How Unlikely?

[Dardedar]

DAR
Mark Arnold, a mathematician, emailed this to us and said it would be fine to post it somewhere. This science forum would be an appropriate place to put it. Sav, do post your response.

***
How Unlikely?

Suppose an event was considered very unlikely, like someone guessing an 8 digit number. The probability that a single random guess is correct
is 1 in 100 million. The expected number of random guesses needed
before getting the correct number is about 50 million. So, if we
guessed 1 number per second, every second, we would expect that it
would take (on average) about 579 days before the correct number was
hit. Of course, if you want to be sure that the correct number gets
hit, then you would need to budget for the worst case of about 1157
days (the correct guess didn’t get hit until all 100, 000, 000
possibilities were exhausted).

Suppose we have a sample of n labeled elements we are selecting from,
and one is “correct”. We select (“guess”) until we get it right (select
the correct element). That the expected number of guesses before
success is about n/2 makes sense; in fact it is exactly (n + 1)/2: If
there are 2 elements, then the possible (and equally likely) outcomes
are {right} and {wrong,right}. If n = 3, then the possible outcomes are
{right}, {wrong,right} and {wrong,wrong,right}. For n = 2 the average
number of guesses is the mean of {1, 2}. For n = 3 the average number
of guesses is the mean of {1, 2, 3}. If you hypothesize that the expected number of guesses before success is the mean of {1, 2, . . . , n}, then you are right, and that mean is g(n) = (n + 1)/2.

Now suppose we have some feedback. For example, suppose that we guess
the 8 digit number one digit at a time, and we are told whether that
digit is wrong or right. Then we have broken our problem into a process
consisting of 8 steps, each of which is to correctly guess one of the
numbers {0, 1, . . . , 9}. What is the expected number of guesses
needed before getting the correct number now? Well, since there are 10
elements in {0, 1, . . . , 9}, we expect that it will take, on average,
g(10) = 11/2 guesses per digit, giving a grand total of 8 11/2 = 44.
Yes, on average, only 44 guesses will be needed to guess this 8 digit number! With feedback we can do in about 44 seconds what it took about 1.5 years to do without! With feedback, the worst case takes 80 seconds (each digit takes 10 guesses), compared to over 3 years without.

Of course this can be generalized. I can guess a random (English)
4-letter string on average in 228,488.5 guesses, but with
letter-by-letter feedback it only takes an average of 54 guesses. Is
this how safe-cracking works? (I don’t know). Did you ever play
“20-questions”? This feedback is certainly not possible in standard
software for passwords, PINs, etc. Feedback may or may not be present
in a given natural system, but when it is evident, we should not be
surprised by that system’s complexity nor impressed by its
“improbability”.

In fact we should be very surprised to find any biological system
without many and various kinds of feedback. The most fundamental type
of biological feedback is probably: “if you have no offspring, then you
don’t pass your genes on”...

--------------------
Mark Arnold Assoc Prof & Graduate Coordinator
Department of Mathematical Sciences
University of Arkansas Fayetteville, AR 72701
***

[Savonarola]

I often find myself rather hesitant and/or intimidated when disagreeing with a bona fide professor of some sort. I also dislike countering material sent by someone who not only supports our cause but also takes steps to help further it. Nevertheless, here's what I found myself typing...

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But genetic coding doesn't really have "digit-by-digit" (in this case, amino acid by amino acid, or even nucleotide by nucleotide) feedback. (i.e. There is no mechanism to determine when any one component is "correct"; the protein works or doesn't work as a whole, although there is obviously a spectrum from "doesn't do anything at all" to "performs some function, with incredible efficiency.") I would imagine that most* selection doesn't fit this paradigm, as the active site of a protein will be largely conserved while the peripheral tertiary structure would be selected on a "works better" or "doesn't work better" basis, not a "nope... nope... nope... perfect-all-in-one-fell-swoop!" basis.

