Sample Space and Probability¶

The main objective for probability is to develop the art of describing uncertainty in terms of probabilistic models, as well as the skill of probabilitic reasoning.

Sets¶

A set is a collection of objects, which are the elements of sets.

Set Notes¶

If S is a set and x is an element of S, we write \(x \in S\);
If x is not an element of S, we write \(x \notin S\);
A set with no elements called empty set, denoted by \(\emptyset\);
If S contains a finite number of elements, we write it as a list of elements in braces:

\[ S = \{ x_1, x_2, \cdots, x_n \} \]

If S contains infinitely many elements which can be enumerated in a list, we say S is countably infinite and write:

\[ S = \{ x_1, x_2, \cdots \} \]

We can alternatively make the set of all x with a certain property P, (the symbol "|" can be read as "such x that satisfy P"):

\[ \{ x | x \text{ statisfies P} \} \]

If every element of S is also a an element of T, we say that S is a subset of T, denoted as \(S \subset T\) or \(T \supset S\);
If \(S \subset T\) and \(T \subset S\), the two sets are equal, and we write \(S = T\);
It's also expedient to introduce a universal set, denoted by \(\Omega\), which contains all objects of interest in a particular context.

Set Operations¶

The Component of a set S, with respect to the universe \(\Omega\) is the set \(\{x \in \Omega | x \notin S \}\), denoted by \(S^c\). \(\Omega^c = \emptyset\);
The Union of two sets S and T is the set of all elements that belong to S or T, denoted as \(S \cup T\);
The Intersection of two sets S and T is the set of all elements that belong to both S and T, denoted as \(S \cap T\);
Several sets are said to be disjoint if no two of them have a common element;
A collection of sets is said to be a partition of a set S if the sets in the collection are disjoint and their union is S.

Sets and the associated operations are easy to visualize in terms of Venn diagrams.

The Algebra of Sets¶

Set operations have several properties:

\(S \cup T\) = \(T \cup S\)
\(S \cup (T \cup U)\) = \((S \cup T) \cup S\)
\(S \cap (T \cup U) = (S \cap T) \cup (S \cap U)\)
\(S \cup (T \cap U) = (S \cup T) \cap (S \cup U)\)
\((S^c)^c = S\)
\(S \cap S^c = \emptyset\)
\(S \cup \Omega = \Omega\)
\(S \cap \Omega = S\)

Two particularly useful properties are given by de MOrgan's laws:

\[ (\bigcup_n S_n)^c = \bigcap_n S_n^c \]

\[ (\bigcap_n S_n)^c = \bigcup_n S_n^c \]

Probabilistic Models¶

A probabilistic model is a mathematical description of an uncertain situation. It has two elements:

The sample space \(\Omega\), which is the set of all possible outcomes of an experiment;
The probability law, which assigns to a set A of possible outcomes(also called an event) a nonnegtive number \(P(A)\)(called the probability of A) that encodes our knowledge or belief about the collective likelihood of the elements of A.

probabilistic model

Sample Space and Events¶

Every probabilistic model involves an underlying process, called the experiment, which produces exactly one out of several possible outcomes. The set of all possible outcomes is called the sample space of the experiment denoted by \(\Omega\). A subset of the sample space, that is, a collection of possible outcomes, is called event.