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Author
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Topic: Universal probability bound?
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RBH
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Member # 380
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posted 15. November 2005 23:47
Unless one is willing to merely pull a number out of one's left ear and call it a "probability", one must have some notion of the shape of the PDF and some notion of whether the probability estimation calculation's assumptions are met by the phenomena being modeled. Absent those, it truly is no more than pulling numbers out of thin air.
RBH
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Melvin H. Fox
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posted 16. November 2005 07:36
Chris,
Suppose we have some rare event A with small probability p and we allow an unlimited series of Bernoulli trials. Let M be the number of failures before the first success [occurrence of event A]. Then we have the following probability function:
P[M = k] = p[1 - p]^k , k = 1, 2, ...
Further, we would expect that the number of failures before the first success to be:
E[M] = [1 - p]/p
This implies that in the average unlimited series of Bernoulli trials the first occurrence of event A would be on trial M* = 1/p. This is consistent with your cumulative geometric distribution? As it turns out, if we allow p --> 0, we force M* --> infinity then the subsequent limit for the probability that event A will occur before the M* trial is given as:
P[M < M*] = 1 - 1/e
I am curious as to any possible implications due to this result?
I am a simple math teacher and appreciate the patience of all as I catch up in this discussion. Would it be correct to say that at present the UPB is merely an indicator when it is the case that there is little chance; chance had anything to do with the occurrence of a given event? Also, given the above limit, why is it surprising that the rare event A would occur before M* trials? Should we not set a neighborhood [+/- n] about M* akin to the standard deviation?
Surprised if: M < M* - n
Or has this neighborhood already been set by:
ln[.5]/ln[1 - p]
As far as the i.i.d. assumption and pulling numbers out of hats, perhaps once the events covered by the central limit theorem are understood with regard to a UPB, then we could turn our attention to those that are not.
[ 16. November 2005, 08:02: Message edited by: Melvin H. Fox ]
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John A. Davison
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posted 17. November 2005 14:45
There are two great biological mysteries, ontogeny and phylogeny. In my opinion only ontogeny still remains, phylogeny having ceased long ago. Both have proceeded on the basis of stored, front-loaded information the source for which remains clouded in mystery and probably will for some time. I agree entirely with Leo Berg when, commmenting on these two great mysteries, he wrote:
"Neither in the one nor in the other is there room for chance."
Nomogenesis or Evolution Determined by Law, page 134
As for the role of selection:
"The struggle for existence and natural selection are not progressive agencies, but being, on the contrary, conservative, maintain the standard."
ibid, page 406
As for the source of the information:
"Evolution is in a great measure an unfolding of pre-existing rudiments."
ibid, page 406
Pierre Grasse, apparently independently, reached similar conclusions:
"A cluster of facts makes it very plain that Mendelian, allelomorphic mutation plays no part in creative evolution. It is, as it were, a more or less pathological fluctuation in the genetic code. It is an accident on the 'magnetic tape' on which the primary information for the species is recorded."
Evolution of Living Organisms, page 243
and
"However that may be, the existence of internal factors affecting evolution has to be accepted by any objective mind..."
ibid, page 210
and
"But according to Darwinian doctrine and Crick's central dogma, DNA is not only the depository and distributor of the information but its SOLE CREATOR. I do not believe this to be true."
ibid, page 223, (his emphasis)
These along with many other considerations are what led to my publication of "A Prescribed Evolutionary Hypothesis."
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kyle7
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posted 18. November 2005 03:50
RBH,
Now suppose in my last post it was observed that the first letter of each sentence formed some intelligent words. How would we know if this was planned or whether it was just due to chance? How would we calculate the probability. The simplist way (if we had 12 first letters) would be to take 1/26^12 which is approximately 1.0479E-17. Now the ID purists may say that this probability is not small enough to be conclusive. It is much larger than Dembski's 1E-150. But I would argue that this small probability would suggest it was designed, though our level of confidence would not be as great as Dembski's.