* Alternatively, one situation where I could imagine that this representation is somewhat appropriate would be the arising of a homologous protein or domain. (However, see also below.) For example, I can imagine that one of the two homologous domains of carbamoyl phosphate synthetase could have arisen (that is, quite probably arose) by a duplication followed by a mutation or two that facilitated a new chemical reaction (in the sense that the digit-by-digit feedback concept of "correct!" in this case means "wow, check out what just a change or two can do!").

While it is certainly important (i.e. essential) to consider the feedback aspect of natural selection, this mathematical representation is flawed for not only the above reason but also because there is no "correct" combination of nucleotides or amino acids. Just as evolution is undirected, there is no ultimate "goal" sequence for any protein. Evolution just works with what.. well, works.

A generalized view of natural selection accounts for the effect of this type of "feedback." In my presentation a couple months ago, I tried to make this conceptual point when talking about self-replicating RNA strands; the effect is not limited to DNA, proteins, or even modern cells. Those [insert any objects here] that are better able to replicate themselves will tend to produce more offspring than others, thereby increasing their relative numbers.


Dr. Plavcan also explained why these "improbability" arguments are mere red herrings, although nobody could ever convince Dumbski-- er, Dembski of that fact.
Also, following the example of an eight digit number taking an average of 579 days to guess randomly... How many times has 579 days elapsed over 4.5 billion years? Or be more forgiving and only count the first billion or so years after the formation of Earth but before the first life arose.
I get 2.8x10^9 times and 6.3x10^8 times, respectively, both of which are incomprehensibly many. If something so improbable would happen nearly three billion times, how does that make it "so improbable"? The moral of the story: 4.5 billion years is a long, long time.

ETA: I certainly should, however, point out that the ultimate message that Professor Arnold is presenting is sound: any feedback mechanism effectively nullifies the "random" models that creationists are so proud to present.

Feedback may or may not be present in a given natural system, but when it is evident, we should not be surprised by that system’s complexity nor impressed by its “improbability”.

I couldn't have put it better myself. (And quite obviously, I didn't.)
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Critiques welcomed.

[Guest]

okay, whew, i didn't read all that because it was way too text-booky for my brain right now.

But...BUT...I did do something a few weeks ago that blew everybody's minds. My mom and a friend of hers were trying to remember some woman's name who died a long time ago. The name "Bonnie" just came to my mind, and I said "Bonnie?" and they were like, you knew Bonnie? And of course, I didn't, I had no idea who she was. But isn't it freakishly amazing that out of the thousands of female names in the world, I just said the right one??!!!!! I was like, WOW, I'm PSYCHIC, but unfortunately I haven't been able to conjure anything like that again.

(BETSY)

[Dardedar]

Well you let us know when you can Betsy and you can go for our thousand dollar prize.

Speaking of coincidence and text booky responses, I just recieved this response today from Mark Arnold regarding an interesting math question. I find it very interesting that Doug and I both have the EXACT same numbers in our phone number (not in the same order obviously). So I asked him what the odds of this are. He said:

***
Well, all exceptions aside, there are n = 10,000,000
possible "phone numbers" (0000000,0000001,...,9999999).
We will need this.

And there are m = 11,440 different selections of 7 numbers
from the digits 0,1,...,9 (we call these the selections
with repetition of r=7 objects from a collection of t=10,
and the formula for the number of such is
C(t+r-1,r) = (t+r-1)! / ((t-1)! r!),where k! = 1*2*...*k).
We will need this.

Not all of these selections have equal probability, however.
For example there is only one "phone number" with
7 ones, but there are 2520 phone numbers with the
digits 2,3,4,4,6,7,8 (your selection). An easy
question to answer is "given a selection (YOUR numbers),
what is the probability that someone else has that
selection (the same numbers)". That is simply
2520 / 9,999,999 = .000252 (about 1 in 4000).
(It is the mathematicians way: make a problem simple
enough to answer, and then try to generalize...)
***

--snip--

Then it got real hard so I'll snip the rest (!). So the answer is, about one in 4,000 not counting the fact that there are a limited number of prefixes for Fayetteville so that would skew the odds a some.

