A probabilistic model is a mathematical description of an uncertain situation.
A probabilistic model have two main ingredients: a sample space and a probability law.
Every probabilistic model involves an underlying process, called the experiments that will produce exactly one out of several possible outcomes.
A sample space $\Omega$ is the set of all possible outcomes of an experiment.
A collection of possible outcomes is called an event, which is a subset of the sample space.
The probability law assigns a set $A$ of possible outcomes (also called an event) a nonnegative number $\b{P}(A)$(called the probability of $A$).
Any probability law must satisfies the following axioms:
- (Nonnegativity) $\b{P}(A)\geq 0$, for every event $A$.
- (Additivity) if $A_1,A_2,…$ is a sequence of disjoint events, then the probability of their union satisfies $\b{P}(A_1\cup A_2 \cup …)=\b{P}(A_1) + \b{P}(A_2)+…$.
- (Normalization) $\b{P}(\Omega)=1$.
To represent the experiment well, $\Omega $ must be collectively exhaustive, In other words, no matter what happens, all outcomes are in $\Omega$, and $\Omega$ consists only of outcomes.
probability law $P$ assigns to each event $A$ a number $\b{P}(A)$, called the probability of $A$.
Two sets $A,B$ are disjoint if $A\cap B=\emptyset$.
We denote the number of elements in a set $A$ as $\abs{A}$.
If a sample space consists of a finite or even just countable number of outcomes, then the probability law is specified by the probabilities of the events that consist of a single element.
In particular, the probability of any event $\set{s_1,s_2,…,s_n}$ is the sum of the probabilities of its elements:
$$\b{P}(\set{x_1, x_2, …, x_n})=\b{P}(\set{x_1})+\b{P}(\set{x_2})+…+\b{P}(\set{x_n})$$
If a sample space consists of $n$ possible outcomes which are equally likely, then $\forall$ event $A$, $\b{P}(A) = \abs A \cdot \frac{1}{n}=\frac{\abs{A}}{\abs \Omega}$.
Consider a probability law, and let $A, B$, and $C$ be events.
- if $A \subset B$, then $\b{P}(A) \leq \b{P}(B)$.
- $\b{P}(\emptyset)=0$
- $\b{P}(A^c) = 1 - \b{P}(A)$
- $\b{P}(A\cup B) = \b{P}(A) + \b{P}(B) - \b{P}(A\cap B)$
- $A_1 \subset A_2 \subset … \subset \Omega \implies \b{P}(\cup ^ \infty _ {i = 1} A_i) = \limu{ i }{ \infty } \b{P}(A_i)$
$\Omega \supset A_1 \supset A_2 \supset … \implies \b{P}(\cap ^ \infty _ {i = 1} A_i) = \limu{ i }{ \infty } \b{P}(A_i)$
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