In defending their claims of an accidental origin of life and an undirected chance origin of species and biological innovations such as new types of protein molecules, Darwinists have often appealed to incorrect principles such as a principle that given enough time, even the most unlikely things will happen. To help clarify reality, it will be useful to list some facts and solid principles regarding the matter of chance construction. I can define chance construction as the arising by unguided events or unguided processes of organized useful states of matter or orderlylooking or usefullooking sequences of numbers, letters or symbolic tokens.
Humans have intuitive "gut feelings" about chance construction, such as the intuition that long and purposefullooking sequences of text do not arise by chance, and that accidents do not construct complex useful inventions. But it is not necessary to merely appeal to "gut feelings" when talking about matters of chance construction. Some very precise principles and facts can be asserted.
Principle #1: The probability of accidentally exactly matching a particular sequence (whether numbers, letters or amino acids) is roughly equal to one divided by a divisor equal to the number of possible items in each position of the sequence multiplied by itself a number of times equal to the length of the sequence.
The principle above can be expressed by this formula:
p(sequence) = ~ 1 / (t^{l})
where p is a probability, sequence is a specific sequence of tokens such as letters, numbers or codons, t is the number of possible tokens in each position of the sequence, and l is the length of the sequence (how many of these tokens the sequence consists of). The ~ symbol above means "roughly."
It is easy to give some examples illustrating this principle. Consider a set of random digits that has a particular length. The probability of having a random number of the same size matching such a sequence is roughly equal to 1 divided by a divisor equal to the number of possible digits in each position (10) carried to the power of l, where l is the length of the numerical sequence. So, for example, the probability of a fivedigit random number equaling 12345 is roughly equal to 1 divided by a divisor equal to the number of possible digits in each position (10) carried to the power of 5. 10 to the fifth power is 100,000. So the probability of a random fivedigit number equaling 12345 is roughly equal to 1 divided by 100,000. The probability roughly equals .00001.
In the language above 1 use the term "roughly" only because the exact probability here is 1 divided by 99,999 rather than 1 divided by 100,000. Rather than clutter up the language above with more cumbersome language yielding so exact a probability, I prefer to use simpler language that gives an almostexactlyright answer.
To give another example, the probability of a fiveletter lowercase random alphabetic sequence equaling "abcdef" is roughly equal to 1 divided by a divisor equal to the number of possible lowercase letters in each position (26) carried to the power of 6, the length of the sequence "abcdef." 26 to the sixth power is 308,915,776 . So the probability of a random alphabetic sequence of six lowercase letters exactly matching "abcdef" is roughly equal to 1 divided by 308,915,776.
We can use this principle to calculate the probability of exactly matching a sequence of amino acids by a random sequence of amino acids. The number of different types of amino acids used by the proteins of living organisms is 20. So, for example, the probability of getting a random sequence of amino acids matching the amino acid sequence of a particular protein having 100 amino acids is roughly equal to 1 divided by a divisor equal to the number of possible amino acids in each position (20) carried to the power of 100. This is equal to a probability of roughly 1 divided by 20 to the hundredth power, which equals a probability of about 1 divided by 10 to the 130th power. You can use a large exponents calculator such as the one at this site to convert numbers stated using one type of exponent to numbers stating using an exponent of 10. For example:
Principle #2: Whenever the number of possible items in each position of a functional sequence is 10 or greater, a small linear increase in the length of the sequence causes a "combinatorial explosion" skyrocketing of the improbability of a random or accidental sequence matching such a sequence (in other words, an exponential or geometric increase in the improbability).
It is because of this principle that computer security experts always advise that you to use a long password such as a 14character password. The math involved goes like this:
Length of alphanumeric password