D.
-----------------------
An example of coincidence: “In Psalm 46 of the King James Bible, published in the year that Shakespeare turned 46, the 46th word is "shake" and the 46th word from the end is "spear."
--"The Power of Coincidence" by David G. Myers, Skeptic Magazine, vol. 9, #2, 2002

[Barbara Fitzpatrick]

Are you guys trying to argue your way out of evolution? I'm not a statistician, but I know there is a formula for calculating coincidences. (And you drastically reduced the chances of phone numbers using the same digits when you restrict the exchange to a maximum of 3 different numbers).

The real problem with discussing this or any other theory is the lay person's understanding of what theory means. They are taught - it's in the science text books - that "theory" means "scientific guess". Since the term guess is used exceedingly loosely mundanely, they extrapolate that to a very loose definition for theory. My 1st response to the "just a theory" crowd is "there's no such thing as 'just' a theory". But explaning what a theory really is won't do any good until we quit teaching "scientific guess" in the schools.

[Savonarola]

Are you guys trying to argue your way out of evolution?

No, I just dislike flawed representations of it, because that's exactly why (1) people don't actually understand it when they think they do, and (2) anti-science rubbish can become mainstream belief.

I'm not a statistician, but I know there is a formula for calculating coincidences.

I'm not a statistician either, but a one in a billion chance is still a one in a billion chance, not an impossibility. But more to the point, if you pull a royal flush of spades out of a randomized deck, you'd call it a coincidence, but if you pulled, say, the four of clubs, deuce of spades, queen of spades, seven of diamonds, and nine of hearts, you'd think nothing of it... Yet the probability of doing each is identical. Every time you're dealt a poker hand, you could look at your cards and think, "wow, getting this particular hand is four times less likely than getting a royal flush."

The real problem with discussing this or any other theory is the lay person's understanding of what theory means. They are taught - it's in the science text books - that "theory" means "scientific guess".

I don't remember ever being taught that a scientific theory is analogous to a guess. But then again, not everyone has a father with a PhD in a science. Regardless, I'm confident that the books used by the students that I see almost every day don't represent a scientific theory in that way.

[Doug]

I'm not a statistician either, but a one in a billion chance is still a one in a billion chance, not an impossibility. But more to the point, if you pull a royal flush of spades out of a randomized deck, you'd call it a coincidence, but if you pulled, say, the four of clubs, deuce of spades, queen of spades, seven of diamonds, and nine of hearts, you'd think nothing of it... Yet the probability of doing each is identical. Every time you're dealt a poker hand, you could look at your cards and think, "wow, getting this particular hand is four times less likely than getting a royal flush."

DOUG
Part of the reason so many people believe in miracles is that they know precious little about the scientific method, statistics, critical thinking, and historiography. Make classes in each of these mandatory in high schools across the country and religion would disappear in a couple of generations.

[Barbara Fitzpatrick]

Savonarola - don't be confident the students you see everyday weren't taught that "scientific guess" is the definition of "theory" - I used to teach 9th grad physical science & was trained in general science (which included evaluating textbooks) at elementary, middle school, jr high school, and high school levels (AR, TX, & WI) - that definition is in the public school text books at all levels.

[Betsy]

I agree with Barbara that theory is taught as a scientific guess - you have a "theory" (an idea of what or how or why something might be) and you then perform an experiment to prove it is true (or not). (unless you're a religious fundamentalist, then you just assume it's true and go around telling everyone it's true)

So, if that isn't a theory, then what is, please?

[Betsy]

I just read something with the word "hypothesis" in it and realized maybe I was confusing theory with hypothesis. But I'd still like to hear your definition of theory...

