Conditional Probabilities

Conditional probability: The outcome B has the prerequisite that A happened before; this is written as:

P(B|A)

If P(A) \neq 0. The conditional probability that B happens under the condition of A is defined through:

P(B|A) = \frac{P(A \cap B)}{P(B)}

Useful properties

Multiplication:

P(A \cap B) = P(A|B) \cdot P(B)

Bayes' theorem: Let A and B be outcomes with P(A) \neq 0 and P(B) \neq 0, the following holds:

P(A|B) = \frac{P(A)}{Pr(B)} \cdot P(B|A)