Probability Using Venn Diagram

Yes, that is actually one way to define "independent" - two events are independent exactly when P(A and B) = P(A) * P(B).



If two independent events each have some possibility of occurring, then it will always be possible that both occur (i.e. P(A)*P(B) > 0 will always be true), so in a Venn diagram, the two circles would always overlap. That overlap can be big or small, depending on how big each probability is. If the probability is 1/10 that A happens, and 1/10 that B happens, then the overlap would be 1/100, so it's tiny because the probabilities are both so small. In the "only A happens" section of the Venn diagram you'd have 9/100, in the "only B happens" section you'd also have 9/100, and in the "neither A nor B happens" section you'd have 81/100.

But if the probabilities were both large, then the overlap would be large. If there's a 90% chance of rain in city X on any given day, and a 90% chance of rain in some distant city Y on any given day, and if those events are independent, the overlap would be 0.81, because it's very likely both events happen.

That said, you wouldn't generally use a Venn diagram to solve a problem with independent events - it might be useful to see how the Venn works just to help with conceptual understanding, but you'd usually just use the multiplication rules directly to solve an actual GMAT question.