There has been a fascinating discussion on a recent post thread concerning abiogenesis, the creation of molecular chemistry in the context of RNA and DNA, and probability.
I thought I would point people to a few articles, now some years old and probably out of date in certain areas of the findings of the chemistry and research of abiogenesis, but well worth reading for ideas concerning probability and how it is misappropriated by theists and creationists.
The first is called “Lies, Damned Lies, Statistics, and Probability of Abiogenesis Calculations” from the TalkOrigins Archive:
Every so often, someone comes up with the statement “the formation of any enzyme by chance is nearly impossible, therefore abiogenesis is impossible”. Often they cite an impressive looking calculation from the astrophysicist Fred Hoyle, or trot out something called “Borel’s Law” to prove that life is statistically impossible. These people, including Fred, have committed one or more of the following errors.
Problems with the creationists’ “it’s so improbable” calculations
1) They calculate the probability of the formation of a “modern” protein, or even a complete bacterium with all “modern” proteins, by random events. This is not the abiogenesis theory at all.
2) They assume that there is a fixed number of proteins, with fixed sequences for each protein, that are required for life.
3) They calculate the probability of sequential trials, rather than simultaneous trials.
4) They misunderstand what is meant by a probability calculation.
5) They seriously underestimate the number of functional enzymes/ribozymes present in a group of random sequences.
I will try and walk people through these various errors, and show why it is not possible to do a “probability of abiogenesis” calculation in any meaningful way.
A primordial protoplasmic globule
So the calculation goes that the probability of forming a given 300 amino acid long protein (say an enzyme like carboxypeptidase) randomly is (1/20)300 or 1 chance in 2.04 x 10390, which is astoundingly, mind-beggaringly improbable. This is then cranked up by adding on the probabilities of generating 400 or so similar enzymes until a figure is reached that is so huge that merely contemplating it causes your brain to dribble out your ears. This gives the impression that the formation of even the smallest organism seems totally impossible. However, this is completely incorrect.
Firstly, the formation of biological polymers from monomers is a function of the laws of chemistry and biochemistry, and these are decidedly not random.
Secondly, the entire premise is incorrect to start off with, because in modern abiogenesis theories the first “living things” would be much simpler, not even a protobacteria, or a preprotobacteria (what Oparin called a protobiont  and Woese calls a progenote ), but one or more simple molecules probably not more than 30-40 subunits long. These simple molecules then slowly evolved into more cooperative self-replicating systems, then finally into simple organisms [2, 5, 10, 15,28]. An illustration comparing a hypothetical protobiont and a modern bacteria is given below….
Note that the real theory has a number of small steps, and in fact I’ve left out some steps (especially between the hypercycle-protobiont stage) for simplicity. Each step is associated with a small increase in organisation and complexity, and the chemicals slowly climb towards organism-hood, rather than making one big leap [4, 10, 15, 28].
Where the creationist idea that modern organisms form spontaneously comes from is not certain. The first modern abiogenesis formulation, the Oparin/Haldane hypothesis from the 20’s, starts with simple proteins/proteinoids developing slowly into cells. Even the ideas circulating in the 1850’s were not “spontaneous” theories. The nearest I can come to is Lamarck’s original ideas from 1803! 
Given that the creationists are criticising a theory over 150 years out of date, and held by no modern evolutionary biologist, why go further? Because there are some fundamental problems in statistics and biochemistry that turn up in these mistaken “refutations”.
It is not just the application of probability to a given thing that is problematic, but it is what that given thing actually is that we are trying to assign a probability to!
As well as aiming probabilities at the wrong things, there is also often the improper use of probability. The author goes on to say:
Coin tossing for beginners and macromolecular assembly
So let’s play the creationist game and look at forming a peptide by random addition of amino acids. This certainly is not the way peptides formed on the early Earth, but it will be instructive.
I will use as an example the “self-replicating” peptide from the Ghadiri group mentioned above . I could use other examples, such as the hexanucleotide self-replicator , the SunY self-replicator  or the RNA polymerase described by the Eckland group , but for historical continuity with creationist claims a small peptide is ideal. This peptide is 32 amino acids long with a sequence of RMKQLEEKVYELLSKVACLEYEVARLKKVGE and is an enzyme, a peptide ligase that makes a copy of itself from two 16 amino acid long subunits. It is also of a size and composition that is ideally suited to be formed by abiotic peptide synthesis. The fact that it is a self replicator is an added irony.
The probability of generating this in successive random trials is (1/20)32 or 1 chance in 4.29 x 1040. This is much, much more probable than the 1 in 2.04 x 10390 of the standard creationist “generating carboxypeptidase by chance” scenario, but still seems absurdly low.
However, there is another side to these probability estimates, and it hinges on the fact that most of us don’t have a feeling for statistics. When someone tells us that some event has a one in a million chance of occuring, many of us expect that one million trials must be undergone before the said event turns up, but this is wrong.
Here is a experiment you can do yourself: take a coin, flip it four times, write down the results, and then do it again. How many times would you think you had to repeat this procedure (trial) before you get 4 heads in a row?
Now the probability of 4 heads in a row is is (1/2)4 or 1 chance in 16: do we have to do 16 trials to get 4 heads (HHHH)? No, in successive experiments I got 11, 10, 6, 16, 1, 5, and 3 trials before HHHH turned up. The figure 1 in 16 (or 1 in a million or 1 in 1040) gives the likelihood of an event in a given trial, but doesn’t say where it will occur in a series. You can flip HHHH on your very first trial (I did). Even at 1 chance in 4.29 x 1040, a self-replicator could have turned up surprisingly early. But there is more.
