We see one of the most famous presentations (though not the first) of the Argument from Design in William Paley’s Natural Theology (1802). William Paley (1743-1805) was an English philosopher of religion and ethics and had written several essays regarding Christian apologetics, such as his Truth of the Scripture History of St Paul he wrote in 1790. However, his most notable work for which he become widely known is his Natural Theology which he wrote just three years before his death. He presents the famous watchmaker analogy as such (I quote at length):
In crossing a heath, suppose I pitched my foot against a stone and were asked how the stone came to be there, I might possibly answer that for anything I knew to be contrary it had lain there forever; nor would it, perhaps, be very easy to show the absurdity of this answer. But suppose I had found a watch upon the ground, and it should be inquired how the watch happened to be in that place. I should hardly think of the answer which I had before given, that for anything I knew the watch might have always been there.
Yet why should not this answer serve for the watch as well as for the stone? Why is it not as admissible in the second case as in the first? For this reason, and for no other, namely, that when we come to inspect the watch, we perceive – what we could not discover in the stone – that its several parts are framed and put together for a purpose, e.g., that they are so formed and adjusted as to produce out motion, and that motion so regulated as to point out the hour of the day; that if the different parts had been differently shaped from what they are, of a different size from what they are, or placed after any other manner or in any other order than that which they are placed, either no motion at all would have answered the use that is now served by it. 
Paley’s argument thus functions as an Argument from Analogy, which can be schemed as such:
- a, b, c, and d all have properties P and Q.
- a, b, and c all have properties R as well.
- Therefore, d has property R too (probably).
Notice the conclusion to the argument – Therefore, d has property R too (probably). Paley’s argument  is what’s known as an inductive argument. This simply means that the argument doesn’t say “that its conclusion follows necessarily from its premises, but only that its premises establish a probability that the conclusion is true” . This simply plays the role of the function of a sort of scientific investigation, where it is “designed to explain the facts of experience, and must be accepted or rejected according to whether it meets the criteria of adequacy by which hypotheses are appraised in science and in everyday life” .
Thence, suppose we were to fill in the variables of the above argument:
- Boats, houses, watches, and the whole experienced world have such properties as “mutual adjustment of parts to whole” and “curious adapting of means to ends.”
- Boats, houses, and watches have the further property of being produced by design.
- Therefore, it is probable that the universe also has this further property, that it too was produced by design.
However, in what way can we consider the universe being produced by design as probable?
Introducing Probability – The Likelihood Approach
I believe philosopher Robin Collins (2012) has interesting insight in respect to this issue of probability and likelihood of the universe being produced by design. Consider the following hypotheses:
- (1) The Existence of a Life-Permitting Universe (LPU).
- (2) The Theistic Hypothesis (T).
- (3) The Naturalistic Single-Universe Hypothesis (NSU).
Although I think (1) is rather self-explanatory , (2) and (3) are rather important. By (2), we simply say the hypothesis that “there exists an omnipotent, omniscient, everlasting or eternal, perfectly free creator of the universe whose existence does not depend on anything outside of itself” . By NSU, we can simply mean that “there is only one universe, the existence of which is an unexplained, brute given, and that within the universe the laws and constants of physics do not significantly vary from one space-time region to another. . . it excludes any transcendental explanation of the universe, be that explanation theistic or nontheistic” .
Consider finally other notable terms relevant to our consideration for the argument:
- (1′) P(A|B) and conditional epistemic probability.
- (2′) Background information k and k’.
- (3′) Other symbols. (“<<” – or, “much, much less than”)
(1′) may require some careful explanation. Consider A and B to represent two propositions – propositions, which, are simply statements that assert something (“Apples are red,” “Dogs can’t meow,” etc.). Thus, P represents the probability of Proposition A in respect to (or given) Proposition B. In respect to (2′), k refers to our background information in a more general sense, while k’ represents our background knowledge on some given particular thing (we will have to designate something k’ in order for it to have a proper function in our argument).
Thus, let us further consider what is known as the Likelihood Principle. This principle states (or can be stated) that where we have two competing hypotheses (say h(1) and h(2)), an observation (E) counts as evidence in favor of h(1) over h(2) if the observation is more probable under h(1) than (2). In a symbolic form, we can state the principle as follows:
- E counts in favor of h(1) over h(2) if P(e|h(1)) > P(e|h(2)).
However, the form of probability that we are concerned with is what is known as Conditional Epistemic Probability, or, “the degree to which Proposition B, in and of itself, supports or leads us to expect A to be true” . Thus, under this light we can probably deter our language from the Likelihood Principle into what is known as the Expectation Principle. Collins explains:
[I]f an event or state of affairs e is more to be expected under one hypothesis, h(1), than another, h(2), it counts as evidence in favor of h(1) over h(2) – that is, in favor of the hypothesis under which it has the highest expectation. The strength of the evidence is proportional to the relative degree to which it is more to be expected under h(1) than h(2). 
Thus, we can structure the argument as follows:
- Given the fine-tuning evidence, LPU is very, very epistemically unlikely under NSU: that is, P(LPU|NSU & k’) << 1, where k’ represents some appropriately chosen background information, and << represents much, much less than (thus making P(LPU|NSU & k’) close to zero).
- Given the fine-tuning evidence, LPU is not unlikely under T: that is, ~P(LPU|T &k’) << 1.
- T was advocated prior to the fine-tuning evidence (and has independent motivation).
- Therefore, by the restricted version of the Likelihood Principle, LPU strongly supports T over NSU.
-  Quoted from Reason and Responsibility, ed. Joel Feinberg and Russ Shafer-Landau (Thomson and Wadsworth: 2005) p. 32
-  David Hume’s Dialogues Concerning Natural Religion (1779) contains a fictional character named Cleanthes who structures the argument similarly.
-  Feinberg and Shafer-Landau (2005), p. 3 – emphasis added
-  Ibid.
-  Yet, if you do prefer an explanation: “This. . . mean[s] the existence of a material spatiotemporal reality that can support embodied moral agents, not merely life of some sort” (Collins 2012, 203)
-  Ibid.
-  Ibid. – emphasis added
-  Ibid., pp. 205-206
-  Ibid., p. 206 – Collins provides further clarification in terms of what version of the Likelihood Principle he is using, which is what he calls the restricted version due to “certain potential counterexamples” (206). I wont be addressing this matter due its technicality relative to our attention here, but should you as the reader have any concerns or comments about this, please feel free to ask.