Conditional Probability

First thing first, conditional probability is probability! As you will next see.

Definition Given probability space $\{ S, \Sigma, \mathbb{P}\}$. let $A, B$ be two event in $\Sigma$. When we say the conditional probability of A given B, denoted by $\mathbb{P}(A|B)$, we mean the following value

$\mathbb{P}(A\cap B)/\mathbb{P}(B)$

So the conditional probability given B means the following function:

$\mathbb{P}(x|B): \Sigma \to [0, 1] \text{ such that } \mathbb{P}(x|B) = \mathbb{P}(x\cap B)/\mathbb{P}(B)$

Given this definition of conditional probability, we see:

1. $\mathbb{P}(\emptyset|B) = \mathbb{P}(\emptyset\cap B)/\mathbb{P}(B) = \mathbb{P}(\emptyset)/\mathbb{P}(B) = 0$
2. Given $A_1, A_2, ... \in \Sigma$, if $A_1, A_2, ...$ are mutually exclusive,

$\mathbb{P}( \cup_{n=0}^\infty A_n |B ) = \mathbb{P}( (\cup_{n=0}^\infty A_n) \cap B ) / \mathbb{P}(B)= \mathbb{P}( \cup_{n=0}^\infty (A_n \cap B) ) / \mathbb{P}(B)$

$= \cup_{n=0}^\infty \mathbb{P} ( A_n \cap B ) / \mathbb{P}(B) = \sum_{n=0}^\infty \mathbb{P}(A_n | B)$

So again, conditional probability is probability!

With the notion of conditional probability, we can define the notion of independence for events.

Definition We say A and B are independent if $\mathbb{P}(A \cap B) = \mathbb{P}(A) \mathbb{P}(B)$

Remark

• If $A$ and $B$ are independent event, $\mathbb{P}(A|B) = \mathbb{P}(A)$ which is very intuitive, because the probability of $A$ has nothing to do with B.
• Even if $A \cap B = \emptyset$, $A, B$ might not be independent.
• When $\mathbb{P}(B) = 0$, how we define $\mathbb{P}(x|B)$ is up the application. It's like it is meaningless to define what is $0^0$ unless we are under some application.