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Let $(\Omega,F,P)$ be a probability space. It is easy to prove that

$\{A_i\}_{i \in I}$ set of independent events $\Rightarrow \{A^c_i\}_{i \in I}$ set of independent events

However, I have been told that it is straightforward to see that every set containing $A_i$'s or $A_i^c$'s (for example $\{A_1,A_2^c,A_3^c,A_4^c,A_5\}$ where $I=\{1,2,3,4,5\}$) follows to be a set of independent events. I don't see how can this be deducted so straightforward.

Numox
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    If you have a set of events that are independent, can you think of how to show that just replacing one of the events in the set by its complement results in another set of independent events? – Joe May 05 '21 at 18:16
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    In the case in which $I$ is finite, you can use the same technique as described here (https://math.stackexchange.com/questions/1192151/prove-complements-of-independent-events-are-independent). Just replace one event at a time. – Dabbler May 05 '21 at 19:57
  • Thank you! Done :D – Numox May 06 '21 at 10:09

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