Basic Ideas

To begin our study on Probability Theory, we must first explore the connection that exists between deterministic and indeterministic processes and their relationship with probabilities and chance. The processes that occur in nature are classified into two mutually exclusive species: either deterministic or indeterministic. A deterministic process is one that, given certain initial conditions, always leads to the same results. On the other hand, a process is indeterministic when, for the same initial conditions, it can provide a multiplicity of possible outcomes. These possible outcomes, whether unique or multiple, are also known as states. In turn, processes are understood through models, which are analogously deterministic or indeterministic.

It is in the context of indeterministic processes that the study of probabilities emerges. This is a formal science (like logic) with which we can extract information about the frequency with which an indeterministic process tends to produce a certain state or set of states.

Deterministic and Indeterministic Processes in Nature

Suppose we want to model the process of launching a projectile. In a controlled environment, Newton’s mechanics tell us that for each velocity and initial position, the projectile will have a determined impact point, representing a deterministic model of reality. However, in real life, absolutely controlled environments do not exist, and what we will see is that the projectile tends to fall within a certain region, implying a multiplicity of possible states.

It is generally said that our limitations in controlling and measuring both the environment and initial conditions prevent us from determining the final states of a system, forcing us to deal with a multiplicity of them and with non-deterministic models. As a process increases in complexity, so does the lack of control over the environment and initial conditions.

Probabilities and Chance

Chance is an inherent characteristic of indeterministic processes. A process is said to be random or chance-based when, for a certain set of initial conditions, none of the possible states seem to occur more frequently than the others. The simplest example is found in the toss of a coin. As the number of tosses increases, it is observed that the frequency with which heads and tails appear tends to be similar. In other words:

\displaystyle \lim_{N\to \infty} \dfrac{C}{N} = \lim_{N\to \infty} \dfrac{S}{N} = \dfrac{1}{2}

where N is the total number of tosses, C and S are the number of heads and tails obtained among the N tosses.

This number is the probability of obtaining heads or tails, and its understanding is obtained by analyzing experiments with equiprobable outcomes. This is the essence of deterministic and indeterministic processes and their relationship with probabilities and chance.