Parent: probability-theory

Bayes rule

Source: blitzstein-hwang

$$P(A|B) = \dfrac{P(B|A) P(A)}{P(B)}$$

Extra conditioning

“Everything is conditioned on $C$.”

$$P(A|B, C) = \dfrac{P(B|A, C) P(A|C)}{P(B | C)}$$ given $P(A\cap E) > 0$ and $P(B\cap E) > 0$.

Alternative approaches for interpretation

1. $B, ~C$ as a single event $B\cap C$
   $$P(A|B,C) = \dfrac{P(A, B, C)}{P(B, C)}$$
2. Swap roles of $B$ and $C$ $$P(A|B, C) = \dfrac{P(C|A, B) P(A|B)}{P(C | B)}$$