[Savonarola]

Barbara, I didn't say that no students were taught about science poorly, I was just pointing out that there has apparently been at least some improvement.

Betsy, here is how I would answer your question:

A scientific theory is a statement or set of statements that

• explains observable natural phenomena
• has been tested repeatedly without being falsified
• is conceptually falsifiable
• is devoid of superfluous parts

The scientific method works by making guesses ("hypotheses") about how the universe works. We then test these guesses, using test results to eliminate or refine guesses. Ideally, a highly specific, highly accurate set of hypotheses are gathered. Upon widespread replication of test results and much discussion of the set among the scientific community for critiquing, the concept "graduates" to the status of a theory.

[Dardedar]

I just read something with the word "hypothesis" in it and realized maybe I was confusing theory with hypothesis.

DAR
Betsy, science has a specialized (and perhaps unfortunate) use of the word theory. In common speech it is normal to equate theory with hunch or guess and even worse, the word is NOT interchangeable with fact. In science this is not the case. Here is a short little blurb I have handy. I have a longer and better version somewhere but this should help for now:

***
A common problem is that people confuse theory with hypothesis, which is an untested conclusion. A scientific theory, on the other hand, is an explanation derived from and supported by established evidence. After passing test after test, it is the best available explanation of the facts.
Many theories have become established as fact. Doctors are certain of germ theory--that various germs cause a number of ailments. Astronomers are certain of the heliocentric theory of the solar system--the sun, not the earth, is the center of the solar system. Geologists are certain of the plate tectonic theory--that continents and sea floor are moving on large chunks of the earth’s crust. And biologists are certain that all living things share a common ancestor.
***

D.

---------------------
This is good:

AAPT Statement on the Teaching of Evolution and Cosmology

The Executive Board of the American Association of Physics Teachers is dismayed at organized actions to weaken and even to eliminate significant portions of evolution and cosmology from the educational objectives of states and school districts.

Evolution and cosmology represent two of the unifying concepts of modern science. There are few scientific theories more firmly supported by observations than these: Biological evolution has occurred and new species have arisen over time, life on Earth originated more than a billion years ago, and most stars are at least several billion years old.

Overwhelming evidence comes from diverse sources - the structure and function of DNA, geological analysis of rocks, paleontological studies of fossils, telescopic observations of distant stars and galaxies - and no serious scientist questions these claims. We do our children a grave disservice if we remove from their education an exposure to firm scientific evidence supporting principles that significantly shape our understanding of the world in which we live.

No scientific theory, no matter how strongly supported by available evidence, is final and unchallengeable; any good theory is always exposed to the possibility of being modified or even overthrown by new evidence. That is at the very heart of the process of science. However, biological and cosmological evolution are theories as strongly supported and interwoven into the fabric of science as any other essential underpinnings of modern science and technology. To deny children exposure to the evidence in support of biological and cosmological evolution is akin to allowing them to believe that atoms do not exist or that the Sun goes around the Earth.

We believe in teaching that science is a process that examines all of the evidence relevant to an issue and tests alternative hypotheses. For this reason, we do not endorse teaching the “evidence against evolution,” because currently no such scientific evidence exists. Nor can we condone teaching “scientific creationism,” “intelligent design,” or other non-scientific viewpoints as valid scientific theories. These beliefs ignore the important connections among empirical data and fail to provide testable hypotheses. They should not be a part of the science curriculum.
School boards, teachers, parents, and lawmakers have a responsibility to ensure that all children receive a good education in science. The American Association of Physics Teachers opposes all efforts to require or promote teaching creationism or any other non-scientific viewpoints in a science course. AAPT supports the National Science Education Standards, which incorporate the process of science and well-established scientific theories including cosmological and biological evolution.
This statement was adopted by the Executive Board of the American Association of Physics Teachers on April 24, 2005.

http://www.aapt.org/Policy/evolutandcosmo.cfm

[Savonarola]

In common speech it is normal to equate theory with hunch or guess and even worse, the word is NOT interchangeable with fact. In science this is not the case.

