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I'm toying with the concept of independence, which I understand to be the idea that "one event's occurrence has no bearing on the other element's occurrence." I know it is wrong to think that two independent events cannot overlap, and I can propose a simple experiment to show this:

If I draw from a deck of cards, then the events that I draw $\leq 7$, and a black card (events $A$ and $B$, respectively) are independent of each other.

That is $P(B|A) = P(B)$

$$P(B) = \frac{26}{52}$$ and $$P(B|A) = \frac{14}{28}$$

What allows both probabilities to be the same, in this case, is that even after limiting the sample size to cards $\leq 7$, the proportion that of black cards that originally existed remains the same.

Is this the only circumstance (that some properties of both sample spaces remain the same) by which two independent events can overlap? Such a circumstance does not always take place.

For example, if $A$ and $B$ were redefined to cards $\leq 7$ and cards $\leq 8$, then by choosing one event, we would drastically redefine the sample space for a subsequent event's occurrence.

On a related note, how should I interpret the quoted, verbal definition of independence, above, to understand that being disjoint is not a requirement for independence? The definition certainly doesn't imply that.

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    $\begingroup$ In fact, disjoint events can NOT be independent unless one of them has probability 0 itself. $\endgroup$ – Ned Jan 26 '16 at 23:34
  • $\begingroup$ @David Hmm, I should add that I tried to understand that definition in the context of a venn diagram. If two circles on a venn diagram intersect, then isn't it the case that if some point in one circle is chosen, then this point also has a chance of being in the other circle. As a result, one event's occurrence would imply that the other could also occur. $\endgroup$ – Muno Jan 26 '16 at 23:39
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    $\begingroup$ It is very rare that two disjoint events are independent. The reason is that if the two events are disjoint, and if one of these events occur, then the other event wil definitely not occur. Thus one event's occurence highly influences the occurence (or non-occurence) of the other event. $\endgroup$ – Mankind Jan 26 '16 at 23:41
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    $\begingroup$ @Muno I'm not too sure about your first question about the direct argument that two independent events must intersect. I would probably go with the argument "If they are independent, then they cannot be disjoint because...", but as soon as I say "...cannot...", the argument stops being direct. In math terms, however, two events $A$ and $B$ are independent, if and only if $P(A\cap B) = P(A)P(B)$. Thus (if both $A$ and $B$ have non-zero probability), we must have that $P(A\cap B)>0$, meaning that $A\cap B\neq\emptyset$. $\endgroup$ – Mankind Jan 27 '16 at 2:17
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    $\begingroup$ @Muno as for your second question, you are correct that if you know that the first event has occured, and you know that the occurence happens in the intersection of the first and second event, then you know that the second event has also occurred. But independence doesn't relate to where an event has occurred, just that it has occurred. Independence of two events is that if you know that the first event has occurred, then you have no extra information that tells you whether or not the second event has occurred. It is not part of this statement that you know how the first event has occurred $\endgroup$ – Mankind Jan 27 '16 at 2:21

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