Definition of Conditional Probability [i.e. P (B|A)]

lamp23

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I've seen the following given as a definition:

\(\displaystyle P (B|A) =\)\(\displaystyle \frac{P(A\cap B)}{P(A)}\)

yet it seems to make more sense to first define

\(\displaystyle P(B|A) = \)\(\displaystyle \frac{n(A\cap B)}{n(A)}\) where A takes on the role of the sample space and then divide both the top and bottom of the fraction by n(S) and getting:

\(\displaystyle P(B|A) = \)\(\displaystyle \frac{\frac{n(A\cap B)}{n(S)}}{\frac{n(A)}{n(S)}}\)

which equals the top equation.

Does this make sense? Also, can anyone recommend a good probability book or set of notes that does these kind of derivations?
 
I've seen the following given as a definition:
\(\displaystyle P (B|A) =\)\(\displaystyle \frac{P(A\cap B)}{P(A)}\)
yet it seems to make more sense to first define
\(\displaystyle P(B|A) = \)\(\displaystyle \frac{n(A\cap B)}{n(A)}\) where A takes on the role of the sample space and then divide both the top and bottom of the fraction by n(S) and getting:
\(\displaystyle P(B|A) = \)\(\displaystyle \frac{\frac{n(A\cap B)}{n(S)}}{\frac{n(A)}{n(S)}}\) which equals the top equation.
Does this make sense? Also, can anyone recommend a good probability book or set of notes that does these kind of derivations?
I think the issue is that your derivation works only when you have a system where each atom of the set is equiprobable. The definition (and definitions cannot be derived) is more general. My opinion. Quite possibly wrong.
Not only is it completely wrong for continuous distributions, it does not even work for infinite discrete distributions.

It works for finite distributions. But there it is redundant because that is exactly how the finite measure is defined.
 
I think the issue is that your derivation works only when you have a system where each atom of a finite set is equiprobable. The definition (and definitions cannot be derived) is more general. My opinion. Quite possibly wrong. Edits in blue: See next post. Pka is of course right that the derivation does not work for infinite sets, whether discrete or continuous. My original statement, interpreted strictly, was wrong. I have fixed it. My point was that the definition covers the case contemplated by the poster of finite sets with equiprobable elements just as well as the poster's derivation does, but the definition has the advantage of covering other cases to which that derivation does not apply, such as finite sets where the elements are not equiprobable. I did not mean to imply that the derivation would work for infinite sets.

Is there a book you can recommend which would cover conditional probability for infinite sets?
 
Is there a book you can recommend which would cover conditional probability for infinite sets?
Any college level probability textbook will have that topic.
But the issue here is not conditional probability the way the probability measure is defined.

The way your teacher defined conditional probability works for all distributions. What you posted was a rehashing of the definitions of a counting measure on a finite set.

If you have access to a college library, I will recommend Jim Pitman's textbook as one of the most readable.
 
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