Definition of Axiomatic probability

Types of Probability

There are several different types of probability, including:

1. Classical probability
2. Empirical probability
3. Subjective probability
4. Axiomatic probability
5. Frequentist probability
6. Bayesian probability
7. Constructive probability
8. Algorithmic probability
9. Theoretical probability
10. Conditional probability
11. Joint probability

Axiomatic probability is a mathematical framework for defining probability that is based on a set of axioms or assumptions. These axioms are used to establish the properties of probability, such as how it is measured and how it behaves under different conditions. The most common set of axioms for probability were introduced by Andrei Kolmogorov in 1933, and are known as Kolmogorov’s axioms. These axioms provide a way to assign probabilities to events in a consistent and logical way, and form the basis for much of modern probability theory.

Axiomatic probability formula

The axiomatic formula for probability is a set of mathematical rules that define how probability is calculated. It is based on the idea that probability is a measure of the likelihood of an event occurring. There are different versions of the axiomatic formula, but a common one is the following:

1. The probability of an event is a non-negative number, between 0 and 1, where 0 represents an impossible event and 1 represents a certain event.
2. The probability of the sample space (the set of all possible outcomes) is 1.
3. If A and B are two mutually exclusive events (events that cannot happen at the same time), then the probability of either A or B happening is the sum of their individual probabilities.
4. If A and B are not mutually exclusive, then the probability of both A and B happening is the sum of the probability of A happening, plus the probability of B happening, minus the probability of A and B happening.
5. The probability of an event not happening is 1 minus the probability of the event happening.

These axioms provide a foundation for calculating probabilities and allow us to reason about probabilities in a logical and consistent way.

Axiomatic probability example

One example of using the axiomatic formula for probability is determining the probability of rolling a specific number on a fair die. Let’s say we want to find the probability of rolling a 4.

1. The probability of rolling a 4 is a non-negative number between 0 and 1. In this case, it is 1/6 because there are six possible outcomes on a fair die and only one of them is a 4.
2. The probability of the sample space (all possible outcomes on a die) is 1.
3. If we want to find the probability of rolling a 4 or a 6 on a die, we can add the probabilities of each event individually. In this case, the probability of rolling a 4 is 1/6 and the probability of rolling a 6 is 1/6, so the probability of rolling a 4 or a 6 is (1/6) + (1/6) = 2/6 or 1/3.
4. If we want to find the probability of rolling a 4 and a 6 on a die, we can find the probability of rolling a 4, which is 1/6, and the probability of rolling a 6, which is 1/6. Then we subtract the probability of rolling a 4 and a 6 at the same time, which is impossible in this case, so the probability of rolling a 4 and a 6 is (1/6) + (1/6) – 0 = 2/6 or 1/3.
5. The probability of not rolling a 4 is 1 – (1/6) = 5/6.

This example shows how the axiomatic formula for probability can be used to determine the probability of specific events occurring when rolling a fair die.