probability: Definition, Synonyms and Much More from Answers.com
- ️Wed Jul 01 2015
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
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Probability is the likelihood that something is the case or will happen. Probability theory is used extensively in areas such as statistics, mathematics, science and philosophy to draw conclusions about the likelihood of potential events and the underlying mechanics of complex systems.
Interpretations
The word probability does not have a consistent direct definition. Actually, there are two broad categories of probability interpretations: Frequentists talk about probabilities only when dealing with well defined random experiments. The relative frequency of occurrence of an experiment's outcome, when repeating the experiment, is a measure of the probability of that random event. Bayesians, however, assign probabilities to any statement whatsoever, even when no random process is involved, as a way to represent its subjective plausibility.
History
- Further information: Statistics
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 of use in those problems only arose much later.
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."[1]
Aside from some elementary considerations made 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. Jakob Bernoulli's Ars Conjectandi (posthumous, 1713) and Abraham de Moivre's Doctrine of Chances (1718) treated the subject as a branch of mathematics. See Ian Hacking's The Emergence of Probability for a history 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. The reprint (1757) of this memoir lays down the axioms that positive and negative errors are equally probable, and that there are certain assignable limits within which all errors may be supposed to fall; continuous errors are discussed and a probability curve is given.
Pierre-Simon Laplace (1774) made the first attempt to deduce a rule for the combination of observations from the principles of the theory of probabilities. He represented the law of probability of errors by a curve y = φ(x), x being any error and y its probability, and laid down three properties of this curve:
- it is symmetric as to the y-axis;
- the x-axis is an asymptote, the probability of the error ∞ being 0;
- the area enclosed is 1, it being certain that an error exists.
He deduced a formula for the mean of three observations. He also gave (1781) a formula for the law of facility of error (a term due to Lagrange, 1774), but one which led to unmanageable equations. Daniel Bernoulli (1778) introduced the principle of the maximum product of the probabilities of a system of concurrent errors.
The method of least squares is due to Adrien-Marie Legendre (1805), who 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,
c and h being constants depending on precision of observation. He gave two proofs, the second being essentially the same as John Herschel's (1850). Gauss gave the first proof which 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.
On the geometric side (see integral geometry) contributors to The Educational Times were influential (Miller, Crofton, McColl, Wolstenholme, Watson, and Artemas Martin).
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 then interpreted or translated back into the problem domain.
In probability theory, a probability is represented by a real number in the range from 0 to 1. An impossible event has a probability of 0, and a certain event has a probability of 1. (However, other events may also have probabilities 0 or 1: events with probability 0 are not necessarily impossible, and those with probability 1 are not necessarily certain. For examples, see Almost surely.)
There are other methods for quantifying uncertainty, such as the Dempster-Shafer theory and possibility theory, but those are essentially different and not compatible with the laws of probability as they are usually understood.
Applications
Two major applications of probability theory in everyday life are in risk assessment and in trade on commodity markets. Governments typically apply probabilistic methods in environmental regulation where it is called "pathway analysis", often measuring well-being using methods that are stochastic in nature, and choosing projects to undertake based on statistical analyses of their probable effect on the population as a whole. 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 effect of such choices, which makes probability measures a political matter.
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 not 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.
It can reasonably be said that the discovery of rigorous methods to assess and combine probability assessments has had a profound effect on 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, utilize reliability theory in the design of the product in order to reduce the probability of failure. The probability of failure is also closely associated with the product's warranty.
It could be said that there is no such thing as probability. It could also be said that probability is a measure of our degree of uncertainty, or that is, the extent of our ignorance in a given situation. Consequently, there might be a probability of 1 in 52 that the top card in a deck of cards is the Jack of diamonds. However, if one looks at the top card and replaces it, then the probability is either 100% or zero percent, and the correct choice can be accurately made by the viewer. Modern physics provides important examples of deterministic situations where only probabilistic description is feasible due to incomplete information and complexity of a system as well as examples of truly random phenomena.
In a deterministic universe, based on Newtonian concepts, there is no probability
if all conditions are known. In the case of a roulette wheel, if the force of the hand and the period of that force are known,
then 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 analysing 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 
) that only statistical description of
its properties is feasible.
A revolutionary discovery of 20th century physics was the random character of all physical processes that occur at microscopic scales and are governed by the laws of quantum mechanics. The wave function itself evolves deterministically as long as no observation is made, but, according to the prevailing Copenhagen interpretation, the randomness of the wave function collapse when an observation is made, is fundamental. This means that probability theory is required to describe nature. Some scientists spoke of expulsion from Paradise.[citation needed] Others never came to terms with the loss of determinism. Albert Einstein famously remarked in a letter to Max Born: Jedenfalls bin ich überzeugt, daß der Alte nicht würfelt. (I am convinced that God does not play dice). Although alternative viewpoints exist, such as that of quantum decoherence being the cause of an apparent random collapse, at present there is a firm consensus among the physicists that probability theory is necessary to describe quantum phenomena.
See also
References
- ^ Jeffrey, R.C., Probability and the Art of Judgment, Cambridge University Press. (1992). pp. 54-55 . ISBN 0-521-39459-7
- Olav Kallenberg, Probabilistic Symmetries and Invariance Principles. Springer -Verlag, New York (2005). 510 pp. ISBN 0-387-25115-4
- Kallenberg, O., Foundations of Modern Probability, 2nd ed. Springer Series in Statistics. (2002). 650 pp. ISBN 0-387-95313-2
Quotations
- Damon Runyon, "It may be that the race is not always to the swift, nor the battle to the strong - but that is the way to bet."
- Pierre-Simon Laplace "It is remarkable that a science which began with the consideration of games of chance should have become the most important object of human knowledge." Théorie Analytique des Probabilités, 1812.
- Richard von Mises "The unlimited extension of the validity of the exact sciences was a characteristic feature of the exaggerated rationalism of the eighteenth century" (in reference to Laplace). Probability, Statistics, and Truth, p 9. Dover edition, 1981 (republication of second English edition, 1957).
External links
- Edwin Thompson Jaynes. Probability Theory: The Logic of Science. Preprint: Washington University, (1996). — HTML index with links to PostScript files and PDF
- Dictionary of the History of Ideas: Certainty in Seventeenth-Century Thought
- Dictionary of the History of Ideas: Certainty since the Seventeenth Century
- Figures from the History of Probability and Statistics (Univ. of Southampton)
- Probability and Statistics on the Earliest Uses Pages (Univ. of Southampton)
- Analytical argumentations of probability and statistics.
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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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