Probability Q: Show A & B Independent

[GreenPrint]

Homework Statement

Suppose
P(A|B) = P(A|B^c)
(P(B) > 0 and P(B^c) > 0 both understood). Show that A and B are independent.

Homework Equations

The Attempt at a Solution

I don't know where to go from here. Thanks for any help

$\frac{P(A \bigcap B)}{P(B)} = \frac{P(A \bigcap B^{c})}{P(B^{c})}$

 

[HallsofIvy]

Please restate (in fact, go back are reread) the problem. As you have it now it says "Suppose A and B are independent. Show that A and B are independent"!

 

[GreenPrint]

Ok sorry about that I fixed that. Thanks for helping me.

 

[HallsofIvy]

Okay, good. Now what is your definition of "independent events"?

 

[GreenPrint]

P(A intersection B) = P(A)P(B)

not sure what to do from here because it's conditional probabilities.

 

[HallsofIvy]

Good! So you are given that $P(A|B)= P(A|B^c)$ and want to prove that $P(A\cap B)= P(A)P(B)$.

You should know, then, that $P(A)= P(A|B)P(B)+ P(A|B^c)B^c$, for any A and B. Further, $P(B^c)= 1- P(B)$.

(The reason I asked about the definition was that some texts use "P(A)= P(A|B)" as the definition of "A and B are independent. Of course it is easy to show that is equivalent to your "$P(A\cap <br /> B)= P(A)P(B)$".)