It should be possible to model this mathematically, at least in an idealized and oversimplified manner. Let's go back to our rabbit example. Reduced to only one trait (fur) and a binary decision of black or white, the chance element of natural selection has a 0.5 probability of picking the "correct" choice: white.

Let's add a gene for thick (as opposed to thin) subcutaneous fat. The natural selection winner would have both white fur and insulating fat, and if we make the simplifying assumption that the fat layer is either thick or thin, with no intermediate alternatives, the probality of getting offspring with white fur AND good insulation in .5 x .5 = 0.25.

Even assuming the simplifying assumption of each trait requiring only a binary choice (black/white fur, thick/thin fat) with a 50% chance of manifesting itself,) the total constellation of traits required to produce a rabbit is the product of all those probabilities. Even if only a few traits are required to fully winterize our rabbit, we're going to need to produce an awful lot of offspring to get the entire organism to adapt. And the situation is complicated by the fact that sometimes poorly adapted animals get lucky and live long enough to reproduce, and well adapted ones sometimes die anyway because of bad luck.

I can't imagine how DNA does this, random selection can't do it with single traits, the probability product quickly gets too tiny, there must be some processing capability in the DNA itself to link up associated traits so they can be tested against the environment as a group, not one at a time.