Number of possible combinations of the letters and numbers

3

46656

6

2.176782336
E+9

9

1.015599566
E+14

12

4.738381338
E+18

14

6.140942214
E+21

The notation in the right column is exponential notation. For example, "E+9" means 10 to the ninth power, and
"E+21" means 10 to the 21st power. You can see from this table why computer security experts advise you to use a 14character password for financial accounts. If someone were to try to break into some account you had that used such a password, it would take roughly 1,000,000,000,000,000,000,000 attempts, far more than any one could ever make even using a large network of computers dedicated to such a task.
It is easy to see a simple relation in the table above: the relation that simple small linear increases in the password length causes a "combinatorial explosion" skyrocketing of the improbability of a random or accidental sequence matching such a sequence (in other words, an exponential or geometric increase in the improbability). The term "combinatorial explosion" refers to a case when the number of possible combinations skyrockets. Here is one famous example of a combinatorial explosion: the number of possible poker hands (consisting of 5 cards out of a deck of 52 cards) is roughly 2,598,960, but the number of possible hands in the game of bridge (in which people are dealt 13 cards) is roughly 635,013,559,600. When making such calculations, you have the additional complication of having to calculate a reduction in the pool of available cards after each time a card is dealt. Thankfully such a complication does not come into play when we are considering the likelihood of a random sequence of some length matching a particular sequence of the same length.
Principle #2 is of very great importance when we have to consider the probability of a random sequence of amino acids matching the sequence in a particular protein. Some of the mathematics involved are below:
Number of amino acids in a molecule