Now you (RBH) may try to argue that I did not take the probability distribution related to each letter. If we wanted to be more stringent in our analysis we could include this probability distribution based on the writing on the ISCID Forum website. We could take all the posts and develop a distribution for each first letter of a sentence and use these probabilities in our calculation -- but we know that this effort would not produce a significant change in the results. In fact it may make the probability even less, depending on the letters used to make the words. Our results may be something like this: Pr = (1/15)*(1/36)*(1/25)*(1/14)*(1/20)*(1/26)*(1/40)*(1/25)*(1/24)*(1/23)*(1/26)*(1/28). So, the increased analysis may change the probability but it would not be significant to rule out the first analysis. I would write my paper stating why I thought the writing was designed and you could then write your paper to challenge my work.
The same argument can be used in regards to the DNA in life. Given the vast number of subcomponents of the DNA, this argument is even stronger. I would say that this argument -- the need for some complex distribution function in order to estimate probabilites -- is disingenuous at heart.
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Christopher D. Beling
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posted 18. November 2005 11:31
Mel,
You pose some interesting questions. I agree with your maths although it took me some time to see the expectation value of M (the number of failures prior to success). One writes:
E[M]=SUM(M,0-inf): Mp(1-p)^M
=SUM(x,1-inf): (x-1)p(1-p)^(x-1)
=SUM(x,1-inf): xp(1-p)^(x-1) - SUM(x,1-inf): p(1-p)^(x-1)
=SUM(x,1-inf):xP(x) - SUM(x,1-inf): P(x)
=1/p - 1 = [1-p]/p
P(x) being the probability of the rare event A occuring on the xth=(M+1)th trial. [Perhaps you have an easier demonstration?]. From this analysis one sees clearly your result that:
M*=E[x]=SUM(x,1-inf):xP(x)=1/p
Moreover as you say:
P[M < M*] = 1 - 1/e
HOWEVER, although this is correct it is not the critical probability chosen by Dr. Dembski in the design inference. Instead we find:
P[M < M'] = 0.5
Of course this is a slightly more stringent requirement - although one might well say at this level of small probability and with computational errors of uncertainty added in this small difference is hardly going to be significant. The rationale behind the 0.5 though is interesting. It is a kind of "more likely than not" sort of argument see sec6.3 page 190 of DI "The Magic Number 1/2". i.e. one is saying that if M < M' then it is more likely than not that the event did not occur by chance .
I think this is basically right. In your approach the M* is an abstract quantity that can never be truly realized - because it only takes on meaning with reference to an average of M over many hypothetical sets of trials. In reality one only has a single trial in which to attain A.
To be honest although I see that Dr. Demsbski is rigorously correct in setting the critical probability as 0.5, I find from a purely practical and reasonable judgement perspective that this is too high. "Alpha" values of 0.05 or 0.01 (values normal in statistical inference - see DI page 201) would be better suited - in which case
P[M < M"] = alpha ~0.01
in which case one could say that if M < M" then it is CONSIDERABLY more likely than not that the event did not occur by chance
quote:
Surprised if: M < M* - n
Or has this neighborhood already been set by:
ln[.5]/ln[1 - p]
This turns out to give quite an amusing result if one plugs in the value of the variance "n" of the geometric distribution P(x):
n=VAR(M)^(1/2)=[1-p]^(1/2)/p
so that
M*-n=(1-[1-p]^(1/2))/p =~ 1/2
and one is left with M<1/2 , i.e. zero! Indeed in taking a single "sigma" one notes that M is reasonably bound between zero and 2/p. For example this makes sense when trying to throw a six on a dice. One would not be surprised if after 6 throws one had not achieved a 6, but one would certainly be surprised, and annoyed?, if one had not achieved success after 12 throws.
quote:
Would it be correct to say that at present the UPB is merely an indicator when it is the case that there is little chance that chance had anything to do with the occurrence of a given event?
Well yes. A very small chance is a very small chance and it could conceivably have happened - the point is, however, that if after factoring in all the probability resources available the probability of event A is still too small (as defined above) then chance is not the most likely inference. Quoting Dr. Dembski:
quote:
the probabilistic apparatus . . . provides just such a systematic and rational basis for why we can "feel practically justified" eliminating chance in a test of statistical significance. As Howson and Urbach rightly note, there is never a logical contradiction in refusing to eliminate chance in a test of statistical significance - save in the case where the rejection region R has probability zero with respect to hypothesis H and the event E happens to fall within R. Short of this, it is always a logical possibility that E is a chance outcome due to the chance hypothesis H, regardless whether E falls within or outside the rejection region R. But precisely because this is a strictly logical point, one does not have to be a statistician to appreciate it. Indeed, it is precisely the statistician's task to tell us how to explain E when the rejection region does not have zero probablity.
