The axiomatic approach defines probability using a set of simple mathematical rules called axioms, introduced by Andrey Kolmogorov. These rules provide a clear and consistent foundation for modern probability theory.
Consider a random experiment whose sample space is S, and P is the probability of the occurrence of any random event.
- This model assumes that P should be a real-valued function with a range between 0 and 1.
- The domain of this function is defined to be the power set of the sample space.
If all these conditions are satisfied, then the function should satisfy the following axioms:
Axiom 1: For any given event X, the probability of that event must be greater than or equal to 0. Thus,
0 ≤ P(X)
Axiom 2: We know that the sample space S of the experiment is the set of all the outcomes. This means that the probability of getting some outcome from the sample space is 100 percent, i.e P(S) = 1. Intuitively this means that whenever the experiment is performed, the probability of obtaining one of the possible outcomes is certain.
Axiom 3: For the experiments where we have two outcomes A and B. If A and B are mutually exclusive,
P(A ∪ B) = P(A) + P(B) and P(A ∩ B) = 0
Here, ∪ stands for union, and ∩ stands for the intersection of two sets. This can be understood as if saying, “If A and B are mutually exclusive outcomes, the probability that either one of these events will happen is the probability of A happening plus the probability of B happening."
These axioms are also called Kolmogorov's three axioms. The third axiom can also be extended to several outcomes, given that all are mutually exclusive.
Let's say the experiment has A1, A2, A3, and ... An. All these events are mutually exclusive. In this case, the three axioms become:
Axiom 1: 0 ≤ P(Ai) ≤ 1 for all i = 1,2,3,... n.
Axiom 2: P(A1) + P(A2) + P(A3) +.... = 1
Axiom 3: P(A1 ∪ A2∪ A3 ....) = P(A1) + P(A2) + P(A3) ....
Applications
- Axiomatic probability provides a clear and logical foundation for probability and statistics.
- It helps in understanding important probability distributions like Normal, Binomial, and Poisson distributions.
- It is widely used in statistics, data science, and artificial intelligence for prediction and analysis.
- It helps explain uncertain events in fields such as physics, genetics, and thermodynamics.
- In computer science, it is used in algorithms, data security, and error detection techniques.
- In finance and insurance, it helps measure risk and predict possible losses or profits.
- In project management and decision-making, it helps people make better choices under uncertainty.
➢Practice: Solved Examples