prove if $E_{1},E_{2},...,E_{n}$ are independent events then:
$p(\cup_{i=1}^{n}E_{i})=1-\prod_{i=1}^{n}[1-p(E_{i})]$
my try: $$ p(\cup_{i=1}^{n}E_{i})=1-p((\cup_{i=1}^{n}E_{i})^{c})=1-p(\cap_{i=1}^{n}E_{i}^{c}) =1-\prod_{i=1}^{n}p(E_{i}^{c})=1-\prod_{i=1}^{n}[1-p(E_{i})] $$ $E_{1},E_{2},...,E_{n}$ are independent,
so i guess $E_{1}^c,E_{2}^c,...,E_{n}^c$ are also independent
(I need to prove it but got little confused with all the Inclusion-Exclusion principle)
Would appreciate any help :)