Science

Probability is defined in mathematics in the context of discrete elements in sets. However, it can be transitioned into an analogous definition in continuous mathematics. Thirdly, it can be represented as a meld of discrete and continuous concepts as a continuous probability function.

The Discrete Context

In discrete mathematics, probability is the fractional concentration of an element in a logical set. It is the ratio of the quantity of elements of the same ID to the total number of elements in the set. If the numerator is zero, the probability is zero. The probability is never negative because the numerator is never negative and the denominator is a minimum of one. Probability reaches a maximum of one, where the set of elements is homogeneous in ID. Probability can have any value from one to zero, because the denominator can be increased to any positive integer. Thus, probability in its discrete definition is itself a continuous variable, having a range of zero to one.

A probability and its improbability are complements of one.

The Continuous Context

Probability is a fraction of a whole set of discrete elements. If, however, we define a whole as continuous, we can then define probability in a continuous context analogous to its definition in a discrete context. One example is had in identifying the area of a circle as a continuous whole and identifying segments of area using different IDs, such as in a pie chart. Another example is that in statistics where the whole is defined as the area under the normal curve. Probability is then a fraction of the area under the curve.

Melding the Discrete and Continuous as a Probability Function

The simplest set of discrete elements is that in which all the elements have the same ID. The next simplest is that in which the elements are either of two IDs, where the quantities of the elements of each ID are equal. Thus, the probability of each element is one-half and the improbability of each is one-half.

If we choose a continuous function which oscillates between two extremes, we could associate the one extreme with an ID and the other extreme with a second ID. We could view the first ID as having a probability of one at the first extreme and having a probability of zero at the other extreme. We would thus be viewing the function as a probability, which transitions continuously through the intermediate values as it cycles between a probability of one and a probability of zero, i.e. between the two IDs.

At this second extreme, the second ID has a probability of one, which is also the improbability of the first ID.

A visual example would be a rotating line segment oscillating at a constant angular velocity between a horizontal orientation as one ID and a vertical orientation as the second ID. The continuous equation for this would be the cos^2 (α). The function oscillates from the horizontal or from a probability of one at α = 0 degrees to the vertical or to a probability of zero at α = 90 degrees. At α = 180 degrees, it is horizontal again with a probability of one. The probability goes to zero at 270 degrees and back to one at α = 360 degrees. The intermediate values of the function are transient values of probability forming the cycle from one to zero to one and back again from one to zero to one.

The improbability of horizontal, namely the probability of vertical, would be the sin^2 (α). The probability of horizontal plus its improbability equals one. Thusly, cos^2 (α) + sin^2 (α) = 1.

The Flip of a Coin

The score was tied at the end of regular time in the NFC Championship Game in 2016 between the Green Bay Packers and the Arizona Cardinals. This required a coin toss between heads and tails to determine which team would receive the ball to start the overtime. However, in the first toss, the coin didn’t rotate about its diameter. The coin didn’t flip. Therefore the coin was tossed a second time.

If, rather than visualizing a line segment, we envision a coin rotating at a constant angular velocity, we wouldn’t choose horizontal vs. vertical as the probability and improbability, because we wish to distinguish one horizontal orientation, heads, from its flipped horizontal orientation, tails.

A suitable continuous function of probability, P, oscillating between a value of one or heads at the horizontal α = 0 degrees and a value of zero, or tails at the horizontal α = 180 degrees, would be
P = [(1/2) × cos (α)] + (1/2), where the angular velocity is constant.

The probability of tails is 1 – P = (1/2) – [(1/2) × cos (α)]

The probability of heads and the probability of tails are both one-half at α = 90 degrees and α = 270 degrees.

The probability of heads plus its improbability, which is the probability of tails, is one

Whether we visualize these functions, the one as oscillating between horizontal and vertical and the other as oscillating between heads and tails, the functions are waves.

We are thus visualizing a probability function as a wave oscillating continuously between a probability of one and zero. The probability is the fraction of the maximum magnitude of the wave as a function of α.

An Unrelated Meaning of Probability

We use the word, probability, to designate our lack of certitude of the truth or falsity of a proposition. This meaning of probability, reflects the quality of our human judgment, designating that judgment as personal opinion, rather than a judgment of certitude of the truth. This meaning of probability has nothing to do with mathematical probability, which is the fraction of an element in a logical set or, by extension, the fraction of a continuous whole.

