Probability of rolling a cuboid dice It's easy to count the probability of events on a regular dice because we know the probabilities ($P(1)$, $P(2)$, $P(3)$, $P(4)$, $P(5)$, $P(6)$) of all the basic outcomes ($P(i)=\frac{1}{6}$).

But... Is there any (simple) way how to determine the probabilities of basic outcomes of a cuboid dice?
Let's suppose for example a cuboid of sizes 1 cm, 1.1 cm, 1.2 cm...
 A: I once made some tests with this seven-sided die, which was claimed to land on each side with an equal probability:

Source: Wiki Commons.
It turned out that I could easily influence the probability of the outcome (more precisely the probabilty that it lands on a certain type of side) with the throwing technique, for example by giving the die a higher angular momentum when throwing it.
An extreme and illustrative example for this are tippe tops: If thrown without any momentum or angular momentum (i.e., if just dropped) from a small height and with a random starting orientation, they have a non-zero probablity of coming to rest in a “stem-down” position (at least some of them). When the angular momentum is very high in the otherwise identical situation, they almost certainly assume their characteristic rotation after some time and thus come to rest in a “stem-up” position.
The only way that the probability of a resting position does not depend on the initial angular momentum is that all resting positions are indistinguishable in respect to the inertia tensor (which is certainly not true for a true cuboid). Though I cannot prove it right now, I am very certain, that this is true for face-uniform solids (Platonic solids, Catalan solids, …), i.e., solids, whose faces are indistinguishable, only.
Analogue considerations should apply for other parameters like friction between die and surface and initial momentum and height. Thus, we have to agree on some distribution for these parameters to make a statement about the probabilities of the outcomes. And there arguably is no straightforward way to so, as for example choosing a uniformly distributed angular momentum would require a uniform distribution on $ℝ$.
While no direct answer to your question, this should illustrate why even defining the probabilities you seek is a problem on its own and why calculating them would be very difficult.
A: Diaconis, Holmes, and Montgomery have shown that when you look closely at the actual dynamics, taking angular momentum, etc., into account, even a coin toss is rather complicated.  
A: The system can be best modeled by creating an analogy with thermodynamics or considering an integral over the solid angle.  Either way this question that has been answered elsewhere:
http://www.riemer-koeln.de/mathematik/quader/cuboid.metrika.pdf
https://physics.stackexchange.com/questions/41297/how-to-determine-the-probabilities-for-a-cuboid-die
