Conditional Probability

Ponder...

For with , the conditional probability of given is

and if then .

Definition 2.11 (Conditional Probability)

Let be a probability space. Let where . The conditional probability of given is

In particular, if , then

Countable additivity for conditional probability

Proposition 2.12 (Countable Additivity For Conditional Probability)

Let is a disjoint sequence in . Then for some where ,

For a disjoint sequence in and with ,

using countable additivity of .