I think this statement is a bit misleading. A theory is not a fact; a theory explains facts.
In this context, the fact is that life has changed over time ("evolution is a fact"). The theory of evolution explains how life has changed over time.

Gravity is a fact in that two masses do attract each other proportionally to their masses and inversely proportionally to the square of the distance between them. Gravitational theory is simply our best explanation of gravity. Gravitational theory may have some flaws, but that doesn't mean that gravity doesn't exist.

[JD Allen]

I'm not a statistician either, but a one in a billion chance is still a one in a billion chance, not an impossibility. But more to the point, if you pull a royal flush of spades out of a randomized deck, you'd call it a coincidence, but if you pulled, say, the four of clubs, deuce of spades, queen of spades, seven of diamonds, and nine of hearts, you'd think nothing of it... Yet the probability of doing each is identical. Every time you're dealt a poker hand, you could look at your cards and think, "wow, getting this particular hand is four times less likely than getting a royal flush."

I think that is a point that is often missed in the Creation/Evolution debate. People often state that the chance that the chain of events that would lead to the world as it is now is so small as to make it almost impossible. Personally, I disagree with that statement but even if it were true, some result would have to come from history. The odds of this one happening may be 1 in gazillion, but the odds of a result is 1 in 1.

[Savonarola]

I think that is a point that is often missed in the Creation/Evolution debate. People often state that the chance that the chain of events that would lead to the world as it is now is so small as to make it almost impossible. Personally, I disagree with that statement but even if it were true, some result would have to come from history. The odds of this one happening may be 1 in gazillion, but the odds of a result is 1 in 1.

Absolutely right. When I shuffle a deck of cards, then look at the order, I could think, "Whoa, there was only a 1 in 8×10^67 probability of that happening!" However, if I'd gotten any other order, I would have wound up thinking, "Whoa, there was only a 1 in 8×10^67 probability of that happening!"
Another response I like to use (except that it passively implies that the math is done correctly) is that the other gazillion-minus-1 "tries" failed, just as the math predicted, therefore those tries did not produce organisms to sit around and ask why it's so improbable that they exist...

[Doug]

Absolutely right. When I shuffle a deck of cards, then look at the order, I could think, "Whoa, there was only a 1 in 8×10^67 probability of that happening!" However, if I'd gotten any other order, I would have wound up thinking, "Whoa, there was only a 1 in 8×10^67 probability of that happening!"

DOUG
Whereas supposedly it would be more likely if a ghost were to shuffle the cards and get that result?

What I like to ask creationists is: Even IF (and that is an unproven if) it is unlikely that the present universe could have arisen without intelligent design, and IF the odds against it are at least X to 1, whatever large number X may be, show me that the odds of a ghost doing it are less than X to 1.

[Savonarola]

What I like to ask creationists is: Even IF (and that is an unproven if) it is unlikely that the present universe could have arisen without intelligent design, and IF the odds against it are at least X to 1, whatever large number X may be, show me that the odds of a ghost doing it are less than X to 1.

This is what I asked Lee Merrill on IIDB. He decided that going into that would be too much of a derail...

[Doug]

What I like to ask creationists is: Even IF (and that is an unproven if) it is unlikely that the present universe could have arisen without intelligent design, and IF the odds against it are at least X to 1, whatever large number X may be, show me that the odds of a ghost doing it are less than X to 1.

This is what I asked Lee Merrill on IIDB. He decided that going into that would be too much of a derail...

DOUG
That doesn't surprise me. So it usually goes something like this.

Fundie: The universe had to be created by design.
Skeptic: Why?
Fundie: The odds of the universe being created by chance are so astronomical that you should believe that it is MORE likely that a spook did it by magic powers.
Skeptic: OK, show that it is more likely that a spook did it by magic powers.
Fundie: Don't change the subject!