A frequent creationist allegation is that "evolution can't happen because a protein with a specific sequence of amino acids is incredibly improbable." Here's an example, from "Answers in Genesis."
However, ignoring all such problems, and many others that could be detailed, what is the probability of getting just 100 amino acids lined up in a functional manner? Since there are 20 different amino acids involved, it is (1/20)100, which is 10-130. To try to get this in perspective, there are about 1080 fundamental particles (electrons, etc) in the universe. If every one of those particles were an experiment at getting the right sequence with all the correct amino acids present, every microsecond of 15 billion years, that amounts to 4.7 x 10103 experiments. We are still 1027 experiments short of getting an even chance of it happening. In other words, this is IMPOSSIBLE!Cytochrome c is a 100 amino acid protein, for all intents and purposes. But I have demonstrated that it is not anywhere near as improbable as 10-130 - in fact, more than forty orders of magnitude more probable - and in a less specified format, as it would have been when it first appeared, it would have been less improbable again.
There is an argument from improbability - but it doesn't start from a single specific sequence of amino acids.
Similarly, the vague darwinist assertion that "there are gazillions of bacteria, each of whom have the opportunity to make random sequences, so really the suggestion that there is any problem with improbability is absurd" is just as flawed. TalkOrigins is careful to avoid this - there are other points at which their argument could be challenged, but this isn't one. I'm not sure that this is so much the case with people who casually engage in this debate, and stop counting when they get to 1020 bacteria.
What I have suggested - and again, I have little doubt that the same calculations have been carried out elsewhere - is that there is an effective probability boundary of 10-60 for proteins generated at around the time of the origin of life, in the conventional evolutionary model. If a protein has to be specified such that it has a lower probability of arising than this, then there aren't sufficient probabilistic resources. A similar analysis can be applied in a similar sort of way to the appearance of proteins and genes at other stages in evolutionary history.
From here, the analysis opens out in several directions. For a start, in determining a minimum for the improbability of proteins when they initially appear, and determining a current improbability, it is possible to determine how much "work" evolution has done, in refining proteins through evolutionary history. Are the mechanisms available to evolution able to do this work?
Also, the figures that I put down can be more accurately determined - for example, the time at which cytochrome c first appeared can be pinned down. More accurate estimates of the current improbability of cytochrome c can be determined. (I notice that in a paper cited on the TalkOrigins page given above, Yockey says that the improbability of cytochrome c is nearer 10-68 - which makes me feel quite smug about my guess of 10-70!) More thought-out figures for the amount of carbon available for forming "proto-proteins" can be determined. Big issues related to how the genetic code and translation systems might appear remain unresolved.
Starting to attach numbers to the specifications of proteins has an impact on the debate about irreducible complexity. Implicit in this concept is that the different components are themselves improbable - and thus the requirement for them to appear simultaneously requires the multiplication of small improbabilities. The probabilities ought to be hardened up. If we can get a handle on the value of these probabilities, we can again move away from the "yes it is, no it isn't" style of debate that is what the current state of the art boils down to.
Both sides in the debate suggest that this sort of analysis is something that the other side ought to be doing. In truth, without attaching numbers to the propositions that are being kicked around, both sides are debating not science, but presuppositions.