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Given an infinite sequence of infinite strings, where each character of each string is chosen uniformly at random, any given finite string almost surely occurs as a prefix of one of these strings. Both follow easily from the second Borel–Cantelli lemma. For the second theorem, let E k be the event that the kth string begins with the given text.
The third event in a series of events becomes "the final trigger for something important to happen." This pattern appears in childhood stories such as "Goldilocks and the Three Bears", "Cinderella", and "Little Red Riding Hood". In adult stories, the Rule of Three conveys the gradual resolution of a process that leads to transformation. This ...
The Greek philosophers were the earliest Western thinkers to address chance and randomness. Around 400 BC, Democritus presented a view of the world as governed by the unambiguous laws of order and considered randomness as a subjective concept that only originated from the inability of humans to understand the nature of events. He used the ...
Randomization is not haphazard; instead, a random process is a sequence of random variables describing a process whose outcomes do not follow a deterministic pattern but follow an evolution described by probability distributions. For example, a random sample of individuals from a population refers to a sample where every individual has a known ...
Traditional probability theory does not state if a specific sequence is random, but generally proceeds to discuss the properties of random variables and stochastic sequences assuming some definition of randomness. The Bourbaki school considered the statement "let us consider a random sequence" an abuse of language. [3]
In literature, internal analepsis is a flashback to an earlier point in the narrative; external analepsis is a flashback to a time before the narrative started. [ 4 ] In film, flashbacks depict the subjective experience of a character by showing a memory of a previous event and they are often used to "resolve an enigma". [ 5 ]
Myles Colvin rebounded Braden Smith's missed fadeaway jumper and scored with 0.5 seconds left to give No. 13 Purdue an 80-78 win over No. 23 Ole Miss in the championship game of the Rady Children ...
The term Bernoulli sequence is often used informally to refer to a realization of a Bernoulli process. However, the term has an entirely different formal definition as given below. Suppose a Bernoulli process formally defined as a single random variable (see preceding section). For every infinite sequence x of coin flips, there is a sequence of ...