(related to Proposition: Replacing Mutually Independent Events by Their Complements)
Assume \(A_1,...,A_n\) are mutually independent events. Without any loss of generality, we can replace \(A_1\) by the complement event \(\overline{A_1}\). It follows from the probability of event difference and the probability of the complement event:
\[\begin{array}{rcll} p(\overline{A_1}\cap A_2\cap\ldots \cap A_n)&=&p(A_2\cap\ldots \cap A_n)-p(A_1\cap A_2\cap\ldots \cap A_n)&\text{probability of event difference}\\ &=&p(A_2)\cdot\ldots \cdot p(A_n)-p(A_1)\cdot p(A_2)\cdot \ldots \cdot p(A_n)&\text{mutual independence}\\ &=&(1-p(A_1))\cdot p(A_2)\cdot\ldots \cdot p(A_n)&\text{distributivity of real numbers}\\ &=&p(\overline{A_1})\cdot p(A_2)\cdot\ldots \cdot p(A_n)&\text{probability of complement event}\\ \end{array} \]
Thus, replacing any \(A_i\) by the complement event \(\overline{A_i}\) results in mutually independent events.
