What is the Probability of A intersect A? What is the P(A $\cap$ A)? Would it still be P(A)?

I know this may seem childish or elementary but I needed some clarification and nothing online says anything about it.
We can assume they are not independent because they describe the same event.

Visualize what P(A ∩ A) actually mean. The probability of A occurring and A occurring is an event inside a larger sample space. These two events perfectly overlap. Therefore if A occurs it is the same as saying A and A i.e. P(A ∩ A) occurs.
Do not think about it as the probability of A intersecting A, which would be obviously 1.
Instead Define an event $B = A\cap A$, so we have the probability of event B, which is 
$$
    P(B) = \frac{\# B}{\# S}
$$
where S is the Sample Space, i.e. the set of all possible outcomes.
And since B=A, then you have 
$$P(B)=\frac{\# A}{\# S}$$