probability

(prŏb'ə-bĭl'ĭ-tē)

1. The quality or condition of being probable; likelihood.
2. A probable situation, condition, or event: Her election is a clear probability.
3. 1. The likelihood that a given event will occur: little probability of rain tonight.
   2. Statistics. A number expressing the likelihood that a specific event will occur, expressed as the ratio of the number of actual occurrences to the number of possible occurrences.

in all probability

1. Most probably; very likely.

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The probability of an event (see sample space) is a number lying in the interval 0≤p≤1, with 0 corresponding to an event that never occurs and 1 to an event that is certain to occur. For an experiment with N equally likely outcomes the probability of an event A is n/N, where n is the number of outcomes in which the event A occurs. For some experiments, such as throwing a drawing pin and seeing whether it lands point up, there is no possible set of equally likely outcomes. In the 'frequentist' view of probability, the probability of getting 'point up' is the limit, in some sense, of the relative frequency as the number of experiments tends to infinity. In the context of Bayesian inference, each observer has his or her own a priori distribution for the probability, which is then modified a posteriori in the light of whatever results have been obtained.

noun

The likeliness of a given event occurring: chance, likelihood, odds, possibility, prospect (used in plural). See likely/unlikely.

Definition: likelihood of something happening
Antonyms: improbability, unlikelihood

Probability measures the likelihood that something specific will occur. For example, a tossed coin has an equal chance, or probability, of landing with one side up ("heads") or the other ("tails"). If you drive without a seat belt, your probability of being injured in an accident is much higher than if you buckle up. Probability uses numbers to explain chance.

If something is absolutely going to happen, its probability of occurring is 1, or 100 percent. If something absolutely will not happen, its probability of occurring is 0, or 0 percent.

Probability is used as a tool in many areas of genetics. A clinical geneticist uses probability to determine the likelihood that a couple will have a baby with a specific genetic disease. A statistical geneticist uses probability to learn whether a disease is more common in one population than in another. A computational biologist uses probability to learn how a gene causes a disease.

The Clinical Geneticist and the Punnett Square

A Punnett square uses probability to explain what sorts of children two parents might have. Suppose a couple knows that cystic fibrosis, a debilitating respiratory disease, tends to run in the man's family. The couple would like to know how likely it is that they would pass on the disease to their children.

A clinical geneticist can use a Punnett square to help answer the couple's question. The clinical geneticist might start by explaining how the disease is inherited: Because cystic fibrosis is a recessive disease caused by a single gene, only children who inherit the disease-causing form of the cystic fibrosis gene from both parents display symptoms. On the other hand, because the cystic fibrosis gene is a recessive gene, a child who inherits only one copy of a defective gene, along with one normal version, will not have the disease.

Suppose the recessive, disease-causing form of the gene is referred to as "f" and the normal form of the gene is referred to as "F." Only individuals with two disease-causing genes, ff, would have the disease. Individuals with either two normal copies of the gene (FF) or one normal copy and one mutated copy (Ff) would be healthy.

If the clinical geneticist tests the parents and finds that each carries one copy of the cystic fibrosis gene, f, and one copy of the normal gene, F, what would be the probability that a baby of theirs would be born with the cystic fibrosis disease? To answer this question, we can use the Punnett square shown in the figure above. A Punnett square assumes that there is an equal probability that the parent will pass on either of its two gene forms ("alleles") to each child.

The parents' genes are represented along the edges of the square. A child inherits one gene from its mother and one from its father. The combinations of genes that the child of two Ff parents could inherit are represented by the boxes inside the square.

Of the four combinations possible, three involve the child's inheriting at least one copy of the dominant, healthy gene. In three of the four combinations, therefore, the child would not have cystic fibrosis. In only one of the four combinations would the child inherit the recessive allele from both parents. In that case, the child would have the disease. Based on the Punnett square, the counselor can tell the parents that there is a 25 percent probability, or a one-in-four chance, that their baby will have cystic fibrosis.

