One of the most persistent criticisms of the Neo-Darwinian synthesis lies in the statistical challenge to the randomness of the emergent processes.

Darwinian evolution is most unlikely to get even one polypeptide right, let alone the thousands on which living cells depend for their survival. This situation is well-known to geneticists and yet nobody seems prepared to blow the whistle on the theory. Evolution from Space, F. Hoyle & C. Wickramasinghe

 The literature is hopelessly confusing here, because it tends to depend on prior assumptions. Beware of quibbles that distract attention from the basic intent of the critiques, which are surely confirmed now that we see that very complex processes must be invoked that aren't random. The critics, with their own mistakes, have won the argument, though noone really says so. The real issue now seems the origin of the basic 'toolkit' and the body plans visible after the Cambrian. One of the most persistent arguments has been the implausibility of random change in DNA molecules. Such arguments have been attacked repeatedly, but they won't go away. The fossil record doesn't look uniform, nor does random tinkering with genomes look promising. What's more, serious scientists don't go to Las Vegas to get rich, except in Darwinian theory.  One of the most notable critiques of the past generation is that of Evolution from Space, by F. Hoyle and Wickramasinghe, with their famous Boeing airplane argument. This argument has been frantically  'refuted' so many times, I tend to yawn when Darwinists speak on the subject.  This is a challenge to natural selection, not necessarily to naturalistic evolution, although Evolution from Space tends to be attacked for the perhaps wrong spiritual conclusion it draws from its hard-to-refute argument.  

Also there is Michael Denton's Evolution: A Theory in Crisis, and more recently on probability issues, Senapathy's Independent Birth of Organisms,  who finds the statistical arguments to be insuperable. Denton's original book is constantly attacked and may be slightly out of date, but its basic reasoning has not been shown to be wrong. 

Noting the way in which selectionism (originally proposed and rejected earlier by Edward Blyth) triumphed, Hoyle and Wickramasinghe note: 

There was no general perception that the real issue of controversy, as it had existed decades earlier between Blyth and Darwin, had still to be resolved.  The difficulty for the few who wished to come to grips with the real question of whether random mutations and natural selection had been sufficient to explain the origin of species, and by implication the origin of life, which Darwin maintained but which Blyth did not, was that in the nineteenth century the theory was impossible to quantify. Before modern microbiology, the evolutionist simply pointed to the long time-scales of geology and there was then no way to demonstrate that it would need a time-scale 10^40,000 times as long to produce the effects that were being claimed. p.133

Again, P. Senapathy, in Independent Birth of Organisms, in a calculation of the probabilities of random changes in gene sequences claims that

1. The genome of an organism is closed and locked with respect to evolution. The variability of a creature is confined to the closed framework of its genome.
2. Random mutational processes cannot lead to the evolution of new genes and genetic networks needed for new organs and appendages. 

Not even one gene!? What to say of thousands. And the issue, as he notes, requires equal independent parallel change in developmental processes of great complexity. 

 The statistical unlikelihood of natural selection has always haunted the Darwinian theory of evolution. The Genome is one of the most elaborated systems of machines imaginable, with every function clearly given a mechanical sequence, yet the crucial part, its genesis, we are to believe, is purely random. Books by experts can be disingenuous here, so it seems a lost cause to clarify the issues. 

 This statistical argument  has been pointed out so many times that it is essential to the history of the debate, in any case, along with the persisting difficulties in the fossil record, notwithstanding many successes as to fossil gaps,  but overall  it is difficult to understand how the theory maintains its hold in such a flagrant disregard of intuitively obvious problems. 

 The answer is that these attacks endanger the whole game and  are the subject of as many counterattacks, none very convincing. A close look shows the nature of the subject has simply moved on, tacilty acknowledging the point, with its emphasis on new processes of development, hox genes, etc... 

 These statistical arguments could be incorrect, but it is strange that textbooks are virtually silent on the problem! 

• Thus, we have  I.L. Cohen, Darwin was Wrong (Greenvale, NY: New Research Publications, 1984), a self-published 'howl of protest', now vanished, and a good treatment of gnomic probabilities. Arguments using Natural Selection seem oblivious to what happens to the reciprocal (one over.. as 1/2) of a large factorial (N!= N factorial= N x N-1 x N-2 x ...3 x 2 x 1).  I.L. Cohen, in Darwin was Wrong, illustrates the issue of the permutations and combinations of the basic CGAT molecules in the DNA code. He indicates that as few as 84 items evenly divided into 21 each of C, G, A, T, (representing nucleotides) translates into a probability of occurrence: 2.08 x 10 ^-51!! As Cohen notes, mathematicians consider any probability  beyond 10^-50 as automatically being zero. Thus any strand of DNA having over 84 nucleotides is beyond the range of random occurrence. Yet, for example, a mammalian cell is estimated to have about 3 billion nucleotides in its DNA.  

Most efforts to overcome these arguments, such as the claim (cf. Richard Dawkins in The Blind Watchmaker) that a cumulative effect defeats these odds, remain unconvincing, and unproven.  Cf. R. Dawkins, Climbing Mount Improbable, p. 75.

Dawkins claims that while variation is random, natural selection is non-random. But that is a distraction, confusing the issue. selection remains 'random' in the sense that no directional macroevolution is invoked. 

  Here, S. Kauffman, cf. At Home in the Universe, chapter 2-3, starts with the theme of self-organization, after acknowledging the problem.  This is a new process, if real, ceding the point about natural selection. This work can be useful for its straightforward acknowledgement of the problem. Too many books beat around the bush and waste your time. 

The argument by statistics has a long history, cf also the report of a symposium at Wistar Institute and Murray Eden’s hard to obtain, "Inadequacies of Neo-Darwinian Evolution as a Scientific Theory", which concludes, p. 109, "It is our contention that if ‘random’ is given a serious and crucial interpretation from a probabilistic point of view, the randomness postulate is highly implausible and that an adequate scientific theory must await the discovery and elucidation of new natural laws," from Mathematical Challenges to the Neo-Darwinian Interpretation of Evolution. Also discussed in David Berlinski's Black Mischief. Cf. also, David Berlinski in Commentary magazine, June 1996, and follow up, September 1996.