I was asked to prove the following:
If $\{ E_1,E_2,\ldots,E_n\}$ is a fully independent set of events, prove that:
$$ P\left(\bigcup_{i = 1}^n E_i \right) = 1 - \prod_{i = 1}^n \left[ 1 - P(E_i) \right] $$
(Hint: use complements)
I got the following:
Let $\alpha = E_1\cup E_2 \cup \cdots \cup E_n = \{E^C_1 \cap E^C_2 \cap \cdots\cap E_n^C\}^C$. Then, $\alpha^C = \{ E^C_1 \cap E^C_2 \cap \cdots \cap E^C\} $.
Assuming that the set of complements of each ${E_i}$ is also fully independent, we have:
\begin{align} & P(\alpha^C) = \prod_{i = 1}^n P(E_i^C) = \prod_{i = 1}^n \left[ 1 - P(E_i) \right] \\ & P(\alpha) = 1 - P(\alpha^C) = 1 - \prod_{i = 1}^n \left[ 1 - P(E_i) \right] \\ & P(\bigcup_{i = 1}^n E_i) = 1 - \prod_{i = 1}^n \left[ 1 - P(E_i) \right] \end{align}
However, even though I know (by Googling it) that my assumption is true, I'm unable to prove it so I could validly use it as a lemma. Is there another way to prove this equality?