DI page 200
-Chris
[ 18. November 2005, 13:37: Message edited by: Christopher D. Beling ]
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RBH
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posted 18. November 2005 13:57
Kyle7 wrote quote:
Now you (RBH) may try to argue that I did not take the probability distribution related to each letter. If we wanted to be more stringent in our analysis we could include this probability distribution based on the writing on the ISCID Forum website. We could take all the posts and develop a distribution for each first letter of a sentence and use these probabilities in our calculation -- but we know that this effort would not produce a significant change in the results. In fact it may make the probability even less, depending on the letters used to make the words. Our results may be something like this: Pr = (1/15)*(1/36)*(1/25)*(1/14)*(1/20)*(1/26)*(1/40)*(1/25)*(1/24)*(1/23)*(1/26)*(1/28). So, the increased analysis may change the probability but it would not be significant to rule out the first analysis. I would write my paper stating why I thought the writing was designed and you could then write your paper to challenge my work. (Bolding added)
But we don't know that taking into account the actual distributions wouldn't change our results significantly. Once again, in order to use a math model (in this case, the probability calculus) to represent a system, one has to establish (not merely assert) that the assumptions, terms, and operators of the math model veridically map the objects and processes of the system being represented by the model. Otherwise, manipulations of the model (e.g., calculations) produce uninterpretable gibberish.
Even in the case of letters in words we know that the probability of ocurrence of a given letter, say "u", is heavily dependent on context -- if the preceding letter is "q" the probability of occurrence of "u" is much higher than if the preceding letter is "x". Similar sorts of dependencies occur in chemistry, so to calculate the probability of a chemical event without taking into account context produces nonsense. In other language, treating molecules (or words) as discrete combinatorial objects under the i.i.d. assumption and under the assumption of a uniform PDF is an invalid procedure in calculating the probability of occurrence of aggregations of constituents. It produces uninterpretable gibberish.
RBH
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John A. Davison
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posted 18. November 2005 16:05
I give up. I denounce as have all my sources every aspect of what is being discussed here and, as in the past, none of us are allowed to exist. We must not exist for if we did it would become indelibly clear that chance never had anything whatsoever to do either with ontogeny or phyogeny. In the meantime, things are going well at my blog:
prescribedevolution.blogspot.com/
At least there I am able to elicit the attention of the participants. Feel free to join in. It is sad that "brainstorms" has become a pale shadow of its former self. I sincerely hope it can be restored.
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Melvin H. Fox
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posted 21. November 2005 11:10
Hi Chris,
I believe that I am starting to get the picture. As far as my demonstration of the expectation of M, no it is not any cleaner than yours. I usually take the long way around to the truth. What’s more transcribing the notation to html is foreign to me [new to this game] but I am seeking assistance in this regard from one of my students. Obviously, I must read Design Inference or No Free Lunch.
My interest is in a more powerful probability bound. For example, with due respect to Kyle, if we encounter a pyramid with specifications similar to the great pyramid on some otherwise baron and desolate distant planet, we can say with surety that this structure of cut stone piled with precision including tunnels, traps, and inner caverns did not happen by chance. The probability is not very small; it is zero. For the sake of science, we must do better than "just so" however.
Recent developments in physics [superstrings] and challenges to the real number system [R*] together point to a discrete universe and a smallest possible positive real number [d*]. Then d* would pose a more formidable probability bound than UPB. Any event A with probability p calculated in R such that p < d* could not occur by chance in R*. If R* proves to be a more accurate model for analysis in the physical universe than does R, we would be forced to conclude event A was designed. This is vague. Shortly I will post a better construction of this proposition.
By the way, I hope you don’t mind that I plan to use your diagram from the thread [2nd and 4th laws of thermodynamics 9 Sep. 2005] in my classes to explain how the maths are used to model the physical universe.
