Recently, a reader who has little familiarity with mathematics and probabilities asked for some clarification of how the maths worked in Lightform Evolution. Here follows a more detailed breakdown of some of the figures discussed in the section on "Alexandra's ammonia lungs."
Lung Development Word Problem, Introduction
Alexandra (you'll remember from the linked post) is a hypothetical species which goes from being an ocean-dwelling predecessor to a land-dwelling creature of the present day. Although the hypothetical species "Alexandra" is an entire species, we'll refer to her using a singular pronoun for purposes of simplicity.
Alexandra is composed of five trillion cells, or 5,000,000,000,000 cells. Somewhere in-between her years as an ocean-dwelling creature and a land-dwelling creature, she developed lungs that were able to respirate air instead of ocean water. In order for that to happen, she had to develop exclusively air-breathing lungs.
The fossil record tells us that life on Earth has existed for about 3.8 billion years, but we'll be generous, and allow for 4 billion years of time for Alexandria's evolution to take place. That's 4,000,000,000 years. (The other variable involved, you'll remember, is 5,000,000,000,000 cells. Four followed by nine zeroes for years; five followed by twelve zeroes for total number of cells.)
In accordance with the principles of the randomized mercantilist take on evolution that is popular in Earth 2014, it's not the cells themselves that evolve, but rather, the DNA which guides the development of those cells. Nonetheless, that is five trillion pieces of genetic code that need to be affected, in order to transform those lungs, over four billion years, from ocean-respirating lungs to air-respirating lungs.
Lung Development Word Problem, Aside on Complexity
Note how incredibly generous we're being, here, to pop-evolution. The fossil record shows about 3.8 billion years of time from the earliest microorganisms to the present day. We're not only giving Alexandra 4 billion years to evolve only a single internal organ (which would also have to happen in conjunction with the replacement of flippers by arms, fins by feet, etc.) we're also letting her start off, from the beginning, as a highly complex 5-trillion-celled organism, rather than starting as a single-celled organism, which is actually what the fossil record shows at 3.8B years ago. So when you see how unlikely her chances are at evolving just one organ in the allotted 4 billion years, you can magnify that unlikelihood more than a millionfold to see how unlikely it is that she could go from single-celled to air-breathing in that same span of time!
Lung Development Word Problem, Breakdown of Variables
Alexandra's ocean-breathing state: 5,000,000,000,000 cells.
Time allotted for the development of air-breathing lungs: 4,000,000,000 years.
Those variables aren't enough to help us figure out the likelihood of Alexandra's evolution. We'd need to know a few other things. Firstly, we'd need to know the rate at which randomized mutations to Alexandra's DNA occur, and we'd also need to know the level of chemical complexity required to fine-tune those lungs so that they can breathe Earth's atmosphere (as opposed to the Martian atmosphere, the Jovian atmosphere, etc.).
Given the complexity of chemicals available on Earth, and the possible combinations thereto (which are discussed in the full post in more detail, if you're able to go back and give another try at reading it), the chemical complexity spectrum that would be tested by natural selection is 98!. This is a factorial; if you don't remember how those work, go here: Introduction to Factorials in Mathematics.
The factorial 98! is 9.426890448883242e+153. If you're not familiar with that kind of mathematical notation, that number is larger than a googol and a half. If you don't remember how large a googol is, here's a reminder: One googol.
Spelled out all the way, the chemical complexity spectrum to which Alexandra would be subjected to by random evolution is the following number:
The last variable we'd need to know for this probability would be the rate at which cells mutate. Out of the DNA designs for the approximately five trillion cells in Alexandra's adult body, how many of them will "randomly mutate" during each reproduction, such that we might eventually evolve Alexandra from ocean-breathing to air-breathing?
This variable is called the mutation rate, or the rate at which observable changes occur in a DNA sequence during reproduction. Pop-biologists have conceded that the mutation rate is somewhere around 0.008% (8 thousandths of one percent); if you'd like to do more research on the subject, a good introduction would be any edition of Molecular Biology of the Cell. Here's a link to the free summary, which discusses, briefly, e. coli and human reproductive rates: Molecular Biology of the Cell.
Summing Up the Variables
Let's recap. The variables we have available are:
Alexandra's ocean-breathing state: 5,000,000,000,000 cells subject to potential change.
Time allotted for the development of air-breathing lungs: 4,000,000,000 years during which the potential change can occur.
Chemical complexity spectrum to which Alexandra's evolution will be subjected: 9,426,890,448,883,242,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000, 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-large spread.
Rate at which Alexandra's cells mutate: 0.008%.
Lastly, we need to know how often Alexandra reproduces each year, in order to know how many times she can reproduce during the 4 billion years available to her. Let's be incredibly, incredibly generous, and say that she reproduces one hundred times a year. For a bacterial organism, or a fruit fly, that isn't generous, but for a complex, five-trillion-celled creature, that is quite generous. One hundred times a year--one hundred potential chances during each year for her to evolve from ocean-breathing lungs to air-breathing lungs.
