Prove or disprove : $P[A|B] = P[B]$, the A and B are independent? Is this right? SOrry if this is extremely easy.
I did the following but I'm a little bit unsure about it:

Let $A=B$, and $P[A]>0$.
Then $$P[A|A] = P[A]$$
But A is not independent with itself:
$$P[AA] = P[A] \neq P[A]^{2} = P[A]P[A] $$
Thus, the preposition does not hold.

I'd appreciate your comments on this one.
Thanks!

In order to avoid special cases let me consider that $P[A] \neq P[B]$ and both $<1$
I suggest you to think about conditioning as information gained.
So even intuitively $P[A|A]=P[A]$ is a nonsense. Indeed this is like to say "If I know that $A$ happen the probability that it happen remain unchanged". While it is obvious that $P[A|A]=1$
Now

Prove or disprove : $P[A|B]=P[B]$, the A and B are independent? Is
this right?

if we known that $P[A|B]=P[B]$ we can conclude that $A$ and $B$ are dependent events.
Indeed this is like to say "If we known that $B$ happen, then $P[A]$ change and become equal to $P[B]$.