The Statistical Geneticist and the Chi-Square Test

Researchers often want to know whether one particular gene occurs in a population more or less frequently than another. This may help them determine, for example, whether the gene in question causes a particular disease. For a dominant gene, such as the one that causes Huntington's disease, the frequency of the disease can be used to determine the frequency of the gene, since everyone who has the gene will eventually develop the disease. However, it would be practically impossible to find every case of Huntington's disease, because it would require knowing the medical condition of every person in a population. Instead, genetic researchers sample a small subset of the population that they believe is representative of the whole. (The same technique is used in political polling.)

Whenever a sample is used, the possibility exists that it is unrepresentative, generating misleading data. Statisticians have a variety of methods to minimize sampling error, including sampling at random and using large samples. But sampling errors cannot be eliminated entirely, so data from the sample must be reported not just as a single number but with a range that conveys the precision and possible error of the data. Instead of saying the prevalence of Huntington's disease in a population is 10 per 100,000 people, a researcher would say the prevalence is 7.8-12.1 per 100,000 people.

The potential for errors in sampling also means that statistical tests must be conducted to determine if two numbers are close enough to be considered the same. When we take two samples, even if they are both from exactly the same population, there will always be slight differences in the samples that will make the results differ.

A researcher might want to determine if the prevalence of Huntington's disease is the same in the United States as it is in Japan, for example. The population samples might indicate that the prevalences, ignoring ranges, are 10 per 100,000 in the United States and 11 per 100,000 in Japan. Are these numbers close enough to be considered the same? This is where the Chi-square test is useful.

First we state the "null hypothesis," which is that the two prevalences are the same and that the difference in the numbers is due to sampling error alone. Then we use the Chi-square test, which is a mathematical formula, to test the hypothesis.

The test generates a measure of probability, called a p value, that can range from 0 percent to 100 percent. If the p value is close to 100 percent, the difference in the two numbers is almost certainly due to sampling error alone. The lower the p value, the less likely the difference is due solely to chance.

Scientists have agreed to use a cutoff value of 5 percent for most purposes. If the p value is less than 5 percent, the two numbers are said to be significantly different, the null hypothesis is rejected, and some other cause for the difference must be sought besides sampling error. There are many statistical tests and measures of significance in addition to the Chi-square test. Each is adapted for special circumstances.

Another application of the Chi-square test in genetics is to test whether a particular genotype is more or less common in a population than would be expected. The expected frequencies can be calculated from population data and the Hardy-Weinberg Equilibrium formula. These expected frequencies can then be compared to observed frequencies, and a p value can be calculated. A significant difference between observed and expected frequencies would indicate that some factor, such as natural selection or migration, is at work in the population, acting on allele frequencies. Population geneticists use this information to plan further studies to find these factors.

The Computational Biologist and Blast

Genetic counseling lets potential parents make an informed decision before they decide to have a child. Geneticists, however, would like to be able to take this one step further: They would like to be able to cure genetic diseases. To be able to do so, scientists must first understand how a disease-causing gene results in illness. Computational biologists created a computer program called BLAST to help with this task.

To use BLAST, a researcher must know the DNA sequence of the disease-causing gene or the protein sequence that the gene encodes. BLAST compares DNA or protein sequences. The program can be used to search many previously studied sequences to see if there are any that are similar to a newly found sequence. BLAST measures the strength of a match between two sequences with a p value. The smaller the p value, the lower the probability that the similarity is due to chance alone.

If two sequences are alike, their functions may also be alike. For BLAST to be most useful to a researcher, there would be a gene that has already been entered in the library that resembles the disease-causing gene, and some information would be known about the function of the previously entered gene. This would help the researcher begin to hypothesize how the disease-causing gene results in illness.

Bibliography

Nussbaum, Robert L., Roderick R. McInnes, and Huntington F. Willard. Thompson & Thompson Genetics in Medicine, 6th ed. St. Louis, MO: W. B. Saunders, 2001.

Purves, William K., et al. Life: The Science of Biology, 6th ed. Sunderland, MA: Sinauer Associates, 2001.

Seidman, Lisa, and Cynthia Moore. Basic Laboratory Methods for Biotechnology: Textbook and Laboratory Reference. Upper Saddle River, NJ: Prentice-Hall, 2000.

