Independent events and their complements

• 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
• ISSN: 0020-739X; 1464-5211
• DOI: 10.1080/0020739X.2020.1807067

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.