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Probability is a discipline of mathematics concerned with numerical explanations of the likelihood of an event occurring or the truth of a claim.
Definition of Probability
The measurement of an event’s possibility is known as probability. Probability can be used to estimate the likelihood of an event occurring because many occurrences are hard to predict with absolute certainty. The probability value is always between 0 and 1 in mathematics. It can be expressed in percentages, decimals, or fractions.
Terms related to Probability
Experiment: An experiment is an action with unknown results. Every experiment has a few positive results and a few negative results. Thomas Alva Edison’s historic efforts included over a thousand failed attempts before he finally succeeded in inventing the light bulb.
Random Experiment: A random experiment is one in which the set of possible outcomes is known but the specific outcome that will occur on a given execution of the experiment cannot be guessed before the experiment is performed. Random experiments include tossing a coin, rolling a dice, and drawing a card from a deck.
Trials: Trials are the various tries made throughout the course of an experiment. In other terms, a trial is any particular outcome of a random experiment. Tossing a coin, for example, is a test.
Event: An event is a trial with a clearly defined outcome. An event is something like obtaining a tail when flipping a coin.
Random Event: A random event is one that is difficult to predict. The chance value for such situations is very much less. A rainbow forming in the rain is a completely random occurrence.
Outcome: There are two distinct results when a sportsperson hits a ball towards the goal post. He may either score or miss the goal.
Possible Outcomes: A possible outcome is a list of all the possible outcomes of an experiment. The probable results of a coin flip are heads or tails.
Equally likely Outcomes: An experiment in which each of the possible outcomes has the same probability is referred to be equally probable. When you roll a six-sided die, you have an equal chance of receiving any number.
Sample Space: The sample space is the collection of results from all of the trials in an experiment. The probable results of rolling a dice are 1, 2, 3, 4, 5, and 6. The sample space is made up of these results. S = {1, 2, 3, 4, 5, 6}
Probable Event: A likely event is one that can be predicted. We can quantify the likelihood of such occurrences. Because the likelihood of a kid getting promoted to the next class can be computed, we may refer to this as a likely occurrence.
Impossible Event: An impossible event is one that occurs outside of the experiment or outside of the sample space of the experiment’s results. In a moderate climate, there is no snowfall. Because the chance of such an event occurring is zero, snowfall might be referred to as an impossible event.
Axiomatic Probability
In mathematics, the axiomatic probability is a unifying probability theory. The axiomatic approach to probability establishes a set of axioms that apply to all probability methods, including frequentist and classical probability. These principles are based on Kolmogorov’s Three Axioms in general. Axiomatic probability establishes mathematical probability’s starting points.
Definition
Probability is defined as a set function P(E) that assigns a value to every event E termed the “probability of E,” such that
1- The probability of a single event P(E) is greater than or equal to zero.
2- The resultant probability of the sample space is 1.
Kolmogorov’s Three Axioms
Andrey Nikolaevich Kolmogorov, a Russian mathematician who lived from 1903 to 1987, pioneered the axiomatic approach to probability. According to him, there are three axioms that may be used to calculate the likelihood of any event (E).
The Three Axioms of Kolmogorov are as follows:
The First Axiom
Axiomatic probability’s first axiom specifies that each event’s probability must fall between 0 and 1.
The number 0 indicates that the event will never occur, whereas 1 indicates that it will occur.
Any occurrence cannot have a negative probability. The probability of any event P (A) has the minimum value of zero, and if probability P (A) =0, event A will never occur.
The Second Axiom
The axiomatic probability of the entire sample space has one as its second postulate (100 per cent).
This is because the sample space S contains all possible results of our random experiment, or something happens if the experiment is run at any moment. As a result, each trial’s result is always part of experiment S’s sample space.
As a result, the event S occurs every time, and P(S) =1.
Let us consider an example: if we roll a dice, Sample space(S) = 1,2,3,4,5,6, and thus the event’s outcome will always fall between 1 and 6, P(S)=1.
The Third Axiom
The most intriguing postulate of probability is the third.
The underlying principle behind this axiom is that if any of the events are disjoint (that is, there is no overlap between them), the probability of their union must be equal to the sum of their probabilities.
Consider the case when A1 and A2 are mutually incompatible occurrences or outcomes: P (A1 ∪ A2) = P (A1) + P (A2).
The ∪ stands for the meaning of ‘union’ here.
Conclusion
Andrey Kolmogorov proposed the Kolmogorov axioms in 1933 as the basis of probability theory. These axioms are still important because they have direct applications in mathematics, physics, and real-world probability. Kolmogorov Axiomatic Probability is based on three axioms.
