); see Fienberg (2005). See

Summary of probabilities

Summary of probabilities

Event	Probability
A	${\displaystyle P(A)\in [0,1]}$
not A	${\displaystyle P(A^{\complement })=1-P(A)\,}$
A or B	${\displaystyle {\begin{aligned}P(A\cup B)&=P(A)+P(B)-P(A\cap B)\\P(A\cup B)&=P(A)+P(B)\qquad {\mbox{if A and B are mutually exclusive}}\\\end{aligned}}}$
A and B	${\displaystyle {\begin{aligned}P(A\cap B)&=P(A|B)P(B)=P(B|A)P(A)\\P(A\cap B)&=P(A)P(B)\qquad {\mbox{if A and B are independent}}\\\end{aligned}}}$
A given B	${\displaystyle P(A\mid B)={\frac {P(A\cap B)}{P(B)}}={\frac {P(B|A)P(A)}{P(B)}}\,}$

Relation to randomness and probability in quantum mechanics

In a

• Contingency
• Equiprobability
• Fuzzy logic
• Heuristic (psychology)

Notes

References

Bibliography

• ISBN 0-387-25115-4
• Kallenberg, O. (2002) Foundations of Modern Probability, 2nd ed. Springer Series in Statistics. 650 pp. 
  ISBN 0-387-95313-2
• Olofsson, Peter (2005) Probability, Statistics, and Stochastic Processes, Wiley-Interscience. 504 pp
  ISBN 0-471-67969-0
  .

External links

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Library resources about
Probability

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• Resources in your library

• Virtual Laboratories in Probability and Statistics (Univ. of Ala.-Huntsville)
• Probability on In Our Time at the BBC
• Probability and Statistics EBook
• Edwin Thompson Jaynes. Probability Theory: The Logic of Science. Preprint: Washington University, (1996). – HTML index with links to PostScript files and PDF (first three chapters)
• People from the History of Probability and Statistics (Univ. of Southampton)
• Probability and Statistics on the Earliest Uses Pages (Univ. of Southampton)
• Earliest Uses of Symbols in Probability and Statistics on Earliest Uses of Various Mathematical Symbols
• A tutorial on probability and Bayes' theorem devised for first-year Oxford University students
• U B U W E B :: La Monte Young pdf file of An Anthology of Chance Operations (1963) at UbuWeb
• Introduction to Probability – eBook Archived 27 July 2011 at the Wayback Machine, by Charles Grinstead, Laurie Snell Source Archived 25 March 2012 at the Wayback Machine (GNU Free Documentation License)
• (in English and Italian)
  ISBN 88-8091-176-7
  (digital version)
• Richard Feynman's Lecture on probability.