Is there a probability interpretation that only allows for probabilities in $\left[0,1\right]\cap\Bbb Q$? Are there probability interpretations that only allow for probabilities that are members of the set $$\left[0,1\right]\cap\Bbb Q?$$
Related, but distinct: Allowed probabilities under frequentism.
 A: If the sample space is finite that it is possible to develop a rather expressive portion of probability theory under the constraint that all probabilities are rational. As soon as the sample space becomes infinite though such an endeavor is next to impossible. More importantly, both in the finite and infinite case there is very little reason (philosophical or mathematical) to consider such an attempt. The reason is that the field of real numbers, as a place to perform computations, is a by far superior to the field of rational numbers. As far as any philosophical considerations, you can view the real numbers as approximations of rational numbers (no, I'm not confused), as follows. Suppose you really want to work with rational number for whatever physically inspired, philosophical, or religion reasons. You quickly find out that a set of rational numbers can be bounded above yet have no supremum. So, while it's easy do arithmetic with the rational numbers, you find that you can't take suprema or infima without a lot of extra care. This makes life terribly complicated. So, you enlarge your system and complete the rationals to obtain the field of real numbers. Now suprema exist, but they may not be rational. Luckily, every real number is as close as you like to some rational, so you can think of that real number you got as the supremum of a bunch of rational numbers, as being an approximation for a rational number (that does not really exist). This turns out to make computations easy. The price is that you now have all those irrational numbers that don't necessarily agree with your physically inspired motivation, philosophy, or religion. But, it's a good approximation and the mathematics works perfectly, so you remind yourself that any interpretation of mathematics to anything outside of mathematics is just an interpretation and everything is fine. I hope this addresses your question. 
