## Abstract

Evolutionary biologists of today are confronted with the problem of explaining how such an enormous amount of information contained in the most basic cell could have been organized into a life form using only chance, material, and time. The historical explanation has been to claim that time on the order of billions of years will result in these complex structures. This article will mathematically model one simple aspect of cell formation and, using mathematical statistics, compute the expected waiting time for this structure to occur. I will also look at the ideas of Emile Borel, Michael Behe, William Dembski, and Emmett Williams as they relate to the effect time has on highly improbable events.

## Introduction

The motivation for this paper came from two conflicting statements that I carried around over several years and finally decided to attempt a resolution. The first is from the evolutionist George Wald claiming time is a great miracle worker in the molecules-to-man evolution process; the second is from the celebrated French probabilist Emile Borel stating that highly improbable events never occur. Since molecules-to-man evolution requires a huge sequence of highly improbable random events, these two statements are in direct conflict. I suspected that time has an approximate linear effect on highly improbable events while the requirement for the structures of life grows exponentially with complexity, thus time cannot be the miracle worker as claimed.

Molecules-to-man evolution can be defined as the natural process that has produced the present-day life forms from matter, energy, chance, genetic modifications and natural selection, and changing environments over vast periods of time. The random combining of basic elements with energy but without outside intelligence is the mechanism by which the first simplest cell is said to have been formed. Vast amounts of time are assumed by the evolutionists to make this process plausible. The objective of this paper is to use probability and mathematical statistics to demonstrate that time can have no significant effect on highly improbable events.

## A Very Improbable Event

The formation of the simplest known cell from molecules, energy, and random processes is a very complex event. Some of the requirements are:

- formation of amino acids from basic elements
- combining the amino acids of only the L-configuration and not the D-configuration (the racemization problem)
- formation of long sequences of amino acids in the correct order to form proteins
- large numbers of chemical bonds to form DNA and RNA
- formation of the remaining parts of the cell

All of these steps are chemically and thermodynamically unstable in the proposed early earth environment and all defy the Second Law of Thermodynamics. Considering only the problem of building L-proteins, the simplest possible self-replicating entity would conservatively contain about 124 proteins of 400 amino acids each (Glass, Assad-Garcia, Alperovich, et al. 2006; Riddle 2006). 19 of the 20 amino acids can be in either the L-form or the D-form, revert back and forth in type, and are statistically centered on a 50:50 mixture by the process called racemization.

Considering only the racemization aspect of 19 of the 20 amino acids, this process would require about

(400 x 19/20) x 124 = 47,120 successive selections each with probability ½.

This probability is approximately .5

^{47,120}= 10^{-14,184}This statistic is only an estimate of the selections necessary to have the right amino acids in the right form (L) to form these 124 proteins and does not take into account the order of amino acids for the proper protein formation or other aspects of cell formation.

The concept of fluctuations to generate small decreases in entropy as a possible driving force for the process of creating complex molecular structures has been argued by Prigogine, Nicolis, and Babloyantz (Prigogine, Nicolis, and Babloyants 1972). Dr. Emmett Williams responded to this idea in a Creation Research Society publication (Williams 1981). Using thermodynamic equations Dr. Williams showed that entropy decreases of 1 part in a million due to fluctuations have a probability of 10^{-19}. Moreover, if minute ordering fluctuations did occur, they would be destroyed immediately upon the next fluctuation. Thus, we cannot expect to build structures in small orderly successive steps. This is analogous on the molecular level to the irreducible complexity concept of Dr. Michael Behe.

## The Myth of Time

One of the biggest myths put over on the public by Carl Sagan in his *Cosmos* television series was the effect time has on the occurrence of evolutionary processes. People were so enchanted with his pronunciation of “billions and billions” that time took on a god-like quality. In one of the latter episodes dealing with the formation of life on earth we were treated to Buddhist philosophy, eastern religious music, and cartoon-like animations of life forms morphing into higher and higher forms.

The prominent evolutionist, George Wald, stated “Time is in fact the hero of the plot. The time with which we have to deal is of the order of two billion years. What we regard as impossible on the basis of human experience is meaningless here. Given so much time the ‘impossible’ becomes possible, the possible probable, and the probable virtually certain. One has only to wait: time itself performs the miracles” (Wald 1955).

Two objections come to mind. First, there are many scientific arguments that our solar system and earth are not even hundreds of million years old, and second, the effect of time is meaningless on highly improbable events. This last point can be demonstrated with mathematical statistics and can be illustrated using an example from the game of Keno.

