Independent events and their complements

Independent events and their complements

Take a family of independent events. If some of these events, or all of them, are replaced by their complements, then independence still holds. This fact, which is agreed upon by the members of the statistical/probability communities, is tremendously well known, is fairly intuitive and has always be...

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Published in:International journal of mathematical education in science and technology Vol. 52; no. 6; pp. 937 - 947
Main Authors: Crispim, Carolina Martins, Mizuno, Gabriel Perez, Pizzinga, Adrian
Format: Journal Article
Language:English
Published: London Taylor & Francis 03-07-2021
Taylor & Francis Ltd
Subjects:
60-02
60A-99
97-02
Basic probability
complement
independent events
Introductory Courses
Intuition
mathematical induction
Mathematical Logic
Mathematics Instruction
Probability
Probability theory
Statistical analysis
Statistics
Validity
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Abstract Take a family of independent events. If some of these events, or all of them, are replaced by their complements, then independence still holds. This fact, which is agreed upon by the members of the statistical/probability communities, is tremendously well known, is fairly intuitive and has always been frequently used for easing probability calculations and allowing important developments in probability theory. However, can this result be proven without advanced elements from probability theory, such as independent σ-fields and related results? This paper aims primarily to fill this possible gap. Two elementary proofs that require only introductory probability content are given in detail and compared with the 'technical' proof found in books on advanced probability theory. A discussion regarding how these proofs could be included in regular probability courses is also included in the survey.
AbstractList Take a family of independent events. If some of these events, or all of them, are replaced by their complements, then independence still holds. This fact, which is agreed upon by the members of the statistical/probability communities, is tremendously well known, is fairly intuitive and has always been frequently used for easing probability calculations and allowing important developments in probability theory. However, can this result be proven without advanced elements from probability theory, such as independent σ-fields and related results? This paper aims primarily to fill this possible gap. Two elementary proofs that require only introductory probability content are given in detail and compared with the 'technical' proof found in books on advanced probability theory. A discussion regarding how these proofs could be included in regular probability courses is also included in the survey.
Take a family of independent events. If some of these events, or all of them, are replaced by their complements, then independence still holds. This fact, which is agreed upon by the members of the statistical/probability communities, is tremendously well known, is fairly intuitive and has always been frequently used for easing probability calculations and allowing important developments in probability theory. However, can this result be proven without advanced elements from probability theory, such as independent s-fields and related results? This paper aims primarily to fill this possible gap. Two elementary proofs that require only introductory probability content are given in detail and compared with the 'technical' proof found in books on advanced probability theory. A discussion regarding how these proofs could be included in regular probability courses is also included in the survey.
Author Mizuno, Gabriel Perez
Crispim, Carolina Martins
Pizzinga, Adrian
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Snippet Take a family of independent events. If some of these events, or all of them, are replaced by their complements, then independence still holds. This fact,...
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SubjectTerms 60-02
60A-99
97-02
Basic probability
complement
independent events
Introductory Courses
Intuition
mathematical induction
Mathematical Logic
Mathematics Instruction
Probability
Probability theory
Statistical analysis
Statistics
Validity
Title Independent events and their complements
URI https://www.tandfonline.com/doi/abs/10.1080/0020739X.2020.1807067
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