Doing the Math
Now that we've set up our variables, we can do the problem. Here's how it looks in word form:
[Alexandra's total number of cells, or five trillion] multiplied by [the average rate at which her cells mutate during reproduction, or 0.008%] multiplied by [the average number of times she successfully reproduces, in a year, organisms which are healthy enough to go on to reproduce themselves, or 100], multiplied by [the total time she has available to evolve just one organ to go from ocean-breathing to air-breathing].
In short form:
Five trillion times point zero zero eight percent times 100 times four billion.
Here are just the numbers:
5,000,000,000,000 x 0.008% x 100 x 4,000,000,000 = ______ ?
And the answer is, if you'll get out your calculator to check my work:
Again, if you don't use that kind of mathematical notation regularly, that expression means "1.6 followed by 22 zeroes," or 160,000,000,000,000,000,000,000, which we can represent verbally as "160 septillion."
What that result means is that, by the end of a 4 billion year study period, in which nothing interrupts Alexandra's reproductive process, and she continues to successfully reproduce reproductively-successful organisms at a rate of one hundred per year, she'll have 160 septillion chances to mutate cellular provisions in her DNA in such a way as to (potentially) develop air-breathing lungs.
The Likelihood of Alexandra's Success
160 septillion seems like a big number, doesn't it? But remember--the chemical complexity spectrum she faces in developing lungs attuned to respirate Earth-air, rather than Saturnian air, Andromedan air, Piscian air, terrestrial oceans, hypothetical Martian oceans, etc. That's a much bigger number. 160 septillion may be 1.6e+22, but the chemical complexity spectrum is 9.426890448883242e+153. That is a much bigger number.
To obtain the likelihood of Alexandra's success during the 4 billion year time period, we take her chances--160 septillion--and divide them by the denominator of 9,426,890,448,883,242,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000, 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.
That's a mouthful, which is why mathematical notation is so helpful. The problem is really just a fraction, which looks like so:
(1.6e+22) / (9.426890448883242e+153)
To complete the division problem, we can easily eliminate the "22" from both the numerator and the denominator, leaving behind:
(1.6) / (9.426890448883242e+131)
Can you see Alexandra's problems now? 1.6 divided by over a googol. The percentage chance of an organism developing, through random mutation and natural selection, even a single internal organ during four billion years is incredibly tiny. Here's her chance of success; the chance of even one organism managing to evolve an atmosphere-respirating lung from an ocean-respirating lung:
If the mathematical notation is unclear, here's what it looks like as a standard percentage:
That's it. That's the percentage chance of a single organism developing an air-respirating lung. (And in order for that organism to then reproduce the lung and create a new species, the lung would have to be a dominant genetic trait--so dominant that it persisted through lots of reproductions with non-air-breathing types, until speciation occurred. Which further lowers Alexandra's chances.)
That is why "random" evolution is so ridiculous. If life had not four billion, but four hundred billion years to evolve, the percentage chance would be 1.697272296390741e-129, or:
If Alexandra had four hundred billion years to evolve, and she successfully reproduced one million times a year, the chance would be 1.697272296390741e-122, or:
How many more ridiculous fantasies should we make up in an attempt to save the idiotic faith of capitalist pop-evolution? We've already eliminated from consideration Alexandra's trachea; the design of her red blood cells; her scales, her spine, her digestive and rectal functions, and many, many thousands of other changes that would have to happen, in coordination with her lung changes, to make the lung evolution successful rather than fatal--including the neural pathways and instinctual responses that would cause her to willingly attempt to breathe air, rather than water, in the first place. What else can we do to make that impossibly tiny probability above look better?
Okay, so Alexandra reproduces a million times a year, and we've multiplied by a factor of one hundred the amount of time she had to evolve even one organ. What else can we do? Let's be even more stupidly generous. Let's say that Alexandra successfully reproduces one billion times per year. No, ten billion. No, a hundred billion! We'll say that Alexandra grows at such an astounding rate that, each year, there are a hundred billion new Alexandras, zero of which die off, and for every year thereafter, there are another hundred billion.
We'll also assume, as we have been throughout, that not a single Alexandra regressively mutates, i.e., that none of the evolutionary mutations regress Alexandra back toward the ocean-breathing stage, and that once a positive change has been made, it stays, unaltered, forever. (Otherwise, we'd have to reduce Alexandra's chances even further, assuming that some of those hundred billion offspring each year might begin developing "backward," thereby ruining their chances at contributing to future air-breathing success.)
Okay, so a hundred billion successful reproductions a year. Our number is still incredibly small. Alexandra's chances are now 1.697272296390741e-117, or:
Even though 9.426890448883242e+153 is a relatively large number in Earthly biochemical terms, it's actually far too generous to pop-evolution to use such a small number of potential chemically-coordinated mutations. For purposes of simplicity, I've been unduly generous to Anglo-American capitalist evolutionary theory, because I've only considered the "lung" organ itself, rather than all the associated changes to air-breathing apparatus--mouth, esophagus, etc.--that Alexandra would also have to simultaneously develop in order to make the lungs worth anything. Realistically, Alexandra's chances at randomly, naturally selecting air-respirating lungs are much, much lower than we've expressed here.
Even so, the failure of pop-evolution to satisfy such a generous requirement is not "meaningless." Rather, it shows us just how wildly improbable and truly insane such a theory would be, to claim that such sophistication could have developed randomly in only 4 billion years.