A very small proportion of much of the flyable atmosphere is occupied by aeroplanes. There are places where this is not true, for example near airports. But these places are few and far between (though all journies fly through them at least twice!).
For the rest of space, the probability of intersection of two volumes the size of an aircraft in the volume of space is really rather small. Pick a bit of atmosphere at random, what's the probability it will contain an aeroplane?
In what I'll call free-air (air not near airports) this is skewed by desired entry and exit points and the shortest path between them. Given free reign, all planes would travel by the most fuel or time efficient route between the two areas of non-free air.
Air traffic control forces airlines to act suboptimally, rising to a particular altitude, travelling along a particular non-minimal route, and so on, a cost in terms of efficeincy to ensure safety and to avoid what you might call the tragedy of the calculus of variations.
However, the algorithmic means by which air traffic control operates, which is predicated on the discretisation of space -- the division of space into discrete quantities, levels, paths, and so on -- combined with the increasingly accurate following of these paths by aeroplanes equiped with sophisticated navigational aids, is an initial step towards greatly increasing the probability of collision. Two appropriately timed planes travelling at 12,000 feet will crash: two planes travelling at a (truely) randomly chosen height between 11,000 and 12,000 will only have a 1% chance of colliding.
Of course, the algorithm layered on top of this discretisation, whatever it might be, is some sophisticated avoidance algorithm, or other. What is implemented, however, is not the algorithm, but a realistion of the algorithm -- a shadow of the pure algorithm cast on the wall of the cave -- with a certain probability of failure due to inevitable imperfections (human or otherwise). When the implementation fails, the accurate following of the discretised space increases the chance of collision greatly.
In freeair, it seems to me, there is a case for random flightpaths. These would be designed to alter the deviation from the mean path at the socially acceptable compromise between safety and price (which there is at the moment and, as I said above, in my opinion inevitable in any realisation, though discussion of this subject is taboo).
Note that I'm not advocating a free-for-all. That would lead to the tragedy of the shortest path again. There might be a box, sealed and sacrosanct, which emits a truely random path which must be followed in all its peversity.
Dispite what some of my more actuarial friends might claim, though, there is (unfortunatley for my safety) more to risk than the sum of probability times severity. It can be seen more clearly in transport accidents than in any other field, that our society firstly requires a moral agent to be given a thing we call "control" of a dangerous device (though that control is always imperfect and transitory, and secondly that a failure through imperfect realisation is actually the preferred form of failure, compared to failure through "random fate". It seems to me that this is largely through the blame structure we use to dispell accidents, which is well-defined in the case of perfect realisation, but is silent for "cold" "unfeeling" random error. An accident caused by random chance would be more difficult to dispell than one caused by imperfect realisation.
Given these societal preferences, it is perhaps best to approach a compromise system which fulfills the requirement of an algorithm which, when perfectly realised is completely safe, but within this constraint attempts to handle errors in realisation through randomising within the constraints which this imposes. For example a number of corridors containing a truely massive bundle of randmly chosen (sometimes intersecting) paths. There /should/ be just one plane in each bundle, but when there is not, then they are likely to be travelling along different paths, and the probability of intersection is small.
Sorry this comment is so long.