-Mel
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Christopher D. Beling
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posted 21. November 2005 21:52
Hi Mel, John, RBH and others,
I want to give a rather definitive example of biological transition where the UPB and LPB (local probability bound) can effectively rule out the chance hypothesis. This is the Histone 4 problem - which is closely related to what may be termed the Prokaryote to Eukaryote transition problem. I was made aware of this simply enormous problem in biology a few years ago when reading Sir Fred Hoyle's book Mathematics of Evolution
(pages 102,103 and 127). I have not seen this problem discussed elsewhere - presumably because it is too great an embarrassment, but it surely needs to be aired.
Histone 4 (H4) is a highly conserved protein of the histone family of proteins that are collectively involved in the tight packing of DNA into nucleosomes during the process of mitosis (cellular division). Histone 4 (and also Histone 3) have to be very accurately shaped to perform their very complex function of providing structures that permit DNA wrapping (and release after cross-over and replication). Let me quote from Hoyle to give a more exact picture:
quote:
The histones are a small class of protein which play a critical role in the process of cell division. Except at times of cell division the chromosomes exist freely and separately in the cytoplasm of the cell. With the approach of cell division, the chromosomes are first duplicated and then condensed into a compact, much more visible structure known as chromatin . . The histones appear to provide physical support for the chromosomes in this process of condensation and in the complex maneuvers, which then lead to crossover and cell division. A form of histone-4 with rogue properties that led to wrong crossover or to chromosomes being torn during cell division would clearly be lethal . . Without histone-4 being exactly right, cells could not divide properly and nothing in the whole biological system would work correctly.
A picture of this amazing DNA wrapping protocol in which H4 participates may be found in this article DNA wrapping that deals with another highly conserved protein Asf1 involved in nucleosome construction.
Well lets get to the point: H4 comprises a chain of 102 amino acids that is very tightly conserved in its sequencing. Quoting from Lehninger "Principles of biochemistry" (4th Ed, p 939)
quote:
The H3 histones are nearly identical in amino acid sequence in all eukaryotes, as are the H4 histones, suggesting strict conservation of their functions. For example, only 2 of 102 amino acids residues differ between the H4 histone molecules of peas and cows, and only 8 differ between the H4 histones of humans and yeast
Thus the problem for the neo-Darwinian hypothesis is the first discovery of H4 (without which eukaryotic reproduction could not have been discovered). Quoting from Hoyle again on how this could have happened.
quote:
Evidently not by random process, for with a chance 1/4 of choosing each of the correct base pairs at random, the probability of discovering a segment of 200 specific base pairs is 4^-200, which is equivalent to 10^-120. Even if one were given a random choice for every atom in every galaxy in the whole visible universe, the probability of discovering H4 would still only be a miniscule 10^-40
Note here that Hoyle is referring to a conserved nucleotide sequence of 200 base pairs. I'm not quite sure where he gets this number but I have checked that the probability he gets agrees well with that obtained for correctly sequencing 102 amino acids as predicted by Yockey's information theory - from which the probability is:
P=2^-NH
where N = 102, and H is the Shannon entropy per amino acid residue (i.e. -log2(1/20)=4.32bits). i.e.
P[H4]=2^(102*4.32)=2^-440=4^-220=10^-120
Now we see that Hoyle is setting the UPB effectively at 10^-80 (based on the number of fundamental particles in the universe~ 10^80) and in so doing he reasons that H4 did not occur by "random process".
If we instead take the UPB to be 10^-150 then Hoyle's conclusion would no longer be justified - but how relevant is the UPB here? After all the speculation is that H4 was discovered in procreating prokaryotes in the earth's oceans - although astrobiologists may disagree. I suggest that we have to look at (i) the probabilistic resources that are available for forming H4 based upon neo-Darwinian theory and (ii) the LPB i.e. the probability bound based on local (not universal) physical resources.
(i) Probabilistic resources
Lets start by assuming a colony of water born prokaryotes to have been the source of the first H4. With a mutation rate of 0.3 per chromosome/generation the rate of throwing new DNA configurations that get fixed in the population would be (according to Kimura's neutral theory) 0.3 per generation (i.e. independent of population size). If we take the fastest rate for bacterial reproduction to be 1/10min then a simple calculation shows that after 10^9 years one would expect to get only 1.6x10^13 different DNA configurations. Factorizing this probability resource in with P[H4] gives us only P'[H4]=1.6x10^-107. This resource saturated probability, however is still too large as we have assumed all the genome configuration changes are focused on a small 300bp section of DNA which they are clearly not in the above calculation.
