independent events

for independent events, why does probability of A given B = P(A)

Almost by definition, thats what it means. Two events are independent if knowledge of one does not change the probability of the other. So p(A | B) = p(A), i.e. knowlege about B (the given part) goes not affect the distribution of A.

Maths using the conditional equation
p(A | B) = p(A & B)/p(B)
This is equal to p(A) when the joint p(A & B) = p(A)p(B), which is the usual statement of independence in terms of the joint probability p(A & B).

P(A I B) is the intersection of both A and B
P(A) is all of circle A
it's likely that they'll never be the same?

It helps to draw the "venn diagram" you're thinking about. However, p(A | B) is not the intersection of A & B which is generally known as the joint probability p(A & B). The conditonal probability is
p(A | B) = p(A & B) / p(B)
So is the joint probability p(A & B) divided by p(B).

They're independent when
p(A |B) = p(A & B) / p(B) = p(A)
as knowledge of B has no effect on the p(A). This occurs when (just rearrange)
p(A & B) = p(A)*p(B)

The ratio of p(A & B) : p(B) is the same as p(A) : 1. So knowning whether A occurs (or not) has no effect on the probability of B. If you;re thinking about it from a venn diagram perspective, this is the view to take.

The "likely" isnt the right way to look at it. Independence (or not) is a property of the problem.