Union of two events

p(AB)=p(A)+p(B)p(AB)p(A \cup B) = p(A) + p(B) - p(A \cap B)

Joint probabilities

p(A,B)=p(AB)=p(AB)p(B)p(A, B) = p(A \cap B) = p(A|B)p(B)

This is known as the product rule.

Given a joint distribution on two events p(A,B)p(A, B), we define the marginal distribution as follows:

p(A)=bp(A,B)=bp(AB=b)p(B=b)p(A) = \sum_{b}p(A, B) = \sum_{b}{p(A|B=b)p(B=b)}

This is the sum rule or the rule of total probability.

Conditional probability

Conditional probability of event AA, given that BB is true:

p(AB)=p(A,B)p(B) if p(B)>0p(A|B) = \frac{p(A,B)}{p(B)} \text{ if } p(B) > 0