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Given the following definitions:

  1. Disjoint means if event A occur ,B cannot possibly occur. Draw in the Venn Diagram, it should not contain any intersection. Hence, mathematically,

    P(A,B) = 0.

  2. However, for independence, it says if we know that A occurs, it tells us nothing whatsoever about what B occurs. Does it mean it does have intersection?? Come to think of it. If A and B are independent,

    P(A n B) = P(A)P(B).

Mathematically speaking, if A and B are disjoint, then 

P(AnB) = 0 /= P(A)P(B)

If P(AnB) = P(A)P(B), then it means there should be an intersection???

Xegara
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    There is always an intersection. However, it can be empty. In that case the events are disjoint. Independence does not guarantee that the intersection is not empty. For instance if $A$ and $B$ are both empty then they are independent and have an empty intersection. – drhab Feb 20 '14 at 13:29
  • @drhab so you mean independence may also mean disjoint? But disjoint will never lead to independence? – Xegara Feb 20 '14 at 13:42
  • Independence does not mean disjoint and disjoint does not mean independence. I am only saying that these concepts do not exclude eachother. You can have a situation where you have independence and disjointness. – drhab Feb 20 '14 at 13:45
  • Can you cite a situation that does have independence and disjointness and give an explanation why is that so? – Xegara Feb 20 '14 at 13:47
  • Let it be that $A$ and $B$ are disjoint and secondly that $P(A)=0$. Then also $P(A\cap B)=0=P(A)P(B)$ wich means independence by definition. – drhab Feb 20 '14 at 13:49
  • Interesting.. Thanks a lot! :) – Xegara Feb 20 '14 at 13:52
  • @drjab Wait. If P(A) = 0, in reality, then there is no event A. Does it still mean that there is a situation where you have independence and disjointness? – Xegara Feb 20 '14 at 13:56
  • p(a) = 0 means the probability of event A occuring is 0, not that the event of A doesn't exist. if you are given the conditions p(a) > 0 and p(b) > 0 then there isn't independence if disjoint – David L Feb 20 '14 at 14:36

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