Number of possible combinations of the molecule's amino
acids

5

3.2
E+6

10

1.024
E+13

20

1.048576
E+26

40

1.099511627
E+52

80

1.208925819
E+104

160

1.461501637
E+208

320

2.135987035
E+416

640

4.562440617
E+832

1280

2.081586438
E+1665

2000

1.148130695
E+2602

Human cells are eukaryotic cells. Using the word "residues" to mean amino acids, the paper here says, "The average eukaryotic protein is 450 residues long." The paper here lists a median amino acid length of 419 in human proteins (Table 2), according to one database, and 471 according to another database. The human body has hundreds of different types of proteins with more than 2000 amino acids each, and more than a thousand different types of proteins with more than 1000 amino acids each.
I will now tell you how to get an authoritative answer about how many human protein molecules have more than 2000 wellarranged amino acid parts. Using the UniProt protein database that anyone can use without a login, you go to www.uniprot.org, and type in the following search phrase (or, using less effort, just click on the link below): (length:[2000 TO 50000]) AND (organism_name:"Homo sapiens")
This gives you a results screen like the one below.
You will see more than 1000 rows in the result set. The results will first show the simplest proteins with more than 2000 amino acids. Click on the Length column header, and the results will be sorted like we see above, with the most complex proteins shown first.
There seem to be some duplicates in the results, or cases of proteins that are minor variations of the same protein. But scrolling through the results, you will be able to see two things:
(1) There are at least hundreds of types of proteins in the human body that each have thousands of amino acids.
(2) The most complex proteins in the human body have more than 10,000 wellarranged amino acids. For example, the Titin protein consists of more than 30,000 wellarranged amino acids.
Using a variation of the search string above, you can get an idea of how many types of human protein molecules have more than 1000 amino acids each. For example, suppose you change the www.uniprot.org search string to be the one below (or just click on the link below):
You will get a result set of more than 8000 rows. Allowing for many duplicates, we can assume that human bodies contain more than 1000 types of "highest complexity" protein molecules, where "highest complexity" means having more than 1000 amino acids.
Principle #3: anything requiring a suitable threedimensional arrangement of a certain number of parts is very gigantically more unlikely (exponentially and geometrically more unlikely) to be achieved by accidental construction than something merely requiring a suitable onedimensional arrangement of such parts.
Principle #3 should come as no surprise to someone who read a recent article in the typically materialist Quanta Magazine. The article discusses something much, much simpler than constructing something useful in three dimensions: the mere task of finding what is called a Hamiltonian path through a set of nodes in 3D space, a path that traverses all nodes without touching any of them more than once. The path might need to traverse a set of points (nodes) such as this:
"Suppose you’re given a graph and asked to find something called a Hamiltonian path — a route that passes through every node exactly once. This problem is clearly solvable in principle: There are only a finite number of possible paths, so if all else fails you can just check each one. That’s fine if there are only a few nodes, but for even slightly larger graphs the number of possibilities spirals out of control, quickly rendering this simple algorithm useless.
There are more sophisticated Hamiltonian path algorithms that put up a better fight, but the time that algorithms require to solve the problem invariably grows exponentially with the size of the graph. Graphs don’t have to be very large before even the best algorithm researchers have discovered can’t find a path 'in any reasonable amount of time,' said Russell Impagliazzo, a complexity theorist at the University of California, San Diego. 'And by "reasonable amount of time," I mean "before the universe ends.” ' "
Let's compare two different cases:
Case #1: You must have a linear sequence of 100 tokens (each of which have t possible values) exactly matching a particular sequence.
Case #2: You must have a threedimensional structure in which 100 structural units (each of which can have t possible values) exist arranged in an exact way in a threedimensional space which has a width, length and depth all roughly equal to about 100 times the width, length and depth of each of the structural units.
It should be intuitively obviously that Case #2 requires an arrangement that is vastly harder to achieve by chance that Case #1. Let me give an example to illustrate the point.
Below is the front of a structure that you could build in threedimensional space using number blocks that can have numbers between 0 and 9.
Let's assume that each of the blocks is a cube, even though some of the blocks in the visual above do not look like cubes.
Let us imagine that the total structure consists of four units looking like the one shown above, with the units being arranged so that a bird's eye view shows something looking like a rectangle. So a person looking at the structure from either the front or the back or the left or the right would see something looking exactly like the visual above (except that all of the blocks would be cubes with equal volume). Note that there is a special pattern here, following the rule that whenever a number appears directly below another number, the number below it is always the same number.
Altogether this threedimensional structure requires 100 wellarranged blocks. It consists of four units, each of which contains 25 blocks. Whenever a block appears in the structure of 100 blocks, the block must be in the right position, and it must also be the right number.
We can calculate the probability of such a structure arising from someone taking a bucket of 100 number blocks and pouring them on to the ground. To simplify the calculation, let us assume that the numbers on the blocks exactly match what is needed to make the structure. So the bucket contains this number of number blocks:
