A proof question involving union and complements of events

[laura_a]

Homework Statement

If X1, X2, X3, ... Xk are independent events, prove that

P(X1 U X2 U X3 U ... U Xk) = 1 - [1 - P(X1)][1-P(X2)]...[1-P(Xk)]

Homework Equations

The Attempt at a Solution

Well I have tried a few methods, but I know it's got something to do with

P(X1) = cc(P(X1)) (complment complement)
P(X1) = 1 - c(P(X1))
P(X1) = 1 - (1-P(X1))

I know that much, but I can't link it in with unions of independent events?? I

 

[AKG]

DeMorgan's Law:

$(A_1 \cup \dots \cup A_k)^c = A_1^c \cap \dots \cap A_k^c$