Natural selection changes the game. Imagine - gross oversimplification - that 500 mutations are needed to go from Hyracotherium to Equus. Each mutation is advantageous and each has the odds of 1.0E-7 of appearing spontaneously per birth in the population. So what are the odds?

(1) Assume it happens all at once. Then the odds of jumping from Hyracotherium to Equus is (1.0E-7)^500 = 1.0E-3500 or utterly mind-numbingly unlikely. Even two mutations being present in the one individual is 100 trillion to one against.

(2) Assume there's 10 million breeding animals per generation. Thus a mutation, of any one of the 500, might occur every generation. Because they're advantageous the mutations can persist in the population, eventually becoming more common. They're selected and thus they're collected in the genome of the species. Say it takes 100 generations for a mutant, but advantageous, gene to dominate its fellow alleles - to become fixed. Thus the odds are good, after a hundred generations, of two advantageous "mutations" (though one is now "fixed" as part of the genome) being present in an individual. And so forth for the rest of the mutations from species A to species B. After 50,000 generations its entirely feasible for all 500 mutations to be present in a significant number of members of the population. Yet in all only 500,000 billion breeding animals have been in existence. But natural selection has preserved each mutation and increased its prevalence, thus dramatically altering the odds.

And the time frames are utterly different. Assume a 5 year generation time. In the first case the number of generations is 10^343 - a ridiculous amount of years being implied. In the second it's just 50,000 - a mere 0.25 million years. A veritable geological eye-blink.

Now several traps conceptually exist in this picture. Firstly, the odds of any particular combination of mutations will be at the end-point of the process is very, very small - the same is true for any draw of all 52 cards in a poker deck. But the process of mutation and natural selection behind any living species implies that the particular sets of genes we see had some advantage and would have been favoured by the odds.

Secondly, there's no such thing as a smooth transition from one species to another - to the exclusion of other possibilities. Every evolutionary lineage is less a vine climbing a wall than a tree branching and branching over and over. At all scales of taxonomy. Sure a smooth transition can be documented for any higher group transition (say from "reptile" to "mammal") - but only by ignoring all the side-branches, living and dead.

That's to be expected, because there's usually no sharp transition between daughter and parent species - parental species can exist in parallel with the daughter species, if some geographic or other barrier stops the lesser being absorbed by the larger. Thus bifurcation is inevitable. Imagine if every group of 10 mutations acquired by our hypothetical species meant a species branch point - thus we have 2^(50) bifurcations producing ~ 1.1 quadrillion different daughter species.

Nothing in nature is ever so simple, but it illustrates that the power of the combinatorial explosion can work both ways - for sceptics of evolution, and its true believers. The true believers have the advantage that biology really does work via mutation and selection, and the difficulty that the sheer complexity of the end-products makes accounting for them very, very difficult in anything but simplistic terms. But the genomics/proteomics boom that's sweeping molecular biology is making the formerly impossible now look very possible...