Tamarin, Robert H. Principles of Genetics, 7th ed. Dubuque, IA: William C. Brown,2001.

Internet Resources

The Dolan DNA Learning Center. Cold Spring Harbor Laboratory. http://vector.cshl.org.

The National Center for Biotechnology Information. http://www.ncbi.nlm.nih.gov.

—Rebecca S. Pearlman

The likelihood of an event occurring. In statistics, probability, p, is expressed as a number ranging from 0—absolute impossibility—to 1—absolute certainty. It may also be expressed as a percentage.

p = 0.05 is the 95% level

p = 0.01 is the 99% level

p = 0.001 is the 99.9% level

The likelihood that a given event will occur. Probability is expressed as values between 0 (complete certainty that an event will not occur) to 1 (complete certainty that an event will occur), or percentage values between 0 and 100%.

LearnThatWord.com is a free vocabulary and spelling program where you only pay for results!

A number between zero and one that shows how likely a certain event is. Usually, probability is expressed as a ratio: the number of experimental results that would produce the event divided by the number of experimental results considered possible. Thus, the probability of drawing the ten of clubs from an ordinary deck of cards is one in fifty-two (1:52), or one fifty-second.

The basis of statistics. The relative frequency of occurrence of a specific event as the outcome of an experiment when the experiment is conducted randomly on very many occasions. The probability of the event occurring is the number of times it did occur divided by the number of times that it could have occurred. Defined as:$$\hbox{p}={\hbox{x}\over (\hbox{x+y})$$

• where — p = probability, x = positive outcomes, y = negative outcomes.
• prior p. — estimation of the probability that a particular phenomenon or character will appear before putting the patient to the test, e.g. testing the probable productivity of a patient by testing its forebears.
• subjective p. — the measure of the assessor's belief in the probability of a proposition being correct.

1. a measure of the increased likelihood that something will occur. n 2. a mathematic ratio of the number of times something will occur to the total number of possible occurrences.

• Quantities, Relationships, and Operations - probability: ratio of number of outcomes in a given series of events to total number of possible outcomes in that series
• Order, Hierarchy, and Systems - probability: chance or likelihood that given event or result will occur

For the Law & Order: Criminal Intent episode, see Probability (Law & Order: Criminal Intent).

Certainty series
Agnosticism Belief Certainty Doubt Determinism Epistemology Estimation Fallibilism Fatalism Justification Nihilism Probability Skepticism Solipsism Truth Uncertainty
v d e

Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we are not certain.^[1] The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The certainty we adopt can be described in terms of a numerical measure and this number, between 0 and 1, we call probability.^[2] The higher the probability of an event, the more certain we are that the event will occur. Thus, probability in an applied sense is a measure of the likeliness that a (random) event will occur.

The concept has been given an axiomatic mathematical derivation in probability theory, which is used widely in such areas of study as mathematics, statistics, finance, gambling, science, artificial intelligence/machine learning and philosophy to, for example, draw inferences about the likeliness of events. Probability is used to describe the underlying mechanics and regularities of complex systems.

Contents 1 Interpretations 2 Etymology 3 History 4 Theory 5 Applications 6 Mathematical treatment 6.1 Independent probability 6.1.1 Mutually exclusive 6.1.2 Not mutually exclusive 6.2 Conditional probability 6.3 Summary of probabilities 7 Relation to randomness 8 See also 9 Notes 10 References 11 External links

Interpretations

Main article: Probability interpretations

The word probability does not have a singular direct definition for practical application. In fact, there are several broad categories of probability interpretations, whose adherents possess different (and sometimes conflicting) views about the fundamental nature of probability. For example:

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.^[3]
2. Subjectivists assign numbers per subjective probability, i.e., as a degree of belief.^[4]
3. Bayesians include expert knowledge as well as experimental data to produce probabilities. The expert knowledge is represented by a prior probability distribution. The data is incorporated in a likelihood function. The product of the prior and the likelihood, normalized, results in a posterior probability distribution that incorporates all the information known to date.^[5]

Etymology

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.^[6][7]

