Almost surely
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| + | [[Image:Chimpanzee Typing (1907) - New York Zoological Society.jpg|thumb|200px|Given enough time, a [[chimpanzee]] punching at [[random]] on a [[typewriter]] would [[almost surely]] type out all of [[William Shakespeare|Shakespeare's]] plays. | ||
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| + | Photo: ''[[Chimpanzee Typing]]'' (1907) - New York Zoological Society]] | ||
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| + | In [[probability theory]], one says that an [[event (probability theory)|event]] happens '''almost surely''' (sometimes abbreviated as '''a.s.''') if it happens with probability one. The concept is analogous to the concept of "[[almost everywhere]]" in [[measure theory]]. While there is no difference between ''almost surely'' and ''surely'' (that is, entirely certain to happen) in many basic probability experiments, the distinction is important in more complex cases relating to some sort of [[infinity]]. For instance, the term is often encountered in questions that involve infinite time, regularity properties or infinite-[[dimension]]al spaces such as [[function space]]s. Basic examples of use include the [[law of large numbers]] (strong form) or continuity of [[Brownian motion|Brownian paths]]. | ||
| - | [[Carlo Rovelli]] adds that certainty, in real life, is useless or often damaging (the idea is that "total security from error" is impossible in practice, and a complete "lack of doubt" is undesirable). | + | '''Almost never''' describes the opposite of ''almost surely''; an event which happens with probability zero happens ''almost never''. |
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| - | * [[Almost surely]] | + | |
| + | == See also == | ||
| + | * [[Convergence of random variables]], for "almost sure convergence" | ||
| + | * [[Degenerate distribution]], for "almost surely constant" | ||
| + | * [[Almost everywhere]], the corresponding concept in measure theory | ||
| + | * [[Infinite monkey theorem]], a theorem using the aforementioned terms. | ||
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Photo: Chimpanzee Typing (1907) - New York Zoological Society
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In probability theory, one says that an event happens almost surely (sometimes abbreviated as a.s.) if it happens with probability one. The concept is analogous to the concept of "almost everywhere" in measure theory. While there is no difference between almost surely and surely (that is, entirely certain to happen) in many basic probability experiments, the distinction is important in more complex cases relating to some sort of infinity. For instance, the term is often encountered in questions that involve infinite time, regularity properties or infinite-dimensional spaces such as function spaces. Basic examples of use include the law of large numbers (strong form) or continuity of Brownian paths.
Almost never describes the opposite of almost surely; an event which happens with probability zero happens almost never.
See also
- Convergence of random variables, for "almost sure convergence"
- Degenerate distribution, for "almost surely constant"
- Almost everywhere, the corresponding concept in measure theory
- Infinite monkey theorem, a theorem using the aforementioned terms.