## Time is Linear

To illustrate the effect of time on improbable events, consider the casino game of Keno. In this example, we select 10 numbers from the board of 80 numbers and require that all 10 numbers come up in the random selection by the casino of 20 numbers. Using basic combinatorics the probability of success P(S) of this random event is

P(S) = (70! x 60! x 20!) / (60! x 10! x 80!) = 1.12212 x 10

^{-7}

If we play this game at a uniform rate, this example satisfies the assumptions required for a Poisson process (Hogg, McKean, and Craig 1970).

Suppose we play 500 of these Keno games per day, then the probability of K successes in a day is distributed as a Poisson random variable (as an approximation to the binomial) with parameter λ

where

λ = N P(S) = 500 x P(S) = 5.61059 x 10

^{-5}

The probability of 0 successes in a Poisson (λ) process

= P(0) = Exp (- λ) = .999943896

The probability of one or more successes = P_{1 (1 day)} = 1 – P(0) = 5.61043 x 10^{-5}

If we play for 100 days, λ_{100}= 100λ = 5.61059 x 10^{-3 }

the probability of 0 successes in 100 days = P(0) = Exp (- λ

_{100}) = .99440512

The probability of one or more successes in 100 days = P _{1 (100 days)}=1 – P (0) = 5.59488 x 10^{-3}

The ratio of probabilities for 100 days and 1 day is

P

_{1 (100 days)}/ P_{1 (1 day)}= 99.72

Thus 100 days has increased the probability by approximately 100 times and the effect of time is very nearly linear.

Next compute this ratio in the limit as the probability of the event approaches zero; which is equivalent to the parameter λ approaching zero.

Lim λ → 0 ( (1 – Exp (-100λ) ) / ( 1 – Exp (-λ) ) ) = 100 using l’Hopital’s rule.

Thus, for Poisson distributed events as the probability approaches 0 in the limit, the effect of time on the probability approaches linearity.

## Waiting Times for a Success

Using the Poisson distribution as an approximation to the binomial distribution allows an easy answer to the question of expected waiting time (t) in days for 1 success. The waiting time for 1 success in a Poisson (λ) process is distributed as an exponential probability law with parameter λ with density (see Hogg et al. 1970)

f (t) = λ exp (-λ t); t ≥ 0 and mean μ = E ( t ) = 1/λ

So playing 500 games per day the average wait time for a win is a little more than 17,823 days or 48.83 years. This result ought to be a discouragement to Keno players!

## The Attack on Borel by Evolutionists

The National Center for Science Education published an attack (NCSE 2000) on Creationists claiming creationist science and mathematics were faulty and that we have misinterpreted Borel’s claim that highly improbable events never happen. He stated clearly that the events were highly improbable not that they have probability = 0. The paper confuses these two ideas. Borel specifically used the word *never,* which indicates time is a factor in what he said. As an example, in the paper they imagine what amounts to a uniform distribution over 10^{100} possible outcomes. One number is specified as a successful trial. All others are a random trial failure. Thus each possible equally likely outcome has a probability well below Borel’s probability bound of 10^{-50} for events that never occur. The claim is made that a computer can select numbers from all possible outcomes thus “all these improbable events can and will occur given infinitely many chances.” Pseudo random number generator algorithms are anything but random as they repeat sequences of digits in cycles at some point in the process, so it is impossible for this “thought experiment” to actually be performed with any degree of fidelity on any computer. The NCSE paper has no real mathematical content and analysis with its example, so let’s do some mathematical statistics with the example given in the paper. We shall thereby clarify what Borel had in mind when he made his probability bound.

As in the example above we can again use the Poisson distribution to model this process and then the Gamma distribution for the waiting time for one success.

We will assume one billion trials per day (N = 10^{9}) which is a large number for physical processes.

Thus, λ = N P(S) = 10

^{9}x 10^{-100}= 10^{-91}

The expected waiting time for a success is therefore

μ = E (t) = 1/λ = 10^{91}days or 2.74 x 10^{88}years

Compare this number with the evolutionist claim that our solar system is less than 5 x 10^{9} years old. This should clarify what Borel meant when he said “improbable events never occur.” Why? There isn’t even close to enough time. Contrary to the statement made in the paper about an eternity of time and infinite time and resources, the physical world we live in has finite time and finite resources. For example, if DNA were somehow constructed with random processes it would have to have occurred during the less-than-five billion years of the Solar system. This process contains information on a level that is almost more than we can imagine and dwarfs the example we just examined. In addition to all the other great contributions Borel made to the mathematics of probability, he certainly showed great insight in dealing with these not-based-in-reality claims that all events with non-zero probability occur eventually.

