dabaobao wrote:
Are events considered independent only when P(A&B) = P(A)P(B)?
Yes, that is actually one way to define "independent" - two events are independent exactly when P(A and B) = P(A) * P(B).
dabaobao wrote:
In that case, would the independent events have a Venn diagram with two disjoint circles or overlapping lapping with extremely small overlap? My understanding is that they will have an extremely small overlap as shown in my uploaded file. Since the overlap for such events (i.e. my example on the right half of the page) is so small [Comparing P(A) = P(B) = 0.1 to P (A&B) = 0.1*0.1 = 0.01), they are called independent events even though they have a minor overlap.
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.
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