Posted on 03/05/2002 9:45:44 PM PST by Southack
This is part two of the famous "Million Monkeys Typing On Keyboards for a Million Years Could Produce The Works of Shakespeare" - Debunked Mathematically.
For the Thread that inadvertently kicked started these mathematical discussions, Click Here
For the Original math thread, Click Here
I'm sure you're right that the lower x is the more expected number of generations will need to pass before the mutation becomes prevalent. With a population around 1000 and rate of 1/1000 I ran it ten times. It took 347, 975, 775, 353, 262, 659, 609, 241, 79, and 204 generations before mutants exceeded 1/2 of the population. With the same population size and a rate of 1/10000 it took 1564, 3551, 4261, 3979, 1047, 7947, 3936, 8336, 747, and 1632 generations - about 10x. With 10000 and 1/10000 it looks like about another 10x. I'm guessing it is linear in both parameters.
I'd be happy to share the program with you, it's really simple.
One more gratuitous ad hominem and I'm going to write you off as too childish to converse with. Grow up.
Let's say your lottery involved picking a number from 1 to 6, in an (ordered) sequence of 10, with replacement. The odds of any single 10-tuplet are about 1 in 60 million. That is calculated as 6 raised to the tenth.
Now tell me something I don't know.
When you say suggest that the odds are certain that someone will win (actually, not quite certain, because there could be no match, and a rollover of the jackpot),
I suggested nothing of the sort. My point had nothing to do with how often someone may or may not win, nor did I make any claim whatsoever about the frequency of winners. Please work on your reading comprehension.
you are implicitly defining the universe of successes as the universe of all 10-tuplets, the probability of which will be 60 million times 1 divided by 60 million, or one.
That's not the "universe of successes", that's the universe of all possible trials.
The probability of any one 10-tuplet is still 1 in 60 million. Let's say you wanted to figure the odds that all the winning numbers are even. That's 1/2 raised to the tenth, or 1 in a thousand in rough numbers. Both computations are figured as P1*P2*P3...PN. You can get from the first computation to the second by figuring the number of 10-tuples in which there are only even numbers (3 to the tenth, or about 59,000, and multiplying by the probability of each 10-tuplet.
Bored now. You don't need to recite basic algebra to me, either.
Thus, it is seen that is useful to separate the question of what the probability is of one 10-tuplet from the question of how many are in the set A for which one is determining P(A).
Yes, which is why I *POINTED THIS OUT TO YOU* earlier. Don't suddenly act like you're trying to teach it to me.
One practical reason why this approach makes sense is the difficulty of determining how to count all functional DNA computations.
"Computations"? Don't you mean configurations?
Because we are talking about bitflip mutations, it seems obvious that PN << 1, and we have assumed some significant minimum size of N,
No, *you* have "assumed some significant minimum size of N". The fact remains that many beneficial mutations are the result of a single nucleotide change.
so in our case P1*P2*P3...PN is pretty small.
Backpedaling so soon? A little while ago it was "next to impossible", now you've toned it down to "pretty small".
As to how dense in our probability space are the functional DNA sequences--by intuition it seems likely to be not dense at all.
"By intuition"? And here you had been trying to give the impression that your conclusion was mathematically sound. Now you admit you're guessing.
In logic, well-formed sentences form a small fraction of all sentences. In all programming languages I have been exposed, the fraction of working programs in the space of all possible binary combinations of a certain set size is also small.
Both true, but neither gives a good guide as to how many DNA changes are possibly useful. Computer programs and sentences in a language are far more rigid in allowable syntax than DNA strands DNA is far more "forgiving" of changes, for a variety of reasons.
Your analysis also overlooks the likelihood that workable DNA sequences are "near" each other in sequence. In other words, while a totally random DNA sequence would likely be useless, and so would any single mutation of it, the same can't be said for mutations of already "working" DNA, where changes would result in only a small adjustment to something that was already functional, in which case the odds are much higher that the results will be functional as well.
Using your own example, while it's true that a random sequence of words is unlikely to form a valid sentence, taking an already valid sentence and randomly substituting a new word for one of the existing words is much more likely to produce something that's readable. The quick brown fox jumped over the lazy accountant.
I would think that if the density of successful DNA combos in DNA space were high, we would irradiating our sperm and ova rather than avoiding high radiation--or at least doing that to our livestock.
Nonsense. Natural mutations occur at close to the "optimum" rate for the introduction of new features in a species without producing so many mutations that each generation consists mostly of mutants. One mutation here or there is grist for natural selection -- thousands (the result of irradiation), on the other hand, is a recipe for disaster.
