De Morgan's Laws:

• $\overline{A ∪ B} = \overline{A} ∩ \overline{B}$
• $\overline{A ∩ B} = \overline{A} ∪ \overline{B}$

Mutual Exclusion: If $A$ happens, $B$ cannot and vice versa

• iff $P(A∩B)=∅$

Product Rule: $P(ABC)=P(A)\,P(B|A)\,P(C|AB)$

• $P(ABC) = P(A∩B∩C)$ (shorthand notation)

Sum Rule: $P(A∪B) = P(A) + P(B) - P(A∩B)$

• Proof by Venn Diagram

Independence: Probability of $B$ changes given that $A$ happened

• iff $P(A∩B∩C∩...)=P(A)P(B)P(C)...$
• iff $P(A|B) = P(A)$ and vice versa

Conditional: Probability of $A$ given that $B$ happened

• $P(A|B) = \frac{P(A∩B)}{P(B)}, P(B)>0$

Bayes' Theorem: Cond. prob. using the reverse condition

• $P(A|B)=\frac{P(B|A)\,P(A)}{P(B)}$