Probability Question Proving?

Suppose that A, B and C are independent events and P(C) ≠ 0. Prove:

a) P (A ∩ B | C) = P(A | C) P(B | C)

b) P (A ∪ B | C) = P(A | C) + P(B | C) - P (A ∩ B | C)

My book only this formula to proved the above question.

p(A | B) =(A ∩ B)/ P(B)

I dont know how to deal with 3 constants?

Any help?

Thanks

Re: Probability Question Proving?

Hello, blackZ!

I'll do the first one . . .

Since are independent,

. . we have, for example:

On the left side we have:

. .

. . . . . . . .

On the right side we have:

. .

. . . . . . . .

Re: Probability Question Proving?

Thanks for the (a).

For (b), I simplified the right side to **P(A)+P(B) - P(A)P(B)**

I am not sure what UNION and CONDITIONAL probability means when A,B and C are independent? Is it the same as INTERSECTION and CONDITIONAL?

Thanks

Re: Probability Question Proving?

Quote:

Originally Posted by

**blackZ** Thanks for the (a).

For (b), I simplified the right side to **P(A)+P(B) - P(A)P(B)**

I am not sure what UNION and CONDITIONAL probability means when A,B and C are independent? Is it the same as INTERSECTION and CONDITIONAL?

Re: Probability Question Proving?

Quote:

Originally Posted by

**Plato**

Thats more confusing. I dont know how the conditional probability **|** will go in there.

Re: Probability Question Proving?

Re: Probability Question Proving?

Quote:

Originally Posted by

**blackZ** Thats more confusing. I dont know how the conditional probability **|** will go in there.

OK, I thought you could work through it.

Divide by both sides by