History

Main article: History of probability

The scientific study of probability is a modern development. Gambling shows that there has been an interest in quantifying the ideas of probability for millennia, but exact mathematical descriptions arose much later. There are reasons of course, for the slow development of the mathematics of probability. Whereas games of chance provided the impetus for the mathematical study of probability, fundamental issues are still obscured by the superstitions of gamblers.^[8]

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."^[9] However, in legal contexts especially, 'probable' could also apply to propositions for which there was good evidence.^[10]

Aside from elementary work by Girolamo Cardano in the 16th century, the doctrine of probabilities dates to the correspondence of Pierre de Fermat and Blaise Pascal (1654). Christiaan Huygens (1657) gave the earliest known scientific treatment of the subject.^{[citation needed]} Jakob Bernoulli's Ars Conjectandi (posthumous, 1713) and Abraham de Moivre's Doctrine of Chances (1718) treated the subject as a branch of mathematics.^[11] See Ian Hacking's The Emergence of Probability and James Franklin's The Science of Conjecture for histories of the early development of the very concept of mathematical probability.

The theory of errors may be traced back to Roger Cotes's Opera Miscellanea (posthumous, 1722), but a memoir prepared by Thomas Simpson in 1755 (printed 1756) first applied the theory to the discussion of errors of observation.^{[citation needed]} The reprint (1757) of this memoir lays down the axioms that positive and negative errors are equally probable, and that certain assignable limits define the range of all errors. Simpson also discusses continuous errors and describes a probability curve.

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.^[12] 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."^[12]

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

Adrien-Marie Legendre (1805) developed the method of least squares, and introduced it in his Nouvelles méthodes pour la détermination des orbites des comètes (New Methods for Determining the Orbits of Comets). In ignorance of Legendre's contribution, an Irish-American writer, Robert Adrain, editor of "The Analyst" (1808), first deduced the law of facility of error,

h being a constant depending on precision of observation, and c a scale factor ensuring that the area under the curve equals 1. He gave two proofs, the second being essentially the same as John Herschel's (1850). Gauss gave the first proof that seems to have been known in Europe (the third after Adrain's) in 1809. Further proofs were given by Laplace (1810, 1812), Gauss (1823), James Ivory (1825, 1826), Hagen (1837), Friedrich Bessel (1838), W. F. Donkin (1844, 1856), and Morgan Crofton (1870). Other contributors were Ellis (1844), De Morgan (1864), Glaisher (1872), and Giovanni Schiaparelli (1875). Peters's (1856) formula for r, the probable error of a single observation, is well known.

In the nineteenth century authors on the general theory included Laplace, Sylvestre Lacroix (1816), Littrow (1833), Adolphe Quetelet (1853), Richard Dedekind (1860), Helmert (1872), Hermann Laurent (1873), Liagre, Didion, and Karl Pearson. Augustus De Morgan and George Boole improved the exposition of the theory.

Andrey Markov introduced^{[citation needed]} the notion of Markov chains (1906), which played an important role in stochastic processes theory and its applications. The modern theory of probability based on the measure theory was developed by Andrey Kolmogorov (1931).

On the geometric side (see integral geometry) contributors to The Educational Times were influential (Miller, Crofton, McColl, Wolstenholme, Watson, and Artemas Martin).^{[citation needed]}

Further information: History of statistics

Theory

Main article: Probability theory

Like other theories, the theory of probability is a representation of probabilistic 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 Kolmogorov formulation and the Cox formulation. In Kolmogorov's formulation (see probability space), sets are interpreted as events and probability itself as a measure on a class of sets. In Cox's theorem, probability is taken as a primitive (that is, not further analyzed) and the emphasis is on constructing a consistent assignment of probability values to propositions. In both cases, the laws of probability are the same, except for technical details.

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 laws of probability as usually understood.

Applications

Probability theory is applied in everyday life in risk assessment and in trade on commodity markets. Governments typically apply probabilistic methods in environmental regulation, where it is called pathway analysis. A good example is the effect 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 trader 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 neither assessed independently nor necessarily very rationally. The theory of behavioral finance emerged to describe the effect of such groupthink on pricing, on policy, and on peace and conflict.^[13]

It can reasonably be said that the discovery of rigorous methods to assess and combine probability assessments has profoundly affected modern society. 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.

