Conditional Probability

\(P(A \mid B)=\frac{P(B \mid A)P(A)}{P(B)}\)

• \(P(A)\): the prior
• \(P(B \mid A)\): the likelihood
• \(P(A \mid B)\): the posterior
• \(P(A \mid B) = P(A)\): A and B are Independent.
• \(P(A \cap B \mid C)=P(A \mid C)P(B \mid C)\): A and B are conditionally independent on the occurence of the event C.
• Often has given that

Law of Total Probability

Formally, if \(B_1, B_2, \ldots , B_n\) form a partition of the sample space (i.e., they are mutually exclusive and exhaustive events), then for any event \(A\):

\[P(A)=\sum_iP(A \mid B_i)P(B_i)\]

It provides a convenient way to think about partitioning events. Often comes with tree of outcomes

Combination

\(\left( \begin{array}{c} n \\ k \end{array} \right) = \frac{n!}{k!(n-k)!}\)

Permutation

\(\frac{n!}{(n-k)!}\)