... apparently, in The God Delusion, "If the design is so improbable, how much more improbable is the designer?"

This sounds neat, but without reading his book, and relying on Allygally's comment in the preceding post for this quote, it is a poor argument.

I assume that what Dawkins is suggesting is that the assessment of the probability of something arising by chance is small, and if it is small, then the probability of there being a designer who is capable of producing it is even smaller.

If this truly reflects his opinion, it represents a change of mind from him. He used to accept that you couldn't go on relying on "lucky breaks" as an explanatory system - and he suggested (if I remember right, in The Blind Watchmaker) that if the probability of life evolving were so small as to be unlikely more than once in the galaxy, then inferring "chance" rather than "design" would be unreasonable. Of course, this was in the good old days when Sagan was confidently telling us that there was likely to be millions of intelligent lifeforms in the galaxy - prior to Brownlee, Ward (Rare Earth), Gonzalez and Richards (Privileged Planet). This original position is, of course, no longer tenable, so perhaps the new position represents the "new" Dawkins.

The argument can be easily dismantled - first by analogy, and second by logic. It is very improbable that an airliner might appear by chance. Dawkins (via Allygally) argues that if an airliner is an improbable design, it is much more improbable that there is a designer.

More logically, a corollary of what Dawkins is saying appears to be that the more designed something looks, the less likely it is to have a designer - an absurd position to take. It misses the point that the "design inference" style argument has an implied "given". "The probability of this feature arising given no designer is small" - from a design inference point of view, using the probability boundary, what this is saying is that the probability is such that you would be lucky to see this in the universe. Therefore (I think the force of the argument is) it is unreasonable to assume it is the product of chance, and more reasonable to assume that it is the product of design, even though the probability that there is a designer is indeterminate.

What is the probability that there is a designer? One way of looking at this is that since at least half the world's population believes that there is one (or more), it can hardly be less than 50:50.

I find it hard to believe that this really represents Dawkins' position. He might (in Christian terms) be a fool, but he's not stupid. If this is not what Dawkins is getting at, then Allygally (or Dawkins) can feel free to come back and correct me.