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Axiomatic Probability

Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true.

Axiomatic probability is a unifying probability theory. It sets down a set of axioms (rules) that apply to all of types of probability, including frequentist probability and classical probability. These rules, based on Kolmogorov’s Three Axioms, set starting points for mathematical probability.

Axiomatic Probability is just another way of describing the probability of an event. The probability of an event is a number between 0 and 1, where, roughly speaking, 0 indicates the impossibility of the event and 1 indicates certainty. The higher the probability of an event, the more likely it is that the event will occur.

Axiomatic Probability, as the word says itself, some axioms are predefined before assigning probabilities. This is done to quantize and hence to ease the calculation of the happening of the event.

History

The Kolmogorov Axioms are the foundations of probability theory introduced by Andrey Kolmogorov in 1933. These axioms remain central and have direct contributions to mathematics, the physical sciences, and real-world probability cases.

Andrey Kolmogorov - Axiomatic Probability
Andrey Kolmogorov (Source)

We assume that all outcomes have an equal chance (probability) to occur. Thus, to define probability, we used equally likely or equally probable outcomes. This is logically not a correct definition.

Thus, another theory of probability was developed by Andrey Kolmogorov in 1933. He laid down some axioms to interpret probability, in his book ‘Foundation of Probability’ published in 1933’.

Axiomatic Probability Conditions

There are three Conditions to axiomatic probability put up by Andrey Kolmogorov also known as Kolmogorov Axioms.

Let ‘S’ be the sample space, ‘E’ be the event, ‘\(\omega\)‘ be the possible outcomes, ‘n’ be no. of subsets & ‘F’ be the event space.

S = {\(\omega_1, \omega_2, …\omega_n\)}

First Axiom

The probability of an event is a positive real number, 

P(E) \(\in R, P(E) \geq 0 ,\hspace{0.5cm} \forall E \in F\)

(OR)

\(0 \leq P(\omega_i) \leq 1 \hspace{0.5cm} \text{for each }  \sum_{i=1}^{n} P(\omega_i) = S\)

Second Axiom

The probability of the sum of all subsets in the sample space is 1.

P(S) = 1

(OR)

\(P(\omega_1) + P(\omega_2) + … P(\omega_n) = 1\)

Third Axiom

If \(E_1\) and \(E_2\) are mutually exclusive events, then

\(P(E_1 \cup E_2) = P(E_1) + P(E_2)\)

See Set Operations for more info

We can also see this true for \(P(\phi)\).

Therefore, \(P(E \cup \phi) = P(E) + P(\phi) = P(E)\)

Here, \(P(\phi)\) is a null set (or) \(P(\phi)\) = 0

Axiomatic Probability Applications

  1. Probability theory is applied in everyday life in risk assessment and modeling. The insurance industry and markets use actuarial science to determine the prices and making decisions. 
  2. Also, probability can be used to analyze trends in biology (e.g., disease spread) as well as ecology.
  3. Probability is used to design games along with old games references and market surveys as well players’ feedbacks.
  4. The cache language model and other statistical language models that are used in natural language processing are also examples of applications of probability theory. 

Axiomatic Probability Examples

Question 1. What is the probability of getting two heads and one tail if an unbiased coin was tossed three times?

 Solution. 

S = { HHH, HHT, HTH, HTT, THH, THT, TTH, TTT }

The no. of events in sample space we can find by the formula \(2^n\), where n is no. of times coin is tossed, i.e \(2^3\) = 8 events.

Favourable outcome (E) = 2 heads and 1 tails

E = { HHT, HTH, THH }

=> P(E) = \(\frac{3}{8}\).

Question 2. If A and B are two candidates seeking admission to an engineering college. The probability that A is selected is .5 and the probability that both A and B are selected is at most .3. What is the possible probability that of B getting selected?

Solution.

\(
P(A) = 0.5\\
P(A)*P(B) \leq 0.3\\
=> (0.5)P(B) \leq 0.3\\
=> P(B) \leq 0.6\\
\)

FAQs

What is the difference between sample space and event space?

An event space contains all possible events for a given experiment. An event is just a set of outcomes of an experiment, combined with their probability.
The sample space of an experiment is the set of all possible outcomes of that experiment.

Why can’t an event be having a probability of more than 1?

P(0) is said to be an impossible event with the chances of it happening is 0%
While P(1) is said to be a sure event with a chance of it happening is 100%.
Hence, it cannot exceed the 100%.

What are mutually exclusive events?

Two events are mutually exclusive or disjoint if they cannot both occur at the same time. A clear example is the set of outcomes of a single coin toss, which can result in either heads or tails, but not both.

What is a Venn diagram?

A Venn diagram is an illustration that uses circles to show the relationships among finite groups of things. Circles that overlap have a commonality whereas the circles that do not overlap do not share those traits. Venn diagrams help to visually represent the similarities and differences between two concepts.

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