How can we prove that union of independent events is equal to 1-intersection of all events?

Question

Prove that if $E_1, E_2, E_3,...,E_n $ are independent events, then $P (E_1 \cup E_2 \cup E_3 ... E_n) = 1 - \prod \limits_{i=1}^{n} [1- P(E_i)]$. The left hand side shows the probability of complete set $E_n$, and the $[1- P(E_i)]$ shows the complement of $E_i$, I am confused by this product $\prod \limits_{i=1}^{n}$ sign, it is the sum of products, which should be the intersection of all $E_i$ ?

Answer 1

$\prod_{i=1}^n a_i$ means the product, i.e. computing $a_1 \times a_2 \times \ldots \times a_n$.

By De Morgan's rule,

$$P \left( \bigcup_{i=1}^nE_i\right)=1-P \left( \bigcap_{i=1}^nE_i^c\right)=1-\prod_{i=1}^nP \left( E_i^c\right)$$

where the last equality is due to independence.