I think the onus is on the evolutionary camp to show that things are different when it comes to DNA.
They already have, try reading some of the literature.
But you yourself made a claim -- that the chances of several neutral mutations combining to perform a useful function was "next to impossible". Therefore, the onus is on *you* to support such a claim. Despite your recitations of basic math, you have yet to do so.
I wouldn't argue with the mathematical results you propose--in a mathematical universe. A mathematical universe has no characteristics within it that were not put into it. For instance, a mathematical keyboard would never have to face the misfortune of keys being jammed after being bombarded for a very, very long time by little pebbles.
In a real universe the keyboard would have to deal with the forces of nature: gravity, the strong force, the weak force and electromagnetism. In a pristine mathematical universe hydrogen could bounce around forever and remain nothing but hydrogen. But in a universe with the natural cosmic forces at play, you start with hydrogen and get--at minimum--the known periodic table...and that's before even thinking about the unknown properties of the universe, like dark matter.
"No, post #557 in a nutshell was that you were trying to change her suggestion (huge number of serial bitwise changes) into something else entirely (a single gigantic cryptographic transformation that makes the transition in one single step). It's pure apples and oranges. The fact that you think your post is relevant in any way to hers shows that you either fail to understand her example, or that you don't understand your own (i.e., the nature of cryptographic transformations)."Don't be simpleminded.
I wouldn't know how.
A cryptographic algorithm does not have to be all one step. One can encrypt an already encrypted file, that was itself already encrypted. Likewise, one can decrypt a file that has already been decrypted. Triple DES does this everyday, in fact.
...and triple DES can be considered a single transformation. f(f(f(x))) is no different from its composite single function, g(x).
The fact remains that a cryptographic transformation with a SINGLE key (you suggested 96 bits as a maximum), no matter how complicated and multi-stepped, is still a transformation that takes the input data and produces the output data in a single leap, which is drastically different from what the person to whom you were responding had suggested as a possible mechanism.
Nice try, but do not pass Go, do not collect $200. You either grossly misunderstood her suggestion, or you were unable to understand your own counterexample well enough to realize why you were arguing oranges to her apples.
You can't salvage it by waving your hands like this.
You erred.
*snort* I'm not the one making ludicrous comparisons that aren't even in the same ballpark, son.
You assumed that extracting the Windows XP message from DOS had to be performed in a single step. That sort of oversimplification of cryptography either means that you don't really know much about the field, or that you are deliberately setting up a strawman to knock down later, perhaps by claiming that since what I asked "had" to be performed in a single step (when it doesn't), that I was changing the game (when I demonstrably was not).
Do your hands get chafed when you wave them that much?
I know cryptography well, don't try to teach your grandma to suck eggs. I know it better than you do, since unlike you I'm not foolish enough to try to split hairs about the irrelevant difference between an iterative algorithm and single-pass one. The fact remains, however, that by requiring a single 96-bit key, as you did, any encryp0tion algorithm whatsoever, no matter how complex, which requires that key for operation is a single-step encryption algorithm -- it takes an input, the computer cranks for a while, and then the output emerges.
Deal with it -- your silly encryption challenge was in no way homologous to her point. Give it up. When you find yourself in a hole, stop digging.
You do get credit for having a bad attitude, but not for being clever.
Yawn. Come back when you have an argument. Being cutesy and insulting just doesn't cut it, and it only makes it painfully clear just how little defense you actually have for your post. I caught you at it, and no amount of dancing around and being snide is going to deflect attention from that. In fact, it only makes it worse.
"It was *your* cited example, don't try to foist it off on me. Is it *really* your contention that because harmful mutations get weeded out, that it is *therefore* not possible for neutral mutations to propagate?" - Dan DayFirst of all, you cited my example in your response, so don't try to distance yourself from it now.
Stop playing word games, you're not very good at it. You called it "my" example -- it wasn't, it was yours. The fact that I examined it and showed how many holes it had in it doesn't make it mine. Nor have I ever tried to "distance" myself from it, other than to point out how flawed it was. So stop trying to be cute.
Second, by my recall this is the first post that you've made to me wherein you add the new qualifier of "neutral" to your original erroneous claim that "mutations have a 100% expected chance" of being passed on to their offspring.
Then you need to work on your reading comprehension *and* your memory. From post #622, written to you, and responded to by you (and thus, one would presume, read by you):
Try to keep up with the discussion, Maro was specifically trying to calculate the odds for a combination of mutations which, and I quote, "make no functional difference".This appears in the VERY SAME post in which I made the 100% expected value claim, and the fact that that claim was made in the context of the neutral mutations being discussed should have been clear to anyone.
