Probability of the Union of Indepedent Events

[Obraz35]

Homework Statement

Show that if A1, A2, ..., An are independent events then
P(A1 U A2 U ... An) = 1 - [1-P(A1)][1-P(A2)]...[1-P(An)]

Homework Equations

If A and B are independent then the probability of their intersection is P(A)P(B).
The same can also be said of A^C and B.

The Attempt at a Solution

I have managed to prove this algebraically for the case where n=2. But I am having trouble trying to do it for the general case since it is not as easy to break down the union of A1 through An into a union of disjoint sets as it was for n=2.

 

[VeeEight]

Apply De Morgan's Law (for n events)