2.1 Conditional probability

Definition:

Conditional probability:

Suppose $A$ and $B$ are two events in the sample $\Omega$ and $\mathbb P(B) \neq 0$.

The probability given by

$$ \mathbb P(A|B):=\frac{\mathbb P(A\cap B)}{\mathbb P(B)} $$

is called the conditional probability of event $A$ given (or condition on) event $B$.

Theorem:

For fixed event $B \subseteq \Omega$, under the conditions of the previous Definition, we have $\mathbb{\tilde{P}}(A):=\mathbb P(A|B)$ is a probability defined on the sample space $\tilde{\Omega}=B$, i.e.,

1. For any event $E(\subseteq\tilde{\Omega})$, $0 \le \mathbb{\tilde{P}}(E) \le 1$;
2. $\mathbb{\tilde{P}}(\tilde{\Omega})=1$ and $\tilde \Omega$ is the unique event which satisfies this identity;
3. For any sequence of pairwise mutually exclusive events $E_1, E_2, \cdots$(that is , events for which $E_i \cap E_j = \empty$ when $i \neq j$) in $\tilde \Omega$,

$$ \mathbb{\tilde{P}}(\bigcup \limits^\infin_{i=1}) = \sum \limits^\infin_{i=1}\mathbb{\tilde{P}}(E_i) $$

2.2 The multiplication principle

Corollary:

Multiplication principle:

Another form of the definition formular: