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Can some of you tell me why the following holds true?:

$ P(A \& B^C )= P(A)-P(A\&B)$ in which: $ B^C$ is $B$ complement. In which: $A$ and $B$ are independent.

thanks..

student441
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    Have you drawn a Venn diagram yet? – whuber Apr 08 '13 at 17:29
  • Thanks...Forgot about that. So $A$ and $B$ do not necessarily have to be independent? What I stated holds true in general? – student441 Apr 08 '13 at 17:33
  • What does your diagram tell you? To derive your result do you need to put $A$ and $B$ into some special configuration or assume some special relationship between their probabilities? Or does it demonstrate the relationship is true regardless of the probabilities of $A$, $B$, and $A\cap B$? (If you would like to see an example of what I'm describing, I used such figures in [this answer](http://stats.stackexchange.com/questions/47671/what-is-the-difference-between-using-the-multiplication-rule-or-using-venn-diagr/47800#47800).) – whuber Apr 08 '13 at 17:35
  • Can I conclude from your answer that independence always means overlap between events? – – student441 Apr 08 '13 at 17:59
  • No, because that would not be true. :-) Independence means, loosely put, that *probabilities multiply.* That is, events $A$ and $B$ are independent when $\Pr(A\cap B)$ = $\Pr(A)\Pr(B)$. I believe this is explained and illustrated in the link provided in my previous comment. – whuber Apr 08 '13 at 18:48
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    **Except** for the special case when at least one of the two events has probability $0$, independence of events $A$ and $B$ **always** implies that the events "overlap" and their intersection $A\cap B$ has positive probability: $P(A\cap B) > 0$. Note that events $A$ and $B$ are independent if and only if $P(\cap B) = P(A)P(B)$, and the right side cannot equal $0$ unless at least one of $P(A)$ and $P(B)$ is $0$. So, when whuber says "that would not be true :-)" he is thinking of the special case mentioned above when student441'a conclusion fails to hold. – Dilip Sarwate Apr 08 '13 at 21:19
  • See [this question](http://stats.stackexchange.com/q/24877/6633) and its answers for more details about independence and overlap, or the lack thereof. – Dilip Sarwate Apr 08 '13 at 21:30

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