## Laws of Probability

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.