My teacher defined conditional probability as:
\(\displaystyle P (B|A) =\)\(\displaystyle \frac{P(A\cap B)}{P(A)}\)
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 by definition of P would yield the equation at the top.
Does this make sense? Also, can anyone recommend a good probability book that does these kind of derivations?
\(\displaystyle P (B|A) =\)\(\displaystyle \frac{P(A\cap B)}{P(A)}\)
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 by definition of P would yield the equation at the top.
Does this make sense? Also, can anyone recommend a good probability book that does these kind of derivations?
