Independent events and their complements. Part II

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.