## The Importance of Irreducible Complexity

Michael Behe has developed the concept of an “Irreducibly Complex System” in his book *Darwin’s Black Box *(Behe 2003). He defines it as “A single system composed of several well-matched, interacting parts that contribute to the basic function, wherein the removal of any one of the parts causes the system to effectively cease functioning.” Darwinian evolution using natural selection requires modified systems that are functionally complete. Darwin recognized that his theory required numerous, successive slight modifications for the development of a complex organ. Irreducible systems do not allow for the possibility of building simpler stable structures on the way to the final system.

This is a very important point in the debate over origins as the prospect of a reducible system can have an enormous simplifying effect on the final probabilities for a complex system. As an example of this simplifying effect, consider again the Keno problem. Suppose the game was reducible by allowing two 5-number successes to be the equivalent of one 10-number success. This is analogous to having a 5-number success acting as a stable, functioning entity on the way to the complete 10-number system. The effect of having this reduced system is to reduce the expected waiting time to just 6.2 days from 17,823 days.

On the molecular level this demonstrates the importance of the analysis of Dr. Williams in refuting thermodynamically stable reduced systems for constructing complex structures, for any small fluctuations away from thermodynamic equilibrium towards increased order have infinitesimally small probabilities and have no way to hold the increased order upon future fluctuations. Thus, a completed structure is required for a new equilibrium point, analogous to the irreducibly complex idea.

## Universal Probability Bounds

Emile Borel proposed 10^{-50} as a universal probability bound for which a random event will never occur. “Random events of probability less than 10^{-50} never happen.” (Borel 1965 and 1962). William Dembski justifies a more stringent universal bound of 10^{-150 }based on the number of elementary particles in the observable universe, duration of the universe, and Planck time (Dembski 1999).

Comparing this universal probability bound to the probability computed in the racemization process of 10^{-14,184} we can conclude that the racemization process alone in cell formation will never occur randomly. This process also involves repeated Bernoulli trials resulting in a binomial distribution and a Poisson process and an easy waiting time computation using the exponential distribution.

## Conclusion

For Poisson-distributed events, as the probability approaches 0 in the limit, the effect of time on the probability approaches linearity. The probabilities for a random process resulting in a living cell approach 0 exponentially with the number of elements and sequencing required. Thus, time has no appreciable effect on the possible random construction of the smallest known cell. The irreducibly complex nature of a cell and of molecular structures provides no reduction in the probabilities required for DNA/RNA creation.

I am reminded of Dr. Henry Morris so often referring to Romans 1:20 “. . . His invisible attributes are clearly seen being understood by the things that are made . . . .” We can see the Creator in what we observe daily or we can see Him in the science and mathematics we have discovered.

Our Creator has indeed created a vast, wonderfully complex universe in both the large scale and the infinitesimally small structures.

## References

Behe, Michael. 2003. *Darwin’s Black Box: The Biochemical Challenge to Evolution,* p.39 (paperback edition). New York: Touchstone.

Borel, Emil. 1962. *Probabilities and Life,* p. 28. New York: Dover.

Borel, Emil. 1965. *Elements of the Theory of Probability,* p. 62. Englewood Cliffs, New Jersey: Prentice-Hall.

Dembski, William. 1999. *Intelligent Design,* Chapter 6. Downers Grove, Illinois: IVP Academic.

Glass, John, Nacyra Assad-Garcia, Nina Alperovich, et al. 2006. Essential genes of a minimal bacterium. *PNAS* 103 no. 2:425–430.

Hogg, Robert V., Joseph W. McKean, Allen Thornton Craig. 1970. *Introduction to Mathematical Statistics,* p. 95. New York: Macmillan Company.

National Center for Science Education. 2000. Creationism and Pseudomathematics. *RNCSE* 20, no. 4. (ncse.com/rncse/20/4/creationism-pseudomathematics).

Prigogine, Ilya, Gregoire Nicolis, and Agnes Babloyants. 1972. Thermodynamics of Evolution. *Physics Today* 25, no. 11: 23–28.

Riddle, Mike. 2008. Chapter 6: Can Natural Processes Explain the Origin of Life? In Ken Ham, ed. *The New Answers Book 2.* Green Forest, Arkansas: Master Books.

Wald, George. 1955. *The Physics and Chemistry of Life, *p. 12. Chicago, Illinois: Simon and Schuster.

Williams, Emmett. 1981. Thermodynamics and the Development of Order, Fluctuations as a Method of Ordering. *Creation Research Society Monograph Series* 1:55–66.

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