Unconditional probabilities underdetermine conditional probabilities

Consider two random processes, R₁ and R₂. Process R₁ randomly and uniformly chooses a real number between 0 and 1, with each number being a possible outcome. Process R₂ works as follows. It starts with the result of R₁. If the result of R₁ is other than 1/2, that is also the result of R₂. If the result of R₁ is 1/2, then the result of R₂ is defined to be 1/4.

Observe that for any measurable subset S of the interval between zero and one, the probability that R₁ generates a number in S equals the probability that R₂ generates a number in S. To see this, observe that unless R₁ results in 1/2, R₂ will result in the same thing as R₁. So, the probability that R₁ yields something in S differs from the probability that R₂ yields something in S by at most the probability that R₁ gives 1/2. But the probability that R₁ yields 1/2 is zero, so the probability that R₁ yields something in S does not differ from the probability that R₂ yields something in S.

Now, intuitively the conditional probability that R₁ results in 1/2 given that it results in either 1/2 or 3/4 is a half. But intuitively the conditional probability that R₂ results in 1/2 given that it results in either 1/2 or 3/4 is zero, since R₂ cannot result in 1/2. So, the conditional probabilities for R₁ and R₂ differ, while the unconditional ones are the same. So we cannot define conditional probabilities in terms of unconditional ones (and in particular, we cannot define conditional probabilities for zero-probability events using the Radon-Nikodym theorem or some other limiting procedure) in a way that does justice to our intuitions about conditional probability. This conclusion is of course the central point of Hajek's "What conditional probability could not be."

Of course, to me when wearing my probability theorist's hat, this doesn't matter. When I wear my probability theorist's hat, probability measures are simply measures with total probability one, and everything of significance is defined only up to almost sure equality (X=Y almost surely iff P(X=Y)=1) so the distinction between R₁ and R₂ is irrelevant. Such probability measures need not have any epistemological or physical interpretation. But there is a non-mathematical notion of conditional probability as well, one tied to epistemological and physical applications, and it is that kind of conditional probability that does not supervene on unconditional probability.