- #1

Rasalhague

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I'm wondering how conditional probability relates to concepts of sample space, observation space, random variable, etc. Using the notation introduced in the OP here, how would one define the standard notation for conditional probability "P(B|A)" where A and B are both subsets of some sample space S?

Suppose P(A) is nonzero. I'm thinking there's an implicit random variable, namely the identity function on S. So S is both sample space and observation space. And the distribution of S-as-observation-space is Q : E --> [0,1] such that Q(B&A) = P(B&A)/P(A). Then P(_|_) is a function R : E x E --> [0,1], defined by:

If P(A) = 0, then R(B,A) = 0,

otherwise R(B,A) = Q(B&A).

Does that work, and is it how conditional probability is usually formalised in this system?

EDIT: On 2nd thoughts, the step of talking about a random variable and an observation space seems a bit superfluous, since we could just define conditional probability directly by the formula. Still, it gaves me a chance to test my understanding of the concepts...

Suppose P(A) is nonzero. I'm thinking there's an implicit random variable, namely the identity function on S. So S is both sample space and observation space. And the distribution of S-as-observation-space is Q : E --> [0,1] such that Q(B&A) = P(B&A)/P(A). Then P(_|_) is a function R : E x E --> [0,1], defined by:

If P(A) = 0, then R(B,A) = 0,

otherwise R(B,A) = Q(B&A).

Does that work, and is it how conditional probability is usually formalised in this system?

EDIT: On 2nd thoughts, the step of talking about a random variable and an observation space seems a bit superfluous, since we could just define conditional probability directly by the formula. Still, it gaves me a chance to test my understanding of the concepts...

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