cheat sheet

2022-03-12 · 1 min read

\[ \renewcommand{\Pr}[1]{\text{Pr}\left[ #1 \right]} \]

Law of Total Probability #

\[ \begin{align*} \Pr{A} &= \sum_n \Pr{A,\, B_n} \\ &= \sum_n \Pr{A \;|\; B_n} \cdot \Pr{B_n} \\ \\ \Pr{A \;|\; C} &= \sum_n \Pr{A \;|\; C,\, B_n} \cdot \Pr{B_n \;|\; C} \\ &= \sum_n \Pr{A \;|\; C,\, B_n} \cdot \Pr{B_n} \qquad \text{(if $B_n$ and $C$ indep.)} \end{align*} \]