Axiomatic Approach to Probability

When we hear the word “probability,” we often think of abstract ideas that are connected to chance or unpredictability. The idea of probability is difficult to explain in a formal sense; however, the intuition that comes the closest to capturing its essence is that it allows us to evaluate the likelihood or chances that a particular event will take place. This analysis sheds light on a great deal of the phenomena that we encounter in the real world. Even the seemingly most random processes or phenomena can be described with the help of probability models, and those descriptions can be predicted, at least to some extent. Because of this, probability is the basis for the algorithms used in artificial intelligence that we actually use in our everyday lives. Let’s take a look at some fundamental terminology before moving on to the formal description of probability laws.

The Sample Space and Its Events

Imagine you are conducting an experiment that requires you to flip a coin. A flip of the coin can only result in one of two outcomes at this point: heads or tails. The purpose of this research is to analyse and compute the probabilities of getting a tail as a result of flipping a coin. This type of experiment is known as a random experiment, and the sample space comprises all the potential outcomes of this experiment. Take, as an illustration, the case where a coin is tossed twice. What are the various outcomes that could occur?

TH, HH, HT, TT

All of these different results make up the sample space.

•Experiment with Random Outcomes An experiment is said to have random outcomes if it is designed so that the results cannot be predicted with absolute certainty.

•Sample Space: The set of all possible outcomes associated with a random experiment A sample space is the set of all possible outcomes associated with a random experiment. It is represented by the letter S in mathematical notation.

In the experiment described above, let’s determine the likelihood of getting a result of two heads. The definition of the probability of this happening is as follows:

P = No.of favourable outcomes/Total no.of possible outcomes

In this particular scenario, the best possible result is HH, and there are a total of four outcomes that are even remotely possible.

Therefore, the probability of getting two heads is equal to ¼.

Several Alternative Methods of Probability Analysis

The previous method of calculating the probabilities relied on the assumption that each of the outcomes had an equal chance of occurring. Take, for instance, the flipping of an impartial coin. Both the head and tails outcomes have an equal chance of occurring. Therefore, this finding cannot be extrapolated to all other experiments. In the early days of the study of probability, there were primarily two schools of thought:

•Classical Probability

•Frequentist Probability

Classical Probability

This strategy operates under the presumption that all of the possible outcomes have the same probability. In the event that our happening can take place in “n” out of a total of “N” different ways. The formula for calculating probability is as follows:

P(event) = n/N

Frequentist Probability

When calculating the probability, this method is more comprehensive than others. It does not make the assumption that all of the outcomes have the same likelihood of occurring. In situations in which the probabilities of the outcomes are not the same, we run the experiment a large number of times—call let’s it M. After that, make a note of the total number of times that particular occurrence took place, say m. After that, arrive at an empirical estimate of the probability by calculating it. Make use of the connection,

P(event) =lim_M→∞ m/M

Both of these strategies fall short when it comes to generalisation and cannot withstand the rigour of mathematics.

Consideration of probability as a function associated with any event is at the heart of the axiomatic method of probability, which adopts the approach of looking at probability as if it were a given.

An Approach to Probability Based on Axioms

Conduct a test with a random sample, where the sample space is denoted by S and the probability of any random event occurring is denoted by P. This model operates under the presumption that P should be a real-valued function that can take values ranging from 0 to 1. The sample space is defined as a power set, which will serve as the domain for this function. If each of these requirements is met, then the function in question should be able to demonstrate compliance with the axioms listed below:

•Axiom 1 states that the probability of any specific event, denoted by the letter X, must be greater than or equal to zero. Thus,

0 ≤ P (X)

•Axiom 2: We are aware that the set of all the results constitutes the sample space denoted by the letter S for the experiment. This indicates that the probability of any single outcome occurring is equal to one, which is denoted by the formula P(S) = 1. Intuitively, this indicates that the probability of obtaining some outcome is one hundred percent whenever this experiment is carried out in any capacity.

P(S) = 1

•Axiom 3: For the experiments in which we can observe both A and B as the results. If A and B cannot occur simultaneously, then

P(A ∪ B) = P(A) + P (B)

Here, ∪ stands for union. This can be interpreted as if it were said, “If A and B are mutually exclusive outcomes, that probability that either one of these events will happen is the probability of A happening plus the probability of B happening.” This interpretation is one way to understand what is being said here.

These precepts are also known as Kolmogorov’s three axioms in some circles. Given that all possible outcomes are incompatible with one another, the third axiom can be extended to cover additional possibilities.

Conclusion

Experiment with Random Outcomes An experiment is said to have random outcomes if it is designed so that the results cannot be predicted with absolute certainty.Sample Space-The set of all possible outcomes associated with a random experiment A sample space is the set of all possible outcomes associated with a random experiment. It is represented by the letter S in mathematical notation.In classical Probability-the strategy operates under the presumption that all of the possible outcomes have the same probability. In the event that our happening can take place in “n” out of a total of “N” different ways.When calculating the probability, this method is more comprehensive than others. It does not make the assumption that all of the outcomes have the same likelihood of occurring.