4 nines, 8 eights, 12 sevens, 16 sixes, 20 fives, 16 fours, 12 threes, 8 twos and 4 ones. We will assume a threedimensional geometric space consisting that is ten blocks wide and five blocks high. This space has a volume of about 50 times 50 times 50 the volume of each block, resulting in a threedimensional geometric space of 125,000 block volumes (50 times 50 times 50 equals 125,000).
Calculating the overall probability of this structure appearing by a chance pouring of the buckets turns out to be quite a complicated affair. We must individually calculate the chance of each type of number appearing in the right place. Below is such a calculation.
Nine blocks: There are four nine blocks (blocks with the number nine on all of their six faces) . The probability of any one nine block ending up in the right position is roughly 1/ (125,000/4), or 1 in 31,250, since any nine block can end up in any of 4 correct positions. The probability of all four nine blocks ending up in the right position is roughly 1 in 9.536743164 E+17.
Eight blocks: There are eight eight blocks (blocks with the number eight on all of their six faces). The probability of any one eight block ending up in the right position is roughly 1/ (125,000/8), or 1 in 15,625, since any eight block can end up in any of 8 correct positions. The probability of all eight eight blocks ending up in the right position is roughly 1 in 3.552713678 E+33.
Seven blocks: There are twelve seven blocks (blocks with the number seven on all of their six faces). The probability of any one seven block ending up in the right position is roughly 1/ (125,000/12), or 1 in 10,416, since any seven block can end up in any of 12 correct positions. The probability of all twelve seven blocks ending up in the right position is roughly 1 in 1.630841125 E+48.
Six blocks: There are sixteen six blocks (blocks with the number six on all of their six faces). The probability of any one six block ending up in the right position is roughly 1/ (125,000/16), or 1 in 7812, since any six block can end up in any of 16 correct positions. The probability of all sixteen six blocks ending up in the right position is roughly 1 in 1.923958738 E+62.
Five blocks: There are twenty five blocks (blocks with the number five on all of their six faces). The probability of any one five block ending up in the right position is roughly 1/ (125,000/25), or 1 in 5000, since any five block can end up in any of 25 correct positions. The probability of all twentyfive five blocks ending up in the right position is roughly 1 in 2.980232238 E+92.
Four blocks: There are sixteen four blocks (blocks with the number four on all of their six faces). The probability of any one four block ending up in the right position is roughly 1/ (125,000/16), or 1 in 7812, since any four block can end up in any of 16 correct positions. The probability of all sixteen four blocks ending up in the right position is roughly 1 in 1.923958738 E+62.
Three blocks: There are twelve three blocks (blocks with the number three on all of their six faces). The probability of any one three block ending up in the right position is roughly 1/ (125,000/12), or 1 in 10,416, since any three block can end up in any of 12 correct positions. The probability of all twelve three blocks ending up in the right position is roughly 1 in 1.630841125 E+48.
Two blocks: There are eight two blocks ( (blocks with the number two on all of their six faces). The probability of any one two block ending up in the right position is roughly 1/ (125,000/8), or 1 in 15,625, since any two block can end up in any of 8 correct positions. The probability of all eight two blocks ending up in the right position is roughly 1 in 3.552713678 E+33.
One blocks: There are four one blocks (blocks with the number one on all of their six faces). The probability of any particular one block ending up in the right position is roughly 1/ (125,000/4), or 1 in 31,250, since any one block can end up in any of 4 correct positions. The probability of all four one blocks ending up in the right position is roughly 1 in 9.536743164 E+17.
Having made these calculations, we can now roughly calculate the probability of you pouring the bucket of 100 blocks and getting the structure I have imagined consisting of four sides that each look like the visual above. To roughly calculate that we must multiply together all of the probabilities calculated above. This is accordance with the probability calculation principle that the probability of a series of causally independent events all occurring can be calculated by multiplying together the probability of each event occurring.
Multiplying together each of the probabilities above, we get a probability equal to roughly 1 in 9.536743164 E+17 times 1 in 3.552713678 E+33 times 1 in 1.630841125 E+48 times 1 in 1.923958738 E+62 time 1 in 2.980232238 E+92 times 1.923958738 E+62 times 1 in 1.630841125 E+48 times 1 in 3.552713678 E+33 times times 1 in 9.536743164 E+17. Very roughly, this is a probability of about 1 in 10 to the 412th power, equal to about 1 in 10 followed by 411 zeroes.
The purpose of this elaborate exercise has been to show how the improbability of a useful accidental construction skyrockets very dramatically (increasing in an exponential and geometric manner) whenever we have something that has to match a threedimensional structure rather than a mere linear onedimensional structure. Let's compare the two using the example discussed here. The linear number sequence corresponding to the structure described above (having four sides looking like the visual above) is 5654765438765432987654321565476543876543298765432156547654387654329876543215654765438765432987654321. The chance of getting such a sequence from a random 100 digit number is 1 in 10 to the hundredth power. But when we put in the requirements for the three dimensional arrangement of the 100 numbers, matching the specification given above, the probability of getting the sequence skyrockets very dramatically. The improbability isn't just three times greater. It becomes about 10 to the 312th power times greater, which is more than 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 times more unlikely.