(ii) Local Probability Bound
Admittedly the LPB is a little subjective, but lets take an extreme of all the water molecules (or atoms - the difference is insubstantial) in the earth's ocean vibrating "throwing" new configurations at thermal frequency (10^13Hz) for a total of 10^9 years. The volume of the oceans is 1.37x10^24 (cm3)volume of oceans giving a total of 4.6x10^46 water molecules and so we obtain a total local number of possible specified events of 10^76 giving us a LPB of 10^-76
In conclusion therefore it is clear that although the saturated probability for H4 formation (i.e. P'[H4]) [10^-107] exceeds the UPB [10^150]it is still MUCH SMALLER than the LPB [~10^-76] --> H4 did not occur by chance (or random access).
The Prokaryote to Eukaryote transition problem.
To the above discussion must be added some "worsening" factors when it comes to the generally supposed fully naturalistic transition from prokaryotes to eukaryotes.
(i) There are many other proteins that have to be coded for to get the highly complex eukaryotic cellular division to occur. There are the other histones such as H3 and Asf1. There is also some timing protocol for the process that must be gene encoded.
(ii) Prokaryotic bacteria are not able to evolve advantageous mutations anyway because without sexual reproduction and crossover disadvantageous mutations always win out (the main thesis of Hoyle's book). Thus we have a kind of chicken and egg situation reminiscent of the origin of life itself - namely without sexual reproduction no H4 (and its companions) and without H4 no sexual reproduction!
Please feel free to critique the above analysis. I like this transition, because it is so less controversial/speculative than the abiogenetic transition - with reproducing prokaryotes in place so too is the neo-Darwinian process (with its established mathematical structure) and thus a test can be made.
-Chris
[ 22. November 2005, 19:25: Message edited by: Christopher D. Beling ]
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Christopher D. Beling
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posted 22. November 2005 02:46
Mel,
quote:
Then d* would pose a more formidable probability bound than UPB.
I would really like to know more about this idea. The idea that space itself is discrete is not new, but there may be some developments in quantum theory that support such ideas - please do keep us posted.
Please feel free to use the diagram - so glad its helpful to you.
A few posts back you made this interesting comment:
quote:
As far as the i.i.d. assumption and pulling numbers out of hats, perhaps once the events covered by the central limit theorem are understood with regard to a UPB, then we could turn our attention to those that are not.
Could you explain to us about the central limit theorem and its application?
-Chris
[ 22. November 2005, 08:39: Message edited by: Christopher D. Beling ]
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Melvin H. Fox
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posted 23. November 2005 08:17
Chris,
Thank you for your detailed analysis of H4 and the related probabilities. I am convinced H4 did not occur by chance. Is there a statistician out there who is not? There is a substantial portion of the academic community who are not. They cling to:
"As Howson and Urbach rightly note, there is never a logical contradiction in refusing to eliminate chance in a test of statistical significance - save in the case where the rejection region R has probability zero with respect to hypothesis H and the event E happens to fall within R." [your quote from DI page 200 posted 18 Nov.]
I have been, to this point in my life, too much the math nerd. If you were to scrutinize the environment in which H4 must have emerged, could it be the case the probability is actually zero? i.e. If I role a die twice the probability I get three 6’s is zero.
As far as d* and the central limit theorem, it is the end of the trimester here at school. If I don’t grade some papers today I will be fired. I will address your interest soon, perhaps on a new thread. I am not web savvy and so I find it cumbersome to insert the proper notation. I have commissioned one of my students to help along these lines.
-Mel
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RBH
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posted 24. November 2005 01:40
As is Melvin, so am I convinced that h4 did not originate by chance. However, my conviction doesn't spring from dubious probability calculations assuming a chance origin that are too small, because those calculations do not represent the distribution of probabilities across possible alternative accounts.
