The concept of Probability

A general model for an experiment could be constructed in the following manner:

1. Take a non-empty set $\Omega$, in such a way that each $\omega \in \Omega$ represents a possible outcome of the experiment.

2. To the ``greatest possible number" of subsets $A \subset \Omega$, each subset $A$ representing a possible event, associate a number $P(A) \in [0,1]$, called the probability of such event.

An event $A$ has been observed when we perform a ``realization of the experiment'' and an outcome $\omega \in A$ is obtained. The probability of the event can be defined as follows (``frequentist approach''):

Repeat the experiment $N$ times, with $N$ large, and observe the results. If event $A$ has occurred $N_A$ times then

$$f_a = \frac{N_A}{N}$$

is the relative frequency of event $A$ and is an approximation of the probability $P(A)$ of occurrence of $A$. Then $P(A)$ can be defined as:

$$P(A)= \lim_{N\rightarrow\infty} \frac{N_A}{N}$$

In practice the probability is a function relating the elements of the family of events to the interval $[0,1]$:

$$P:{\cal A} \rightarrow [0,1]$$

where ${\cal A} \subset \Omega$ and the following properties hold:

$$P(\Omega) = 1$$

$$P(\sum_{n=1}^{\infty} A_n) = \sum_{n=1}^\infty P(A_n)$$

for any family ${\cal A} = \{A_n\}$ of events for which $A_n\bigcap A_m = \emptyset$ if $n\neq m$.

In mathematical terms the ordered triple

$$(\Omega, {\cal A}, P)$$ (1.1)

is called a probability space if

• $\Omega$ is a non-empty set (the space of outcomes),
• ${\cal A}$ is a family of subsets of $\Omega$ (the family of events),
• $P : {\cal A} \rightarrow [0,1]$ is a probability.

The structure (1.1) constitutes the basis for a mathematical model of an experiment, keeping in mind the non-reproducibility which is often encountered in empirical sciences. For instance, in a coin toss the outcomes are ``heads'' ($H$) or ``tails'' ($T$) and we can take

$$\Omega = \{ H, T \}$$

All possible events are elements of

$${\cal A} = \{ \emptyset, \{ H \} , \{ T \} , \Omega \}$$

Lastly, if the coin is ``fair", intuitively the probability $P$ is given by

$$P(\emptyset) = 0 P(\Omega) = 1$$

$$P(H) = 1/2 P(T) = 1/2$$

All the properties of probability, without exception, are derived from the mathematical model (1.1).