The type of considerations I have discussed here have very great relevance to the question of whether it is credible to postulate an unguided origin of species such as our own species. For you to end up with organisms such as human beings, it requires a vast amount of construction of very long purposeful sequences such as the gene sequences in DNA. In each case that the body requires a protein with hundreds of wellarranged amino acids, that requires a very special arrangement of tokens in a gene, an arrangement so special that it must be as wellarranged as the letters in a wellwritten paragraph of about 400 characters. The human body has more than 20,000 such genes, each of which could only be accidentally constructed by luck similar to the luck needed for a typing monkey to produce a wellwritten functional paragraph of abut 400 characters. For you to get such luck it would require a total amount of luck roughly comparable to the luck needed for accidental ink splashes to produce a good, correct, useful 100page instruction manual.
But the odds of purposeful chance construction get almost infinitely worse when you consider the matter of threedimensional arrangement. You don't get protein molecules by just stringing together amino acids into a chain that is like a necklace of beads. Functional protein molecules require wellarranged and very complex threedimensional shapes, which are vastly harder to get. A large fraction of all protein molecules only work when they are part of protein complexes which require a great deal of special threedimensional arrangement to produce. And on and on the special threedimensional arrangement requirements go. Protein complexes must be threedimensionally wellorganized to make organelles, which must be threedimensionally wellorganized in the right way to make cells, which must be threedimensionally wellorganized in the right way to make organs, which must be threedimensionally wellorganized in the right way to make bodies.
The unlikelihood of a successful chance construction of something useful requiring a very complex threedimensional arrangement of parts is gigantically and exponentially greater than the unlikelihood of the successful chance construction of something useful requiring a mere onedimensional arrangement of parts. Realizing such a principle while avoiding stating it, biologists have often tried to evade the consequences of such a principle by trying to make bogus claims that in biology the threedimensional merely requires the onedimensional. Such incorrect claims have taken two forms:
(1) The assertion of something called Anfinsen's Dogma, asserting that the threedimensional structure of a protein molecules is a consequence of the onedimensional structure of the sequence of amino acids in the gene corresponding to that protein. Anfinsen's Dogma never made any sense. Claims that there was experiments providing support for the dogma were never wellfounded, and convincing followup experiments replicating such experiments never occurred. Read my post here for a lengthy discussion of some of the reasons why Anfinsen's Dogma is not a wellestablished scientific result, and why there are very strong reasons for thinking it cannot be true (such as the fact that a large fraction of all protein molecules require the help of other protein molecules  what are called chaperone proteins  to successfully reach their functional threedimensional states).
(2) The assertion of the bogus claim that a DNA molecule is a blueprint or recipe or program for making the body of an organism. DNA is no such thing, and does not specify any of these things: how to make a full body, how to make an organ, how to make a cell, how to make the organelles that are the building blocks of cells, or how to make the protein complexes that are the building blocks of organelles. The diagram below tells us about the hierarchical organization of the human body, and which levels of that organization are not specified by DNA.
The implications of Principle #3 above are enormous: that the odds are utterly prohibitive against any unguided origin of the enormous organization in human bodies, with the odds being prohibitive in not just one way but many different ways.
Principle #4: Any requirement for purposeful mobility means the odds against chance construction undergo an additional skyrocketing, with the odds against chance construction rising exponentially once such a requirement is added.
It is obvious that getting any structure very organized and functional by chance construction becomes exponentially more improbable as soon as you add the requirement that the structure not merely exist, but move around in a purposeful manner. To clarify that, we may consider the case of a bridge. Most bridges across rivers are static. But in a lovely little town to visit, there is an amazing type of bridge: the drawbridge in Mystic, Connecticut. That bridge has a remarkable ability for dynamic movement. At 40 minutes past the hour, the bridge rises up high in the air. This is so that fishing boats can pass through the gap. The video here shows the bridge in action, and discusses the very special and ingenious engineering that was required to make such a bridge:
The chance construction of a bridge such as the drawbridge at Mystic would be exponentially more improbable than the chance construction of a static bridge with no moving parts. What we have in the human body and the bodies of other mammals are not just systems of vast organization, but also systems capable of purposeful mobility. Any system requiring purposeful mobility is exponentially harder to achieve by chance construction than some static system. You might get a rough idea of the improbability of the unguided appearance of a mammal body by comparing it to the improbability of the chance construction of some "roving ability" bridge across a wide river with an astonishing ability to move itself from one spot in the river to some other distant place in the river, wherever it was most needed on a particular day.