The heart of Dawkins argument is in effect that God is very improbable. The greater the improbability of any occurrence, the more improbable a putative designer must be - and if such a designer is improbable, then he/she/it probably doesn't exist -
However statistically improbable the entity you seek to explain by invoking a designer, the designer himself [sic] has got to be at least as improbable. (TGD, paperback edition, p.138)Dawkins describes this "counter-argument" - which he describes as "the statistical demonstration that God almost certainly doesn't exist" (p.137) - as "the Ultimate Boeing 747 gambit," after Fred Hoyle's likening of the appearance of life to being a whirlwind in a junkyard spinning up an airliner.
Actually, I don't think it's a statistical demonstration of anything, other than the fact that Dawkins can't string together either any coherent mathematics or any coherent philosophy. What I find really mindblowing is that Dawkins thinks that here he has found the great decisive argument that will basically wrap it up for God in one fell swoop. Honestly, if it were that good, does he not think that some of the really great atheist minds of the past might have got there before him?
It is worth noting that one of the benefits of this argument is that it seems to make Dawkins completely irrefutable. Is life likely to appear? If so, it is a matter of ease for darwinian processes. Is life unlikely to appear? Then as God is even more unlikely to be present than life, it must be down to a non-theistic process. Again, if it were really that easy to make a case against God, does he not think that somebody might have got there before? And since they haven't, can't he harbour a few grammes of uncertainty about the soundness of his argument?
Dawkins reduces the design argument to "these things are improbable". That may be a fair representation of the design argument as it is presented in the Watchtower tracts he uses to illustrate his point, but it is hardly reflective of "Darwin's Black Box" or "The Privileged Planet" - I'll come to this in more detail later. But as is pointed out - as indeed Dawkins himself points out - the improbability of life or the universe on its own doesn't constitute an argument for anything. If you can't infer that life is designed solely on the grounds of its improbability (which you can't - and indeed, for that matter, proponents of ID don't - due to anthropic effects), then neither can you infer that belief that there isn't a God is reasonable on the grounds of such a God's improbability. God either exists, or he doesn't. The improbability of the proposition has little bearing on its truth. Conan Doyle places the phrase "Once you have eliminated the impossible, then whatever remains, no matter how improbable, must be the truth," in the mouth of Sherlock Holmes. The same applies here - the fact that something is improbable has little bearing on its truth.
Secondly, I think Dawkins is confusing the ideas of improbability and specified complexity. Opponents of ID may express frustration and disagreement with the way Dembski uses such terms in his arguments - but Dawkins makes no attempt to define them at all, or even express any awareness of what the differences between them might be. A random sequence of letters picked from a Scrabble bag is improbable, but it is not complex. The sequence of letters "whatdoyougetwhenyoumultiplysixbynine" is just as improbable as any other sequence of letters with the same length, but it signifies more than a random sequence. That is why Arthur Dent and Ford Prefect recognise it as significant in "The Hitchhiker's Guide to the Galaxy". Life isn't just improbable, it is complex. Likewise, surely it isn't simply the improbability of a hypothetical God that is the issue, it is the complexity of such a God. If you are framing a philosophical argument, such issues need to be carefully defined.
However, this doesn't deal with Dawkins' argument - it just highlights the fact that he doesn't apparently really understand what he is talking about, or is deliberately simplifying things for the sake of his readers. This latter seems unlikely, given how he resents how religions patronise their followers in this way.
Another approach that doesn't deal with Dawkins' argument is to raise the question of how improbable something is. Dembski placed a bound on this - he suggested that there was a limit to the amount of specified complexity that life could have. It is worth noticing that Dawkins was prepared to attach numbers as well. On the issue of abiogenesis, he suggests that there may be a billion billion planets in the universe, and life may arise on a billion of them. One might suggest that if there are a billion billion planets in the universe and abiogenesis only arises on one (ours), or evolution only apparently produces complex life on one (ours), then Dawkins would consider himself corrected. It would be interesting to compare Dawkins' figures here with the figures he put forward in "The Blind Watchmaker", during the height of the Sagan era - unfortunately, I don't have a copy to hand. However, the empirical assessment of these numbers will have to wait, for now. Dembski was far more generous than Dawkins - offering not one in a billion billion (1018) chances but one in 10150 chances as the statistically significant boundary.
Most significantly, though, there is the issue of whether the fact that a supposed God was more complex than anything in the universe would actually be an argument against such a God's existence. I have pointed out before that everything in the universe is contingent - right back to the Big Bang - even if we aren't in a position to determine all those contingencies (and no, I'm not proposing to put God in those gaps in our knowledge, Dawkins and Bonhoeffer will be pleased to know). But there is no reason to think that a hypothetical cause of the universe - which, unlike the staggering quantum fluctuation which some suggest bootstrapped the universe into existence, would be external to the universe, not bound by space-time or energetic constraints - need also be contingent. The fact that the complexity that we observe in the universe is contingent does not mean that the complexity of a cause for the universe would also have to be contingent. This is why Dawkins' argument doesn't stand up.