Another significant application of probability theory in everyday life is reliability. Many consumer products, such as automobiles and consumer electronics, use reliability theory in product design to reduce the probability of failure. Failure probability may influence a manufacture's decisions on a product's warranty.^[14]

The cache language model and other statistical language models that are used in natural language processing are also examples of applications of probability theory.

Mathematical treatment

See also: Probability axioms

Consider an experiment that can produce a number of results. The collection of all results is called the sample space of the experiment. 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 give an odd number on the die. Thus, the subset {1,3,5} is an element of the power set of the sample space of die 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 if you look at a collection of mutually exclusive events (events with no common results, e.g., the events {1,6}, {3}, and {2,4} are all mutually exclusive), the probability that at least one of the events will occur is given by the sum of the probabilities of all the individual events.^[15]

The probability of an event A is written as P(A), p(A) or Pr(A).^[16] 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); its probability is given by P(not A) = 1 - P(A).^[17] As an example, the chance of not rolling a six on a six-sided die is 1 – (chance of rolling a six) . See Complementary event for a more complete treatment.

If both events A and B occur on a single performance of an experiment, this is called the intersection or joint probability of A and B, denoted as .

Independent probability

If two events, A and B are independent then the joint probability is

for example, if two coins are flipped the chance of both being heads is ^[18]

Mutually exclusive

If either event A or event B or both events occur on a single performance of an experiment this is called the union of the events A and B denoted as . If two events are mutually exclusive then the probability of either occurring is

For example, the chance of rolling a 1 or 2 on a six-sided die is

Not mutually exclusive

If the events are not mutually exclusive then

For example, when drawing a single card at random from a regular deck of cards, the chance of getting a heart or a face card (J,Q,K) (or one that is both) is , because of the 52 cards of a deck 13 are hearts, 12 are face cards, and 3 are both: here the possibilities included in the "3 that are both" are included in each of the "13 hearts" and the "12 face cards" but should only be counted once.

Conditional probability

Conditional probability is the probability of some event A, given the occurrence of some other event B. Conditional probability is written , and is read "the probability of A, given B". It is defined by

^[19]

If P(B) = 0 then is undefined. Note that in this case A and B are independent.

Summary of probabilities

Summary of probabilities

Event	Probability
A
not A
A or B
A and B
A given B

Relation to randomness

Main article: Randomness

In a deterministic universe, based on Newtonian concepts, there would be no probability if all conditions are known, (Laplace's demon). In the case of a roulette wheel, if the force of the hand and the period of that force are known, the number on which the ball will stop would be a certainty. Of course, this also assumes knowledge of inertia and friction of the wheel, weight, smoothness and roundness of the ball, variations in hand speed during the turning and so forth. A probabilistic description can thus be more useful than Newtonian mechanics for analyzing the pattern of outcomes of repeated rolls of roulette wheel. Physicists face the same situation in kinetic theory of gases, where the system, while deterministic in principle, is so complex (with the number of molecules typically the order of magnitude of Avogadro constant 6.02·10²³) that only statistical description of its properties is feasible.

Probability theory is required to describe nature.^[20] A revolutionary discovery of early 20th century physics was the random character of all physical processes that occur at sub-atomic scales and are governed by the laws of quantum mechanics. The objective wave function evolves deterministically but, according to the Copenhagen interpretation, it deals with probabilities of observing, the outcome being explained by a wave function collapse when an observation is made. However, the loss of determinism for the sake of instrumentalism did not meet with universal approval. Albert Einstein famously remarked in a letter to Max Born: "I am convinced that God does not play dice".^[21] Like Einstein, Erwin Schrödinger, who discovered the wave function, believed quantum mechanics is a statistical approximation of an underlying deterministic reality.^[22] In modern interpretations, quantum decoherence accounts for subjectively probabilistic behavior.

