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If A and B are mutually exclusive events  
\[P(A \cap B)=0, \: P(A \cup B)=P(A)+P(B)\]
Also,  
\[P(A | B)=0\]
.
\[P(A | B)\]
  means 'probability of A happening given that B has happened. If A and B are mutually excusive and A has happened, then of course B cannot happen.
If A and B are independent events  
\[P(A \cap B)=P(A) \times P(B)\]
.
Also,  
\[P(A | B)=P(A)\]
.
This means that the probability of A happening does not depend on whether B has happened since A is independent of B.
\[P(A | B)=\frac{P(A \cap B)}{P(B)}\]
.
Also,  
\[P(A \cup B)=P(A)+P(B)-P(A \cup B)\]
.
These last two statements are true for all events A and B, independent or not.
\[P(A')=1-P(A)\]
.
This means that the probability of A not happening is one minus the probability of A happening.