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Basic Concepts

Overview

When studying specific random phenomena, we typically focus on the following elements:

  • Sample space \(\Omega\), indicating all possible outcomes of the random phenomenon.
  • Event space \(\mathcal{F}\), representing all events of interest.
  • Probability \(P\), describing the likelihood of each event occurring.

Sample Space and Random Events

Definition

In a random phenomenon, an indivisible possible outcome is called a sample point. The set of all sample points is called the sample space, usually denoted by \(\Omega\).

A random event is a subset of the sample space \(\Omega\), consisting of several sample points, and is denoted by uppercase letters \(A, B, C, \cdots\).

For an outcome \(\omega\) of a random phenomenon and a random event \(A\), we say event \(A\) occurs if and only if \(\omega \in A\).

For example, the point obtained from rolling a die once is a random phenomenon. Its sample space can be represented as \(\Omega=\{1,2,3,4,5,6\}\). Let random event \(A\) be "the point obtained is greater than \(4\)", then \(A = \{ 5, 6 \}\). If the point obtained from a die roll is \(\omega = 3\), since \(\omega \notin A\), event \(A\) does not occur.

Operations on Events

Since we define random events as subsets of the sample space \(\Omega\), we can apply set operations (such as intersection, union, complement, etc.) to random events. The notation is consistent with set operations.

Specifically, the union of events \(A \cup B\) can also be denoted as \(A + B\), and the intersection of events \(A \cap B\) can also be denoted as \(AB\). These are sometimes called the sum event and product event, respectively.

Event Space

When studying a specific random phenomenon, we need to clarify which events are of interest. According to the definition of random events, it is clear that \(\mathcal{F} \subset 2^{\Omega}\) (the notation \(2^{\Omega}\) denotes the power set of \(\Omega\)), but \(\mathcal{F} = 2^{\Omega}\) is not necessary. This might be somewhat difficult to understand when the sample space \(\Omega\) is finite, as \(2^{\Omega}\), though larger, is still finite. However, when \(\Omega\) is an infinite set, the cardinality of \(2^{\Omega}\) becomes much larger, and it inevitably contains some "poorly behaved" events that we are not interested in. In such cases, sacrificing certain properties to accommodate these events seems not worthwhile.

Although \(\mathcal{F} = 2^{\Omega}\) is not necessary, this does not mean any subset of \(2^{\Omega}\) can serve as an event space. We are typically interested in the probabilities of events resulting from operations on other events. Therefore, we want the event space \(\mathcal{F}\) to satisfy the following conditions:

  • \(\varnothing \in \mathcal{F}\);
  • If \(A \in \mathcal{F}\), then the complement event \(\bar{A} \in \mathcal{F}\);
  • If there is a sequence of events \(A_n \in \mathcal{F}, n = 1, 2, 3\dots\), then \(\bigcup A_n \in \mathcal{F}\).

In short, the event space \(\mathcal{F}\) is closed under complement and countable union operations and contains the element \(\varnothing\).

It can be proven that an event space \(\mathcal{F}\) satisfying the above three conditions is also closed under countable intersection.

Taking the die roll as an example, when the sample space is \(\Omega=\{1,2,3,4,5,6\}\), the following two sets can serve as event spaces:

  • \(\mathcal{F}_1 = \{ \varnothing, \Omega \}\)
  • \(\mathcal{F}_2 = \{ \varnothing, \{1, 3, 5\}, \{2, 4, 6\}, \Omega \}\)

But the following two sets cannot:

  • \(\mathcal{F}_3 = \{ \varnothing, \{1\}, \Omega \}\) (not closed under complement)
  • \(\mathcal{F}_4 = \{ \{1, 3, 5\}, \{2, 4, 6\} \}\) (does not contain \(\varnothing\) and is not closed under union)

Probability

Definition

Classical Definition

In the early practice of probability theory, due to the simplicity of the involved random phenomena—specifically, the sample space \(\Omega\) being finite and all sample points intuitively being equally likely—people summarized the following definition:

If a random phenomenon satisfies:

  • There are only a finite number of basic outcomes;
  • Each basic outcome is equally likely;

Then for each event \(A\), define its probability as

\[ P(A)=\frac{\#(A)}{\#(\Omega)} \]

where \(\#(\cdot)\) denotes the measure of the size of a random event (a set).

Later, it was found that this definition could be directly extended to some scenarios where \(\Omega\) is infinite, leading to the so-called Geometric Probability.

Axiomatic Definition

The above definition based on intuitive understanding has a major logical flaw: it uses the concept of "likelihood" when defining "probability", creating a circular definition. Additionally, "equally likely" leads to ambiguity when the sample space is infinite, resulting in a series of problems including Bertrand's paradox.

After continuous exploration, the Soviet mathematician Kolmogorov first gave the axiomatic definition of probability in his 1933 book Foundations of the Theory of Probability:

A probability function \(P\) is a mapping from the event space \(\mathcal{F}\) to the closed interval \([0, 1]\), satisfying:

  • Normalization: The probability of event \(\Omega\) is \(1\), i.e., \(P(\Omega)=1\).
  • Countable Additivity: If a sequence of events \(A_1, A_2, \cdots\) are pairwise disjoint, then \(P\left( \bigcup_{i \geq 1} A_i \right) = \sum_{i \geq 1} P(A_i)\).

Properties of the Probability Function

For any random events \(A, B \in \mathcal{F}\), we have:

  • Monotonicity: If \(A \subset B\), then \(P(A) \leq P(B)\).
  • Inclusion–Exclusion Principle: \(P(A+B) = P(A) + P(B) - P(AB)\).
  • \(P(A - B) = P(A) - P(AB)\), where \(A - B\) denotes the set difference.

Probability Space

As mentioned at the beginning, when studying a specific random phenomenon, we typically focus on the sample space \(\Omega\), the event space \(\mathcal{F}\), and the probability function \(P\). We call the triple \((\Omega, \mathcal{F}, P)\) a probability space.

Probability only has meaning when discussed within a defined probability space. The Bertrand paradox mentioned earlier ultimately arises from an ambiguous definition of the sample space \(\Omega\).

References and Notes

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