See also

Logic portal

Main article: Outline of probability

• Chance (disambiguation)
• Class membership probabilities
• Equiprobable

This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (November 2007)

Notes

1. ^ Kendall's Advanced Theory of Statistics, Volume 1: Distribution Theory, Alan Stuart and Keith Ord, 6th Ed 2009
2. ^ An Introduction to Probability Theory and Its Applications, William Feller. 3rd Ed 1968
3. ^ Hacking, Ian (1965). The Logic of Statistical Inference. 
4. ^ Finetti, Bruno de (1970). "Logical foundations and measurement of subjective probability". Acta Psychologica 34: 129–145. doi:10.1016/0001-6918(70)90012-0. 
5. ^ Hogg, Robert V.; Craig, Allen; McKean, Joseph W. (2004). Introduction to Mathematical Statistics (6th ed.). Upper Saddle River: Pearson. ISBN 0130085073. 
6. ^ The Emergence of Probability: A Philosophical Study of Early Ideas about Probability, Induction and Statistical Inference, Ian Hacking, Cambridge University Press, 2006, ISBN 0521685575, 9780521685573
7. ^ The Cambridge History of Seventeenth-century Philosophy, Daniel Garber, 2003
8. ^ Freund, John. “Introduction to Probability”. 1973, p. 1.
9. ^ Jeffrey, R.C., Probability and the Art of Judgment, Cambridge University Press. (1992). pp. 54-55 . ISBN 0-521-39459-7
10. ^ Franklin, J., The Science of Conjecture: Evidence and Probability Before Pascal, Johns Hopkins University Press. (2001). pp. 22, 113, 127
11. ^ Ivancevic, Vladimir; Tijana Ivancevic. "Quantum Leap". 2008. p 16 ^{[Full citation needed]}
12. ^ ^{a b} Wilson EB (1923) First and second laws of error. JASA 18, 143
13. ^ Singh, Laurie. "Whither Efficient Markets? Efficient Market Theory and Behavioral Finance". The Finance Professionals' Post, 2010.
14. ^ Gorman, Michael. "Management Insights". Management Science, 2011.
15. ^ Ross, Sheldon. A First course in Probability, 8th Edition. Page 26-27.
16. ^ Olofsson, Peter. (2005) Page 8.
17. ^ Olofsson, page 9
18. ^ Olofsson, page 35.
19. ^ Olofsson, page 29.
20. ^ Burgi, Mark. ” Interpretations of Negative Probabilities”. 2009, p. 1.
21. ^ Jedenfalls bin ich überzeugt, daß der Alte nicht würfelt.
22. ^ Moore, W.J. (1992). Schrödinger: Life and Thought. Cambridge University Press. p. 479. ISBN 0-521-43767-9. 

References

• Kallenberg, O. (2005) Probabilistic Symmetries and Invariance Principles. Springer -Verlag, New York. 510 pp. 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

Wikiquote has a collection of quotations related to: Probability

Wikibooks has a book on the topic of Probability

• Virtual Laboratories in Probability and Statistics (Univ. of Ala.-Huntsville)
• Probability on In Our Time at the BBC. (listen now)
• 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
• Probability Homework Help, Definitions, Distribution Calculators and Study Guides
• A tutorial on probability and Bayes’ theorem devised for first-year Oxford University students
• pdf file of An Anthology of Chance Operations (1963) at UbuWeb
• Probability Theory Guide for Non-Mathematicians
• Understanding Risk and Probability with BBC raw
• Introduction to Probability - eBook, by Charles Grinstead, Laurie Snell Source (GNU Free Documentation License)

