Properties of Independent Events: "Let A be an event such that P[A] ∈ (0, 1). Show that..."

[Rayleigh]

Let A be an event such that P[A] ∈ (0, 1). Show that A and AC are not independent. Does this still hold if P[A] ∈ {0, 1}?

The solution is: We have P[A∩A^C] = P[∅] = 0. On the other hand, P[A]·P[A^C] = P[A](1−P[A]). The events are independent if, and only if, these two quantities are equal, i.e., if, and only if, P[A] = 0 or P[A] = 1.

Are the events independent, because for both P[A] = 0 or 1, P[A]·P[A^C] = P[A](1−P[A]) = 0?

 

[blamocur]

Am I the only one who does not know what "A^C" means?

 

[Dr.Peterson]

Am I the only one who does not know what "A^C" means?

I think it means [imath]A^c[/imath], the complement of A. Evidently AC is a typo for the same; my first question would have been, what is C?

Are the events independent, because for both P[A] = 0 or 1, P[A]·P[A^C] = P[A](1−P[A]) = 0?

Yes, that's what they said.

It's by the definition of independence: Two events A and B are independent when [imath]P(A\cap B)=P()P(B)[/imath].