In order to estimate the probability of occurrence of H4, in addition to the chance assembly hypothesis one must take into account proposed naturalistic accounts of its origin. One must assign some probability to them. For example, this paper proposes that eukaryotic histones originated with archaean histone-like precursor proteins, and evolved into their present eukaryotic forms: quote:
Among the nuclear ESPs there are four histones: H2A, H2B, H3, and H4. However, the eukaryotic histones share the same three-dimensional structure with the archaeal histone-like proteins of the Euryarchaeota (methanogens, etc.) (19, 20). Unlike actin, tubulin, ubiquitin, and the GTP-binding proteins, whose three-dimensional homologs are found throughout the Archaea and Bacteria, the histone fold is found only in the Euryarchaeota and not in the Crenarchaeota or the Bacteria. The simplest explanation at the present time for the evolution of histones is that a histone-like protein came in with an ancient archaeal endosymbiont and subsequently evolved into the full eukaryotic complement of histones.
This paper proposes a roughly similar account. What probability does Christopher's analysis assign to that possibility? Zero? Some number less than the probability calculated on the assumption of pure chance assembly? How about the precursors hypothesized? What probability is assigned to them?
You see the problem? In order to eliminate non-chance naturalistic hypotheses you have to somehow assign probabilities to them and defend those assignments.
RBH
[ 24. November 2005, 01:41: Message edited by: RBH ]
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kyle7
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posted 28. November 2005 04:36
RBH,
But how many letters would convince you that something was designed? Certainly, a probability of finding a "u" after a "q" may be greater. But when you are considering a large number of letters this is not important. It really doesn't matter. This is the nature of information.
When someone analyzes the stress in a machine there are different levels of analysis.You can do hand calculations to get an approximate value for the stress. A deeper level of analysis would be to do a finite element analysis. Now you could even go deeper by examining the problem at the molecular level -- even though this may be complete overkill requiring significant time. The same can be said of a probability analysis. You can take it to the nth degree, but it is often unneeded. So if you are talking about a thousand letters, does a "u" after a "q" really matter. I think not. How about a million letters?
Now with ID you can always revisit a problem with a greater depth of analysis. But the first cut can often be done using common sense in a simple fashion, providing significant insight into the problem.
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RBH
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posted 28. November 2005 13:38
kyle7 wrote quote:
But how many letters would convince you that something was designed? Certainly, a probability of finding a "u" after a "q" may be greater. But when you are considering a large number of letters this is not important. It really doesn't matter. This is the nature of information.
It kind of appears that you're appealing to something analogous to the central limit theorem here, and it's an invalid appeal. I know of no "nature of information" that implies that one can disregard the shape of the probability density function over alternatives when calculating a probability.
The situation one is in when trying to calculate these probabilities is precisely analogous to the following: I ask you to calculate the probability of getting a "4" when I toss a many-sided die. I don't tell you how many sides the die has, I don't tell you how many of the sides have a "4" on them, and I don't tell you whether (and how) the die is loaded or whether it's symmetrical or asymmetrical. No matter what number you come up with, it will bear no necessary relationship to the actual probability, since your estimate must be performed in ignorance ot the very factors that are required for your calculation. You could assume that the die is 6-sided, has "4" on only one side, and is a fair die, but without actually having some basis for those assumptions your estimate has an unknowable relationship with the actual probability. You know no more after the calculation than you knew before it, and in fact, you may be in worse shape after doing the calculation to the extent that you believe your calculation told you something valid, when it cannot do that.
In order to calculate a probability one must have a reasonable estimator of the numerator (how many sides have "4" on them), a reasonable estimator of the denominator (how many sides the die has), and a defensible assumption about the shape of the probability density function that is induced by the structure and process that generates the outcome (whether and how the die is loaded and/or asymmetrical). Unless one can provide defensible grounds for all three, one's probability calculations are just so much smoke. They give one the illusion of knowing something that one does not in fact know.
RBH
[ 28. November 2005, 13:40: Message edited by: RBH ]
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Irving
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posted 10. December 2005 23:36
I suppose some things are known...
We know that sides is greater than 1. And we can assume that at least one side not equal to 4 (or the whole exercise is moot), and that the time required from beginning to end of the throw is not equal to zero. Thus regardless of the other factors, there is a finite limit to the number of throws possible in the allowable time...
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