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Abstract
In a previous classroom note (Crispim et al., 2021), we addressed the proving (and the teaching) of a former and famous probability fact: if n random events are independent, then changing one, some, or all of them by the corresponding complements ends up again with independent events. Now, we explore the consequences. The first, for example, is the extension to when the number of events are infinite – which, indeed and perhaps surprisingly, stands as a corollary of the finite case. Also, and beyond covering only two events, we deal with the logical and intuitive consistency between independence and conditional probability. The theoretical content concludes with a proof that two seemingly different definitions of independent events found in the literature actually compare. Then, we finish the paper by discussing some classroom possibilities.
Acknowledgments
We are sincerely in debt with three anonymous Reviewers, whose comments, suggestions and requirements led to a very improved version of this paper. All remaining errors are ours.
Disclosure statement
No potential conflict of interest was reported by the authors.
Notes
1 Even though not exactly the most frequent in the literature, this quite lean definition of σ-field implies, fortunately, that both the sure event Ω and the impossible event are random (why?). The former has probability 1 (by the very definition of P) whereas the latter, zero probability (why?).
2 In fact, this is how occurrence of an event has been defined in some well-established literature. See, for instance, Feller, Citation1968, Hoel et al., Citation1972, Larson, Citation1982, and Casella & Berger, Citation2002.
3 Also in that very page of his book, Chung says, and from his words the emphasis is quite undeniable: ‘It may be said that no one could have learned the subject (probability theory) properly without acquiring some feeling for the intuitive content of the concept of stochastic independence, and through it, certain degrees of dependence’.
4 Actually we have a bit more. In proving the converse, we are allowed to replace and by or in (b), since each of the identities alone implies (a) already. If one chooses, at her/his discretion, to proceed this way (like we do in our classes), it is also worth suggesting to add quotes to some words: ‘reciprocally’, ‘conversely’, etc.
5 The four types of proof proposed in Weber's taxonomy are: (i) proofs that convince; (ii) that explain; (iii) that justify structure; and (iii) that illustrate technique.