# Probability Axioms: Deriving Version 2 from Version 1

• eddo
In summary, the conversation discusses two versions of probability axioms that are equivalent. The first version has three axioms, including the probability of the sure event being 1 and the probability of the impossible event being 0. The second version has four axioms, with the addition of the probability of the union of two events being equal to the sum of their individual probabilities minus the probability of their intersection. The discussion focuses on how to derive axiom 4 from version 2 using version 1. Ultimately, the method involves partitioning A union B into different combinations of intersections and applying boolean algebra to simplify the events into A and B.
eddo
In my probability class we were given two versions of probability axioms which are equivalent. Let S be the sure event, A and B any arbitrary events, I the impossible event. I will use u to denote union, and n to denote intersection:

Version 1
1. P(S)=1
2. P(A)>=0
3. If AnB=I, than P(AuB)=P(A)+P(B)

Version 2
1. P(S)=1
2. P(A)>=0
3. P(I)=0
4. P(AuB)=P(A)+P(B)-P(AnB)

It is easy to go from version 2 to version 1, and I can see how to show that P(I)=0 using version 1. The problem I'm having is how to derive axiom 4 of version 2 from version 1. I know I will have to consider two events made up of unions and intersections of A and B, to which I can apply axiom 3 of version 1, but I can't quite figure out how to do it. Thanks for any help.

After some trial and error I got this: partition A u B into A n ~B, A n B, and B n ~A. If you work that out you can get the right expression; one of the terms you'll get is P(A n B), and there is a way to go from P(A n B) to -P(A n B) and the other terms will fall into place.

I tried this but wasn't able to work it out. How do you go from P(AnB) to -P(AnB)? Once I get to the second step I'm not sure how to write the two terms other than P(AnB) in any other way, since they involve intersections, and the axioms version 1 don't say anything about intersections. You can rewrite the other two terms (before you make them into 2 separate terms) as (An~B)u(Bn~A)=(AuB)n(~Bu~A), but once again this involves intersections, not unions which doesn't help much.

Nevermind I got it. You just have to turn P(AnB) into 2P(AnB)-P(AnB), then group one of each of the positive terms with your other two terms. Axiom 3 of version 1 can than be used in reverse to turn each of these into a single probability, where the events in question simplify, using boolean algebra, to A and B. Thanks for the help Bicycle Tree.

## 1. What are probability axioms?

Probability axioms are fundamental principles that serve as the basis for mathematical probability theory. They provide a set of rules for calculating the likelihood or chance of an event occurring.

## 2. What is the difference between Version 1 and Version 2 of probability axioms?

Version 1 of probability axioms, also known as Kolmogorov's axioms, is the most commonly used set of axioms for probability theory. Version 2, also known as Cox's axioms, is a more generalized and flexible version of probability axioms.

## 3. How can Version 2 be derived from Version 1 of probability axioms?

Version 2 of probability axioms can be derived from Version 1 by introducing the concept of belief functions and using them to define probability. This allows for a more intuitive and flexible approach to probability calculations.

## 4. What are the advantages of using Version 2 of probability axioms?

The advantages of using Version 2 of probability axioms include its flexibility and applicability to a wide range of situations, as well as its ability to handle uncertain and incomplete information. It also allows for a more intuitive understanding of probability.

## 5. How are probability axioms used in scientific research?

Probability axioms are used in scientific research to calculate the likelihood of different outcomes and to make predictions about future events. They are also used to determine the significance of experimental results and to analyze complex systems and data.

Replies
36
Views
3K
Replies
1
Views
267
Replies
8
Views
426
Replies
2
Views
712
Replies
3
Views
1K
Replies
2
Views
1K
Replies
7
Views
1K
Replies
9
Views
1K
Replies
5
Views
3K
Replies
1
Views
1K