1 chance in 4.29 x 1040 is still orgulously, gobsmackingly unlikely; it’s hard to cope with this number. Even with the argument above (you could get it on your very first trial) most people would say “surely it would still take more time than the Earth existed to make this replicator by random methods”. Not really; in the above examples we were examining sequential trials, as if there was only one protein/DNA/proto-replicator being assembled per trial. In fact there would be billions of simultaneous trials as the billions of building block molecules interacted in the oceans, or on the thousands of kilometers of shorelines that could provide catalytic surfaces or templates [2,15].
Let’s go back to our example with the coins. Say it takes a minute to toss the coins 4 times; to generate HHHH would take on average 8 minutes. Now get 16 friends, each with a coin, to all flip the coin simultaneously 4 times; the average time to generate HHHH is now 1 minute. Now try to flip 6 heads in a row; this has a probability of (1/2)6 or 1 in 64. This would take half an hour on average, but go out and recruit 64 people, and you can flip it in a minute. If you want to flip a sequence with a chance of 1 in a billion, just recruit the population of China to flip coins for you, you will have that sequence in no time flat.
So, if on our prebiotic earth we have a billion peptides growing simultaneously, that reduces the time taken to generate our replicator significantly.
Okay, you are looking at that number again, 1 chance in 4.29 x 1040, that’s a big number, and although a billion starting molecules is a lot of molecules, could we ever get enough molecules to randomly assemble our first replicator in under half a billion years?
Yes, one kilogram of the amino acid arginine has 2.85 x 1024 molecules in it (that’s well over a billion billion); a tonne of arginine has 2.85 x 1027 molecules. If you took a semi-trailer load of each amino acid and dumped it into a medium size lake, you would have enough molecules to generate our particular replicator in a few tens of years, given that you can make 55 amino acid long proteins in 1 to 2 weeks [14,16].
So how does this shape up with the prebiotic Earth? On the early Earth it is likely that the ocean had a volume of 1 x 1024 litres. Given an amino acid concentration of 1 x 10-6 M (a moderately dilute soup, see Chyba and Sagan 1992 ), then there are roughly 1 x 1050 potential starting chains, so that a fair number of efficent peptide ligases (about 1 x 1031) could be produced in a under a year, let alone a million years. The synthesis of primitive self-replicators could happen relatively rapidly, even given a probability of 1 chance in 4.29 x 1040 (and remember, our replicator could be synthesized on the very first trial).
Assume that it takes a week to generate a sequence [14,16]. Then the Ghadiri ligase could be generated in one week, and any cytochrome C sequence could be generated in a bit over a million years (along with about half of all possible 101 peptide sequences, a large proportion of which will be functional proteins of some sort).Although I have used the Ghadiri ligase as an example, as I mentioned above the same calculations can be performed for the SunY self replicator, or the Ekland RNA polymerase. I leave this as an exercise for the reader, but the general conclusion (you can make scads of the things in a short time) is the same for these oligonucleotides.
It is well worth reading the whole piece.
It is also worth noting that research from 2011 shows that the universe is already full of more complex matter than we had thought:
Researchers from Hong Kong report that organic compounds of unexpected complexity exist throughout the Universe. They indicate that an organic substance commonly found throughout the Universe contains a mixture of aromatic and aliphatic components. The results suggest that complex organic compounds are not the sole domain of life but can be made naturally by stars.
In 2016, Scientific American ran an article “New Equation Tallies Odds of Life Beginning” that detailed such a calculation, about which it said:
The value Pa, which is the probability that life will assemble out of those particular building blocks over a given time, is murkier—and much more interesting. If the value of Pa is very low, it’s extremely unlikely that life will form even when the ingredients are there—potentially explaining why humans haven’t yet happened to create life in the lab, even if scientists have used the right ingredients, Scharf said. But a planet-wide “lab” would increase the odds that life-creating events will occur.
“We might have to wait 100 million years for it to fall into place just in a test tube,” Scharf said. “Whereas on a planet scale, you’ve got a trillion test tubes—probably even more than that. It’s conceivable that, using this equation, playing these games, is hinting at a possible explanation for why we haven’t seen life miraculously appearing in our laboratories, that … there’s some subtle thing that has to happen that really doesn’t happen often.”
And if the scale is larger than planetary, Scharf said, that could further increase the likelihood of life forming. Early Earth and Mars, for example, were cultivating their own, separate chemistries, but the early solar system was chaotic; impacts with other solar system bodies could have resulted in material exchanges between the two planets. That would have led to even more “test tubes”—the chemical mixing could have allowed even more interactions to occur, potentially hitting the right combination, Scharf said.
If multiple planets exchange materials, it could lead to a sort of “chemical amplification [that] could, in principle, be hugely important,” he said. “It could be all the difference between getting life to occur or not, especially when we’re dealing with such tiny, tiny probabilities on the microscopic scale of something going right,” he added.
But there is still an admission that many variables are unknown, and so accuracy is obviously limited. This is linked to the previous post that detailed the Argument from Ignorance. It is more honest, I would argue, to say, “I am not yet sure,” rather than “I don’t know; therefore, God.”
The fact is, scientists the world over are not doing the whole “give up and defer to God” as their explanatory maxim for phenomena in the world. Take the 2012 research from Weber University, as one picked out of a potential multitude:
A team of scientists announced that they have discovered AEG within cyanobacteria, which are believed to be some of the most primitive organisms on Earth. AEG is a small molecule that when linked into chains forms a hypothetical backbone for peptide nucleic acids, which have been theorized as the first genetic molecules.
The point being that there are so many theists who make huge claims about probability concerning abiogenesis and who don’t appear to have a good grasp on probabilities involved, or the plethora of recent science surrounding such claims.