Dawkins explains the utter improbability of what he calls "single-step" evolution in which living things are created in a single chance event, the principal argument by defenders of creationism and intelligent design.

We have seen that living things are too improbable and too beautifully 'designed' to have come into existence by chance. How, then, did they come into existence? The answer, Darwin's answer, is by gradual, step-by-step transformations from simple beginnings, from primordial entities sufficiently simple to have come into existence by chance. Each successive change in the gradual evolutionary process was simple enough, relative to its predecessor, to have arisen by chance. But the whole sequence of cumulative steps constitutes anything but a chance process, when you consider the complexity of the final end-product relative to the original starting point. The cumulative process is directed by nonrandom survival. The purpose of this chapter is to demonstrate the power of this cumulative selection as a fundamentally nonrandom process.

The Blind Watchmaker, p. 43

Dawkins says that each successive change was simple enough, relative to its predecessor, to have arisen by chance. But like Darwin, Dawkins says little specific about this chance, whether it is epistemic and only human ignorance about the workings of the molecular world or whether it is ontological chance beyond the assumed deterministic nature of classical physical laws.

Dawkins summarizes the argument of his book, The Blind Watchmaker.

Let us hear the conclusion of the whole matter. The essence of life is statistical improbability on a colossal scale. Whatever is the explanation for life, therefore, it cannot be chance. The true explanation for the existence of life must embody the very antithesis of chance. The antithesis of chance is nonrandom survival, properly understood. Nonrandom survival, improperly understood, is not the antithesis of chance, it is chance itself. There is a continuum connecting these two extremes, and it is the continuum from single-step selection to cumulative selection. Single-step selection is just another way of saying pure chance. This is what I mean by nonrandom survival improperly understood. Cumulative selection, by slow and gradual degrees, is the explanation, the only workable explanation that has ever been proposed, for the existence of life's complex design.

The whole book has been dominated by the idea of chance, by the astronomically long odds against the spontaneous arising of order, complexity and apparent design. We have sought a way of taming chance, of drawing its fangs. 'Untamed chance', pure, naked chance, means ordered design springing into existence from nothing, in a single leap. It would be untamed chance if once there was no eye, and then, suddenly, in the twinkling of a generation, an eye appeared, fully fashioned, perfect and whole. This is possible, but the odds against it will keep us busy writing noughts till the end of time. The same applies to the odds against the spontaneous existence of any fully fashioned, perfect and whole beings, including - I see no way of avoiding the conclusion - deities.

To 'tame' chance means to break down the very improbable into less improbable small components arranged in series. No matter how improbable it is that an X could have arisen from a Y in a single step, it is always possible to conceive of a series of infinitesimally graded intermediates between them. However improbable a large-scale change may be, smaller changes are less improbable. And provided we postulate a sufficiently large series of sufficiently finely graded intermediates, we shall be able to derive anything from anything else,

The Blind Watchmaker, p. 317

One must read Dawkins carefully to distinguish the "properly understood" from the "improperly understood" nonrandom survival.

Dawkins says "the explanation for life... cannot be chance" but also "The antithesis of chance is nonrandom survival, properly understood. Nonrandom survival, improperly understood, is not the antithesis of chance, it is chance itself."

"Single-step selection ...is pure chance." But "Cumulative selection, by slow and gradual degrees, is the explanation."

Dawkins could be more clear that each individual step includes "pure chance." These are "Darwin's...gradual, step-by-step transformations."

Dawkins says "cumulative selection [is] a fundamentally nonrandom process." But it would be clearer if Dawkins included the indeterministic chance event, the genetic mutation that precedes the adequately determined selection in each of the "sequence of cumulative steps."

Each cumulative step in evolution is a two-step process (as Ernst Mayr maintained), similar to the cosmic creation process and the two-stage model of free will.

As we write in our book on Einstein,

Charles Darwin recognized that chance was the driver of genetic variation in biology, but he did not say so explicitly because in his time chance was still considered atheistic. It would have embarrassed his wife.

William James was not embarrassed. He told the graduating class of Harvard Divinity School in 1884

“As soon as we begin to talk indeterminism to our friends, we find a number of them shaking their heads. This notion of alternative possibility, they say, this admission that any one of several things may come to pass is, after all, only a roundabout name for chance.”

Albert Einstein saw chance as a “weakness in the theory.” But the important thing is that Einstein was the first person to see ontological “real” chance in physics. Chance in classical physics had previously been regarded as epistemological, as human ignorance.

Dawkins says the improbability of a singe-step chance event is epitomized by the cosmologist Fred Hoyle's "Ultimate 747" event.

The argument from improbability, properly deployed, comes close to proving that God does not exist. My name for the statistical demonstration that God almost certainly does not exist is the Ultimate Boeing 747 gambit.

The name comes from Fred Hoyle’s amusing image of the Boeing 747 and the scrapyard. I am not sure whether Hoyle ever wrote it down himself, but it was attributed to him by his close colleague Chandra Wickramasinghe and is presumably authentic. Hoyle said that the probability of life originating on Earth is no greater than the chance that a hurricane, sweeping through a scrapyard, would have the luck to assemble a Boeing 747.

The God Delusion, p. 113