Re: Axioms for elementary probability

Re: Axioms for elementary probability

[A. MANI]

On Thursday 07 May 2009 08:14:01 David Espinosa wrote:
> Here's a question about elementary (naive, finitist) probability.
> The proper, self-dual axioms for elementary probability are presumably
>
>   P(0) = 0
>   P(X) = 1
>   P(A u B) + P(A n B) = P(A) + P(B)
>
> P's domain is a boolean algebra.  P's codomain is [0,1].
> I'm wondering, what kind of algebraic structure is [0,1] in this case?

It is a partial algebra with partial operations \wedge, v, +, o, 0, 1
(the order can be written with \wedge, v)

a+b is defined iff a+b =< 1 in R
a o b  is always defined (multiplication)

plenty of strong weak equalities hold.

What is the generalization to categories?

Best

A. Mani





-- 
A. Mani
CLC, ASL, AMS, CMS
http://amani.topcities.com

Axioms for elementary probability

[David Espinosa]

Here's a question about elementary (naive, finitist) probability.
The proper, self-dual axioms for elementary probability are presumably

  P(0) = 0
  P(X) = 1
  P(A u B) + P(A n B) = P(A) + P(B)

P's domain is a boolean algebra.  P's codomain is [0,1].
I'm wondering, what kind of algebraic structure is [0,1] in this case?

BTW, from these axioms we can prove nice things like inclusion / exclusion:

  P(A u B u C) = P(A) + P(B) + P(C) - P(A n B) - P(A n C) - P(B n C) + P(A n
B n C)
  P(A n B n C) = P(A) + P(B) + P(C) - P(A u B) - P(A u C) - P(B u C) + P(A u
B u C)

David