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The equation in question is: P(A, B) = P(A | B) * P(B), and P(A,B) = P(A) * P(B). You can see in the image the portion of A ∩ B. Venn diagram of A intersect B

In A independent from B case, you can see that the A ∩ B portion is empty space. So according to the image, P(A ∩ B) = 0, P(A | B) = 0, which is clearly wrong.

The way I intuitively visualize the diagram is: each pixel is a event. If a pixel is in circle A, it belongs to A. If a pixel is in both A and B, it is in A ∩ B. This interpretation is clearly wrong.

So how to interpret the Vein diagram correctly? How to visualize P(A ∩ B), P(A | B) in the case where A is independent of B ? If possible, could anyone draw an image ?

Thank you for your help.

Quickmath
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  • Venn diagrams do not visualize independence. – kludg Dec 13 '19 at 12:31
  • If $P(A\cap B)=0$ then A and B are not correlated. It does not necessarily imply that A and B are independent. – callculus42 Dec 13 '19 at 12:34
  • In your second picture you see the case A∩B=0, disjoint sets. Independent sets are never disjoint, because if you know you are in A, then from this info you know you are not in B, so A dictates about B. – mz71 Dec 13 '19 at 13:02
  • @callculus They are correlated. – mz71 Dec 13 '19 at 13:04
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    Do not confuse the words "Independent" with "Mutually exclusive." These mean completely different things. $A\cap B=\emptyset$ (which would imply that $P(A\cap B)=0$) is the property that we call being "Mutually exclusive." In lay terms, it means that the circles for $A$ and $B$ in the venn diagram do not overlap. On the other hand, $P(A\cap B)=P(A)P(B)$, which is equivalent to the property that $P(A)=P(A\mid B)$, is the property that we call being "Independent." In lay terms, the ratio of the size of $A$ to the whole space is the same as the ratio of $A$ and $B$'s intersection to $B$. – JMoravitz Dec 13 '19 at 13:25

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