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!

To prove this first let's talk about the independent and not independent events:
Independent events is when we roll a die and then flip a coin. The number showing on the die has no effect on the coin that was tossed. Therefore these two events are independent.
Not independent would be the gender of each baby in a set of twins. If the twins are identical, then both of them will be male, or both of them would be female.
Since A and B are independent so:
= P(A) x P(B)
so
P[A | B] =  / P(B) = (P(A) x P(B)) / P(B) = P(A)

In other hand A is not independent with B:
 = P(A) x P(B|A)
P[A|B] =  / P(B) = P(A) x P(B|A) / P(B)

So A and B must be independent.
In case A and B independent and A = B:



