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Starting with a few statements and building up more mathematics from these claims is one technique in mathematics. Axioms are the foundational statements. A mathematical axiom is often anything that is self-evident. Deductive logic is used to establish further claims, known as theorems or propositions, based on a small set of axioms.
Probability, a branch of mathematics, is no exception. Three axioms can be used to define probability. The mathematician Andrei Kolmogorov was the first to do so. The axioms that underpin probability can be utilised to derive a wide range of findings.
To comprehend the probability axioms, we must first go through some basic definitions. Assume that we have a sample space S of possible outcomes. This sample space can be considered the universal set for the situation under investigation. The sample space is divided into subsets known as events E1, E2, and so on.
We further suppose that any given occurrence E may be assigned a probability. This is a function that takes a set as an input and returns a real number as an output. P stands for the probability of the event (E).
Axiom one:
The probability of any event is a nonnegative real number, according to the first axiom of probability. This indicates that a probability can never be smaller than zero and can never be infinite. Real numbers are the type of numbers that we may use. Both rational numbers, usually known as fractions, and irrational numbers that cannot be expressed as fractions are included in this category.
It’s worth noting that this axiom says nothing about the size of an event’s probability. Negative probability are not possible because of the axiom. It reflects the idea that the smallest probability is 0, which is designated for impossible events.
Axiom two:
The probability of the entire sample space is one, says the second axiom of probability. P(S) = 1 is written symbolically. The idea that the sample space is everything imaginable for our probability experiment and that there are no events outside of it is implicit in this axiom.
This axiom does not, by itself, impose an upper limit on the probabilities of events that do not occur throughout the entire sample space. It does imply that something with absolute certainty has a 100% likelihood.
Axiom three:
Mutually exclusive events are dealt with in the third axiom of probability. P(E1 U E2) = P(E1) + P(E2).
The axiom actually covers a situation in which there are multiple (even infinite) events, each of which is mutually exclusive. The chance of the union of the events is the same as the sum of the probabilities as long as this occurs:
P(E1 U E2 U… U En) = P(E1) + P(E2) +… En
Although this third axiom may not appear to be particularly beneficial, when paired with the other two, it becomes rather powerful.
Axiom application:
The three axioms define an upper bound for any event’s probability. Ec stands for the complement of the event E. E and Ec have an empty intersection in set theory and are mutually exclusive. Furthermore, the complete sample space is E U Ec = S
When we combine these facts with the axioms, we get:
1 = P(S) = P(E U Ec) = P(E) + P(Ec).
We may see that P(E) = 1 – P(E) by rearranging the equation above (Ec). We now have an upper bound on the chance of any event of 1 because probabilities must be nonnegative.
We get P(Ec) = 1 – P(E) by rearranging the formula once again (E). This formula also tells us that the likelihood of an event not happening is one minus the probability of it happening.
Conclusion:
Probability cannot be negative, according to the first axiom. P(A) has the least value of zero, and if P(A)=0, the event A will never occur.
Axioms are the foundational statements. A mathematical axiom is often anything that is self-evident. Deductive logic is used to establish further claims, known as theorems or propositions, based on a small set of axioms.Three axioms can be used to define probability.
The probability of any event is a nonnegative real number, according to the first axiom of probability. The probability of the entire sample space is one, says the second axiom of probability. P(S) = 1 is written symbolically.
Mutually exclusive events are dealt with in the third axiom of probability. P(E1 U E2) = P(E1) + P(E2).
The three axioms define an upper bound for any event’s probability. Ec stands for the complement of the event E. E and Ec have an empty intersection in set theory and are mutually exclusive.
