We know that $P(A\mid B) = \frac{P(A \cap B)}{P(B)}$. So $P(B) = P(A\mid B)P(A \cap B)$. Thus are all probabilities conditional probabilities? Can one make a probability more accurate by introducing a conditional component? For example, the probability of rolling a six on a fair die is $1/6$. But can we make this more accurate by assuming prior events?

We know that $P(A\mid B) = \frac{P(A \cap B)}{P(B)}$. So $P(B) = P(A\mid B)P(A \cap B)$. Thus are all probabilities conditional probabilities? Can one make a probability more accurate by introducing a conditional component? For example, the probability of rolling a six on a fair die is $1/6$. But can we make this more accurate by assuming prior events? More generally,
is it possible to develop probability theory based on conditional probability?

We know that $P(A|B) = \frac{P(A \cap B)}{P(B)}$. So $P(B) = P(A|B)P(A \cap B)$. Thus are all probabilities conditional probabilities? Can one make a probability more accurate by introducing a conditional component? For example, the probability of rolling a six on a fair die is $1/6$. But can we make this more accurate by assuming prior events?

We know that $P(A\mid B) = \frac{P(A \cap B)}{P(B)}$. So $P(B) = P(A\mid B)P(A \cap B)$. Thus are all probabilities conditional probabilities? Can one make a probability more accurate by introducing a conditional component? For example, the probability of rolling a six on a fair die is $1/6$. But can we make this more accurate by assuming prior events?