# Depdendent, Independent, and Complementary Events

For two events ##A## and ##B##, where ##A∩B ≠ ∅##, is it possible to deduce from a venn diagram whether or not those two events are dependent? Or is such information unattainable from a mere venn diagram? Also, if we define ##C## as the complement of the union of ##A## and ##B##, why must ##C## be dependent on ##A'##? Why is ##P(C)## not equal to ##P(C|A')##?

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Stephen Tashi
For two events ##A## and ##B##, where ##A∩B ≠ ∅##, is it possible to deduce from a venn diagram whether or not those two events are dependent? Or is such information unattainable from a mere venn diagram?
It isn't possible to determine ithe independence from the usual sort of Venn diagram because you need quantitative information. If probability is represented on a scale proportional to area, you need to determine if the ratio of the area of $A$ to the total area is the same as the ratio of $A \cap B$ to the area of $B$.

Also, if we define ##C## as the complement of the union of ##A## and ##B##, why must ##C## be dependent on ##A'##? Why is ##P(C)## not equal to ##P(C|A')##?
What are you claiming? - "not always equal to" or "never equal to" ?

Homework Helper
If the Venn diagram is labeled - yes, you can determine independent or not - but with the same dull calculations you'd do if you were simply provided the numerical information itself.

Are you asking why, if $C = A \cup B$, that $C' = A' \cap B'$ ? If so, you can convince (not form a proof) yourself of why that is by looking at your hypothetical Venn Diagram. The set C consists of the entire region inside the regions that represent A and B: the complement of C is the entire region inside the diagram but outside the two regions for A and B: that region outside is $A' \cap B'$.

If that is what you meant - why did you pick on the complement of A alone?

It isn't possible to determine ithe independence from the usual sort of Venn diagram because you need quantitative information. If probability is represented on a scale proportional to area, you need to determine if the ratio of the area of $A$ to the total area is the same as the ratio of $A \cap B$ to the area of $B$.

What are you claiming? - "not always equal to" or "never equal to" ?
Never equal to.

If the Venn diagram is labeled - yes, you can determine independent or not - but with the same dull calculations you'd do if you were simply provided the numerical information itself.

Are you asking why, if $C = A \cup B$, that $C' = A' \cap B'$ ? If so, you can convince (not form a proof) yourself of why that is by looking at your hypothetical Venn Diagram. The set C consists of the entire region inside the regions that represent A and B: the complement of C is the entire region inside the diagram but outside the two regions for A and B: that region outside is $A' \cap B'$.

If that is what you meant - why did you pick on the complement of A alone?
What I was asking was: why is the probability of event ##C## different from the probability of event ##C## given ##A'## (or ##B'## really). It does make sense to me; intuitively, since ##P(C|A') = \frac{P(C∩A')}{P(A')}##, and the intersection of ##C## and ##A'## is ##C##, it is clear that ##P(C)## and ##P(C|A')## differ by a factor of ##P(A')##.
But using this same logic on a venn diagram, can't we just as well come to the wrong conclusion that whenever there is an intersection between two events ##A## and ##B##, ##A## must always be dependent on ##B## since the occurrence of ##B## implies the occurrence of the intersection, which is also a part of ##A##?

Stephen Tashi
Never equal to.
$C$ might be the null set.

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