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v d e Statistics   Descriptive statistics Continuous data Location Mean (Arithmetic, Geometric, Harmonic) Median Mode Dispersion Range Standard deviation Coefficient of variation Percentile Interquartile range Shape Variance Skewness Kurtosis Moments L-moments Count data Index of dispersion Summary tables Grouped data Frequency distribution Contingency table Dependence Pearson product-moment correlation Rank correlation (Spearman's rho, Kendall's tau) Partial correlation Scatter plot Statistical graphics Bar chart Biplot Box plot Control chart Correlogram Forest plot Histogram Q-Q plot Run chart Scatter plot Stemplot Radar chart   Data collection Designing studies Effect size Standard error Statistical power Sample size determination Survey methodology Sampling Stratified sampling Opinion poll Questionnaire Controlled experiment Design of experiments Randomized experiment Random assignment Replication Blocking Factorial experiment Optimal design Uncontrolled studies Natural experiment Quasi-experiment Observational study   Statistical inference Statistical theory Sampling distribution Sufficient statistic Meta-analysis Bayesian inference Bayesian probability Prior Posterior Credible interval Bayes factor Bayesian estimator Maximum posterior estimator Frequentist inference Confidence interval Hypothesis testing Likelihood-ratio Specific tests Z-test (normal) Student's t-test F-test Pearson's chi-squared test Wald test Mann–Whitney U Shapiro–Wilk Signed-rank Kolmogorov–Smirnov test General estimation Bias Robustness Efficiency Maximum likelihood Method of moments Minimum distance Density estimation   Correlation and regression analysis Correlation Pearson product-moment correlation Partial correlation Confounding variable Coefficient of determination Regression analysis Errors and residuals Regression model validation Mixed effects models Simultaneous equations models Linear regression Simple linear regression Ordinary least squares General linear model Bayesian regression Non-standard predictors Nonlinear regression Nonparametric Semiparametric Isotonic Robust Generalized linear model Exponential families Logistic (Bernoulli) Binomial Poisson Partition of variance Analysis of variance (ANOVA) Analysis of covariance Multivariate ANOVA Degrees of freedom   Categorical, multivariate, time-series, or survival analysis Categorical data Cohen's kappa Contingency table Graphical model Log-linear model McNemar's test Multivariate statistics Multivariate regression Principal components Factor analysis Cluster analysis Copulas Time series analysis Decomposition (Trend, Stationary process) ARMA model ARIMA model Vector autoregression Spectral density estimation Survival analysis Survival function Kaplan–Meier Logrank test Failure rate Proportional hazards models Accelerated failure time model   Applications Biostatistics Bioinformatics Biometrics Clinical trials & studies Epidemiology Medical statistics Engineering statistics Chemometrics Methods engineering Probabilistic design Process & Quality control Reliability System identification Social statistics Actuarial science Census Crime statistics Demography Econometrics National accounts Official statistics Population Psychometrics Spatial statistics Cartography Environmental statistics Geographic information system Geostatistics Kriging Category Portal Outline Index

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Common misspelling(s) of probability

• probalibity

Dansk (Danish)
n. - sandsynlighed, mulighed

idioms:

• in all probability    efter al sandsynlighed

Nederlands (Dutch)
waarschijnlijkheid, kans, statistiek

Français (French)
n. - chances, risques, probabilités, probabilité, (Math, Stat) probabilité

idioms:

• in all probability    selon toute probabilité

Deutsch (German)
n. - Wahrscheinlichkeit

idioms:

• in all probability    aller Wahrscheinlichkeit nach

Ελληνική (Greek)
n. - πιθανότητα

idioms:

• in all probability    κατά πάσαν πιθανότητα

Italiano (Italian)
probabilità

idioms:

• in all probability    con tutta probabilità, con ogni probabilità

Português (Portuguese)
n. - probabilidade (f)

idioms:

• in all probability    provavelmente

Русский (Russian)
вероятность, возможность

idioms:

• in all probability    по всей вероятности

Español (Spanish)
n. - probabilidad

idioms:

• in all probability    según toda probabilidad

Svenska (Swedish)
n. - sannolikhet, rimlighet

中文（简体）(Chinese (Simplified))
可能性, 机率, 或然率

idioms:

• in all probability    很可能, 多半

中文（繁體）(Chinese (Traditional))
n. - 可能性, 機率, 或然率

idioms:

• in all probability    很可能, 多半

한국어 (Korean)
n. - 있음 직함, 일어남 직함

idioms:

• in all probability    아마, 십중팔구는

日本語 (Japanese)
n. - 見込み, 公算, 蓋然性, ありそうな事柄, 確率, ありそうなこと

idioms:

• in all probability    たぶん

العربيه (Arabic)
‏(الاسم) احتمال, احتماليه‏

עברית (Hebrew)
n. - ‮קירבה לוודאות, סיכוי, אפשרות, הסתברות, ייתכנות‬

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