Probabilities and independence

[donys]

Hi...Can you plz check if my proof is correct?

 

Exercise:

A1,A2,.....An are independently events.

Prove that :

P(A1[union]A2[union]...[union]An) = 1-Πi[element-of]I(1-P(Ai))

note for this (Πi[element-of]I(1-P(Ai))

I={1,2,....n)

P([intersect]Ai)= Π P(Ai)

for 3 events A1,A2,A3

means: P(A1[intersect]A2)=P(A1)*P(A2)

P(A2[intersect]A3)=P(A2)*P(A3)

P(A2[intersect]A3)=P(A2)*P(A3)

P(A1[intersect]A2[intersect]A3)=P(A1)* P(A2) * P(A3)

 

Now my proof:

We know that P([intersect]Ai)= Π P(Ai)

if A1,A2,...,An are independent then and the complements

are independent

P([intersect]Ai)complement = Π P(Aicomplement)

P([union](Ai compl) ) = Π(1-P(Ai))

1-P([union]Ai)= Π(1-P(Ai))

-P([union]Ai)=-1+Π(1-P(Ai))

Finally ... we got our proof

P([union]Ai)=1-Πi[element-of]I(1-P(Ai))

Is it correct?

 

 

And one more....

but i dont know how to prove this:

A,B,C are independent

We must prove that A and B[union]C are independent too

...?