Key Properties of Probability Measure
Summary
In this class, we present the basic definitions of probability theory, including probability measure and sigma-algebra. Additionally, we explore the notion of probability as a limit of relative frequencies, using the example of rolling a six-sided die. Through the relative frequency function, it is demonstrated how to obtain a probability measure through the limit, known as “probability as a limit of relative frequencies”. Key properties of the probability measure are presented, and it is shown how the probability of events can be calculated through favorable cases over total cases. This knowledge is fundamental for understanding probability theory and applying it in decision-making in different contexts.
LEARNING OBJECTIVES:
Upon completing this class, the student will be able to:
- Know the definition of probability measure and its fundamental properties.
- Understand the notion of probability as a limit of relative frequencies.
- Calculate the probability of events through “probability as favorable cases over total cases”.
TABLE OF CONTENTS
BASIC DEFINITIONS
PROBABILITY AS A LIMIT OF RELATIVE FREQUENCIES
Basic Definitions
The part that condenses the essence of a probability space is precisely the idea of probability measure. This is the function that ultimately quantifies the likelihood of a certain event occurring.
A probability measure P is a function such that, given a sigma-algebra \Sigma=(\Omega,\mathcal{A}), it assigns a probability to each event E\in\mathcal{A}. This probability measure satisfies the following properties:
- (E\in \mathcal{A}) \rightarrow (0 \leq P(E) \leq 1)
- P(\Omega) = 1
- \left[E_1,_2 \in \mathcal{A}\right] \rightarrow \left[ (E_1 \cap E_2 = \emptyset) \rightarrow \left(P\left(E_1 \cup E_2 \right) = P(E_1) + P(E_2) \right) \right]
- \displaystyle \left[E_1 \subseteq E_2 \subseteq \cdots \subseteq E_n \in \mathcal{A}\right] \rightarrow \left[P \left(\bigcup_{n=1}^\infty E_n \right) = \lim_{n\to\infty}P(E_n)\right] (continuity)
Probability as a Limit of Relative Frequencies
This notion of probability can be understood through the intuitive idea of probability as “limit of relative frequencies”. For example, consider the experiment of rolling N times a six-sided die. The sample space of each roll is \Omega_{1d6} = \{1,2,3,4,5,6\}. If A is any event from \Omega_{1d6}, then we can measure the frequency f_N(A) with which the event A occurs in N rolls. By doing this we verify that:
0 \leq f_N(A) \leq f_N(\Omega_{1d6}) = N,
and if we consider a second event B from \Omega_{1d6}, then it will happen that:
(A\cap B = \emptyset ) \rightarrow (f_N(A\cup B) = f_N(A)+f_N(B) )
From this, we can define a new “function” g_N, which we call relative frequency, through the expression
g_N(A) = \displaystyle \frac{f_N(A)}{N}
We will notice that the function g_N complies with the first three properties, only the fourth (continuity) cannot be obtained in this way, but is introduced for rather “technical” reasons since some results cannot be obtained without this property.
Even with this, g_N is still not a probability measure, because it is not yet a function. This is because, regardless of the value of N, g_N does not necessarily deliver a single value, as a function would. To solve this, we add a step to the limit to finally have a probability measure, obtaining:
\displaystyle P(A) = \lim_{N\to\infty} g_N(A)
This is known as Probability as a Limit of Relative Frequencies.
Example: Rolling a Six-Sided Die
Continuing the analysis of rolling a six-sided die, it follows that:
\displaystyle\left(\forall x\in\{1,2,3,4,5,6\}\right) \left( g_N(\{x\}) = \frac{f_N(\{x\})}{N}\right)
and if the experiment is repeated a large number of times, it can be verified that
\displaystyle \left(\forall x\in\{1,2,3,4,5,6\} \right) \left( \lim_{N \to \infty} g_N(\{x\}) = \displaystyle \frac{1}{6} \right)
\displaystyle \lim_{N \to \infty} g_N(\Omega_{1d6}) = 1
Therefore, if we define P(\{x\}) = \lim_{N\to\infty} g_N(\{x\}), then it will occur that:
\displaystyle P(x=\{1,3,5\}) = P(\{1\}\cup\{3\}\cup\{5\}) = P(\{1\})+P(\{3\})+P(\{5\}) = \frac{1}{6}+\frac{1}{6}+\frac{1}{6}= \frac{1}{2}
From this example, it is also simple to infer that, given an event E of a sample space \Omega
\displaystyle P(E) = \frac{\# E}{\# \Omega}
This is known as the “probability as favorable cases over total cases” and is fundamental for understanding a large number of reasonings about the space of probabilities and the measure of probability.