Second Axiom

The axioms are a reduced version of those proposed by mathematician Andrey Kolmogorov in 1933. The core concepts of probability theory had previously been “thought to be somewhat unique,” therefore his goal was to place them in their “natural home, among the general notions of modern mathematics.” To this purpose, Kolmogorov provided a mathematical explanation of what a “random event” is (in terms of sets). We skipped over this part when expressing the axioms above, but it’s discussed in Kolmogorov’s original publication Foundations of the Theory of Probability.

Kolmogorov’s axioms placed probability in the context of measure theory. You are assigning a number to some form of mathematical object when you measure something (such as length, area, or volume) (a line segment, a 2D shape, or a 3D shape). Probability is a method of assigning a number to a mathematical object in a comparable way (collections of events). Because of Kolmogorov’s definition, the mathematical theory of measures could include probability theory as a specific case.

If you’re familiar with probability theory, you’ll notice that two key concepts are lacking from the preceding axioms. One is the concept of independent occurrences, which states that the probabilities of all potential mutually exclusive outcomes of a process add up to one.

The axioms of probability:

Mathematicians dodge these difficult concerns by quantitatively defining an event’s probability without delving into its deeper significance. Three requirements, known as the axioms of probability theory, lay at the heart of this concept.

Axiom 1: A real number larger than or equal to 0 is the probability of an event.

Axiom 2: The likelihood of at least one of all conceivable outcomes of a process (such as rolling a die) occurring is 1.

Axiom 3: If two events A and B are mutually exclusive, the chance of either A or B occurring is equal to the likelihood of both A and B occurring.

The first is a direct result of the axioms. Assume that a procedure (such as rolling a die) can produce a number of mutually incompatible primitive occurrences (rolling a 1, 2, 3, 4, 5, or 6). The chance of at least one of these occurrences occurring is 1 according to postulate 2. The likelihood that at least one of them occurs, according to Axiom 3, is the total of the individual probabilities of the elementary events. To put it another way, the total probability of the elementary events is 1.

Moving on to the second missing characteristic, note that the concept of independence does not relate to the events that can occur as a result of a single process (such as rolling a die), but rather to processes: we must define what we mean by two processes (such as rolling a die twice) being independent. Kolmogorov presents the famous notion of independence after carefully defining what we mean by “two processes” mathematically. When the likelihood of both occurrences occurring is equal to the product of their individual probabilities, they are said to be independent.

Discrete probability models:

The underlying principle is that in discrete probability models, the probability of occurrences may be computed by adding all of the associated possibilities, however in continuous probability models, integration must be used instead of summing.

Consider the space S as an example. A discrete probability model is referred to when S is a countable set. Because S is countable, we can list all of its elements in this case:

S={s₁,s₂,s₃,⋯}.

If AS is an event, then A is also countable, and we can express A using the third axiom of probability.

To find the probability of an event in a countable sample space, we simply add the probabilities of each element in that set.

Conclusion

The likelihood of at least one of a process’s conceivable outcomes (such as rolling a die) occurring is 1. The axioms are a reduced version of those proposed by mathematician Andrey Kolmogorov in 1933. The core concepts of probability theory had previously been “thought to be somewhat unique,” therefore his goal was to place them in their “natural home, among the general notions of modern mathematics.” Probability is a method of assigning a number to a mathematical object in a comparable way (collections of events).

Three requirements, known as the axioms of probability theory, lay at the heart of this concept.

A real number larger than or equal to 0 is the probability of an event. Second, The likelihood of at least one of all conceivable outcomes of a process (such as rolling a die) occurring is 1. Third, If two events A and B are mutually exclusive, the chance of either A or B occurring is equal to the likelihood of both A and B occurring.

The underlying principle is that in discrete probability models, the probability of occurrences may be computed by adding all of the associated possibilities.