Definition and Properties 10/12

Definition of the conditional probability    of A given B:

$$P(A\vert B)=\frac{P(A \cap B)}{P(B)}$$

Averaging rule:

P(E)=P(E|F)P(F)+P(E|F^c)P(F^c)

General Averaging Rule:
Let $F_i,i=1\ldots n$ be a partition of $\Omega$, by which I mean that the events F_i are all mutually exclusive and that their union is $\Omega$, then:

$$P(E)=\sum_{i=1}^n P(E\vert F_i)P(F_i)$$

Bayes Rule:(how to find the opposite conditional probability than the ones given):

$$P(B\vert A)=\frac{P(B\cap A)}{P(A)}=\frac{P(A\vert B)P(B)}{P(A\vert B)P(B)+P(A\vert B^c)P(B^c)}$$

General Bayes Rule:
Let $F_i,i=1\ldots n$ be a partition of $\Omega$, then $\{E \cap F_i\}_{i=1..n}$ is a partition of E and:

$$P(F_i\vert E)=\frac{P(E\vert F_i)P(F_i)}{\sum_{j=1}^nP(E\vert F_j)P(F_j)}$$