To lower the odds of the possibility of God's existence, Dawkins utilizes the idea of irreducible complexity (IC).  Succinctly, this proposition asserts that there are many, many examples in the world around us of things that are incredibly complex - such as the human eye.  These examples are too complex to have evolved over time, because if any of the gazillions of complex pieces/parts were removed, the entity would no longer function.  The eye would not see, for example.  The argument is against the natural selection idea that randomly selected mutations that provided benefits were selected for in the breeding process by making their hosts better able to survive and reproduce, thereby passing on the mutation while more inferior genetic arrangements eventually died out.  But if some things are irreducibly complex, then they couldn't be the result of random mutations because they would have had to have come into existence as a complete package, else there would be no incremental benefit to the hosts along the way.

Dawkins of course disagrees with this critique, and has he own theories about how this argument can be deconstructed - all of them hypothetical but plausible from a certain perspective.  

Dawkins then tweaks this argument to his own purposes.  If IC argues that the existence of life as we know it is so improbable that it demands a creator as an explanation for it, then the Creator must be even more complex than the life he has created - and therefore even less likely of existing. If Christians want to rule out natural selection because of the odds against it, they must rule out the existence of God on the same grounds. 

That's it.  

Really.

If a complex universe requires a complex creator, then the odds of that creator simply existing as is - and particularly existing as the simplest essence in the universe (something that Thomas Aquinas argues for in his Summa Theologica) then the odds are overwhelmingly against the existence of such a God.  God cannot be simple, but rather highly complex in order to create a highly complex creation.  

There are several problems with this.  Most notably, Dawkins argues from creation backwards to the creator, but not in a way that Christianity actually portrays God.  Dawkin's presumption is that, since in creation around us, complex things must come (by the definition of evolution and natural selection) from simpler things, then anything capable of creating a complex thing must itself be complex.  What this does is effectively move God into the realm of creation, subjecting him to the same characteristics that we observe around us.

The Bible argues that there are two categories to existence.  In the first category you have God.  In the second category, you have everything else.  Koala bears, ducks, redwood trees, pulsars, angels, demons, and Pat Sajak.  This organization of creation differs markedly from the predominate division of the world derived from Greek philosophy, which creates a dualistic universe made up of spirit and ideas in one category, and physical, material objects and persons in the other.  And it is different from the secular philosophy that there is only one category, and God isn't in it.  The Bible posits that God is outside of creation - fundamentally separate and different from it.  God is eternal, the universe is finite.  God has no creation or creator, the universe has God as it's creator.  

What Dawkins' argument does is erase this distinction.  He places God within the confines of the created order, subjects him (by hypothesis) to the same issues that Dawkins sees in creation, and announces that such a God is nigh on impossible.  What he does in the process is import his interpretation of how things in the natural world must be (simple things evolving into more complex things) and then ports this understanding to his interpretation of God.  None of which is Biblical.

The result is that Dawkins is not dealing with the God of the Bible, but with another god that he has created from his own assumptions and observations about the universe.  It's a version of a straw man argument, a form of logical fallacy where you deliberately misrepresent the position of your opponent in order to more easily tear it down.  I can't say if Dawkins' misrepresentation is intentional or not.  It could be just ignorance, but if that's the case, then Dawkins is tilting at the wrong windmills, and perhaps his view of the Judeo-Christian God would be fundamentally different based on a more correct (Biblically speaking) understanding of him.

But I doubt it.

I'm rather surprised that he feels this is the silver bullet that almost - but not entirely - negates the idea of God.  I'm surprised that he feels this is what should win over agnostics and weaken the faith of Christians.  No doubt, some of this will be accomplished, particularly among folks who have less of a familiarity with what Scripture has to say about God.  But it's hardly the weighty and intellectual dismantling of the faith that Dawkins sets out to accomplish.