[snip]
Again, the Maro discussion concerned "neutral" mutations which did not express as anything.
[snip]
It would help if you checked the context of a discussion before you decided to nitpick it, most of your points are moot.
And if *that* wasn't enough, I mentioned this to you *again* in post #718:
Because detrimental mutations like having two heads strongly tend to be weeded out (by interfering with propagation), whereas neutral mutations (the sort being discussed) don't.I again repeat my admonition -- try *reading* something before you reply to it.
[snip]
I even pointed that out in the part of my post which you quoted in your post, try actually reading it next time.
And even when one considers "neutral" mutations (an oxymoron, perhaps, since by definition a "neutral" mutation makes no change in a beast),
If you consider that an oxymoron, it only shows that you have no real idea what a mutation is.
even "neutral" mutations do not have a 100% expected chance of being passed on to their offspring.
I didn't say "expected chance". I said "expected value". I even enclosed it in quotes when I used it to signal that it was a special term and not a loose layman's use of language. "Expected value" has a particular mathematical meaning, of which you appear to be unaware.
Broadly speaking, not every life form lives to successfully propagate itself. Even if a mutation manifested itself in a newborn, there is not a 100% chance that the newborn will live to breed, that the breeding will be successful, and that there will even be a birth, much less that said mutation would actually survive the whole process internally to the genes of said offspring.
No s**t, Sherlock. But then, I already explicitly acknowledged this in the very same paragraph as the "100% expected value" claim -- try reading for content next time:
The odds of the mutation appearing in subsequent offspring? The "expected value" is 100%, in a species that is neither growing significantly in population nor declining, although the actual results can vary from zero offspring carrying the gene, to a large number depending on the fecundity of the species. On average, though, the gene will be passed on to one offspring by each parent that carries it.Yet you pretend in post after post that first mutations, and then later "neutral" mutations all have a 100% expected probability of being passed on to offspring.
I "pretend" nothing. I make supportable claims. The fact that you repeatedly misunderstand them despite having your misconceptions corrected over and over again doesn't count as adequate rebuttal.
In short, you are uneducated, wrong, and busted.
I see. Look, *I'm* not the one who can fail to read, or remember, what has been said to him two posts ago in an exchange.
Your claim (or claims, plural, if one considers your new use of the "neutral" qualifier) can not stand even cursory, much less rigorous scrutiny.
Yawn. Try again when you have a case -- and actually have a clue what I've actually written.
Even Nebullis would agree that you are wrong on this claim, and she's on your side...
Funny, in post #753 she replies:
I don't have a problem with Dan Day's statement. He explicitly states "on average". There's a variation around that mean, but random genetic drift follows simple allele sorting.So it appears that you're wrong about *THAT*, too. That seems to be a habit of yours.
Furthermore, it's clear that she actually understands what I write, which is refreshing.
"Funny, in post #753 she replies:
I don't have a problem with Dan Day's statement. He explicitly states "on average". There's a variation around that mean, but random genetic drift follows simple allele sorting.
So it appears that you're wrong about *THAT*, too. That seems to be a habit of yours. Furthermore, it's clear that she actually understands what I write, which is refreshing." - Dan Day
Humorously, Nebullis changed her mind several posts later. You simply didn't read far enough down in the thread to see that.
No, I did not change my mind. You need to look closely at the terms used.
Grin. That silly analogy aside, mutations aren't passed on at a 100% rate for a variety of reasons even in an ideal environment. First and foremost, most observed mutations are due to changes in context, not changes in the DNA coding sequences. Add the right chemicals to frogs and you get frogs with both male and female organs, but when they breed without the presence of said chemical, you do not see 100% of their offspring with the reproductive organs of both genders, for instance. Breed a two-headed snake and you once again do not see a 100% success rate in that mutation being passed to its offspring.
Until the DNA itself is changed and copied, you don't even approach a 100% expected value for mutations to propagate.
Even when DNA is changed, there is no guarantee that the copying process will always (i.e., 100%) pass on the change. The copying process isn't that efficient, moreover, the mutation could very well harm the copying process itself presuming that we are speaking more broadly than a "neutral" mutation (an oxymoron - a more accurate term would be a non-sequencing change, such as when no-op instructions are "mutated" rather than when active code is changed).
Where did that come from?
Are you even aware that you are mixing a number of concepts in your posts? You've jumbled mutation rates, epigenetic mutations, teratogenic mutations, and transmission fidelity (which is what I was speaking to) together as if they all have the same probabilities.
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