Driven by our love of quantification, we frequently characterize our personal opinion as a fraction of certitude. This, however, itself is a personal or subjective judgment. A common error is to mistake this quantitative description of personal opinion to be the fractional concentration of an element in a mathematical set.

Errors Arising within Material Analogies of Probability

A common error is to identify material analogies or simulations of the mathematics of probability as characteristic of the material entities employed in the analogies. In the mathematics of probability and randomness the IDs of the elements are purely nominal, i.e. they are purely arbitrary. The probability relationships of a set of six elements consisting of three elephants, two saxophones and one electron are identical to those of a set of three watermelons, two paperclips and one marble. This is so because the IDs are purely nominal with respect to the relationships of probability.

In analogy, the purely logical concepts of random mutation and probability are not properties inherent in material entities such as watermelons and snowflakes. This is in contrast to measureable properties, which, as the subject of science, are inherent in material entities.

The jargon employed in analogies of the mathematical concepts also leads us to confuse logical relationships among mathematical concepts with the properties of material entities. In the roll of dice we say the probability of the outcome of boxcars is 1/36. We think of the result of the roll as a material event, which becomes a probability of one or zero after the roll, while it was a probability of 1/36 prior to the roll of the dice. In fact, the outcome of the roll had nothing to do with probability and everything to do with the forces to which the dice were subjected in being rolled. The analogy to mathematical probability is just that, a visual simulation of purely logical relationships.

We are also tempted to think of the probability 1/36 as the potential of boxcars to come into existence, which after the roll is now in existence at an actuality of one, or non-existent as a probability of zero. In this, we confuse thought with reality. Probability relationships are solely logical relationships among purely logical elements designated by nominal IDs. Material relationships are those among real entities, whose natures determine their properties as potential and as in act.

Quantum Mechanics

In quantum mechanics, it is useful to treat energy as continuous waves in some instances and as discrete quanta in others. It is useful to view the wave as a probability function and the detection or lack of detection of a quantum of energy as the probability function’s collapse into a probability of one or of zero, respectively.

An Illustration

As an aid to illustrate the relationship of a probability function as a wave and its outcome as one or zero in quantum mechanics, Physicist, Stephen Barr, proposed the analogy,

“This is where the problem begins. It is a paradoxical (but entirely logical) fact that a probability only makes sense if it is the probability of something definite. For example, to say that Jane has a 70% chance of passing the French exam only means something if at some point she takes the exam and gets a definite grade. At that point, the probability of her passing no longer remains 70%, but suddenly jumps to 100% (if she passes) or 0% (if she fails). In other words, probabilities of events that lie in between 0 and 100% must at some point jump to 0 or 100% or else they meant nothing in the first place.”

Problems with the Illustration

The illustration fails to distinguish the purely logical relationships of mathematical probability from the existential relationships among the measurable properties of material entities. The illustration identifies probabilities as being of events rather than identifying probabilities as logical relationships among purely logical entities designated by nominal IDs. It claims that probability must transition from potency to act or it is undefined. In contrast, probability is the fractional concentration of an element in a logical set. The definition has nothing to do with real entities, whose natures have potency and are expressed in act.

Another fault of the illustration is that it is not an illustration of mathematical probability, but an illustration of probability in the sense of personal opinion. Some unidentified individual is of the opinion that Jane will probably pass the French exam. The unidentified individual lacks human certitude of the truth of the proposition that Jane will pass and uses a tag of 70% to express his personal opinion in a more colorful manner.

It is a serious error to pick an example of personal opinion to illustrate a wave function, viewed as a probability function. A wave function, such as that associated with the flip of a coin oscillating between heads as a probability of one and tails as a probability of zero, would have served the purpose well.

Of course, a wave, viewed as a probability function, is not the probability of an event. It is the continuous variable, probability, whose value oscillates between one and zero, and as such assumes these and the intermediate values of probability transiently. The additional condition is that when the oscillation is arrested, the wave collapses to either of the discrete values, one and zero, the presence or absence of a quantum. The collapse is the transition of logical state from one of continuity to one of discreteness.

In an exchange of comments with Phil Rimmer on the website, StrangeNotions.com, I attempted to explain the distinction between probability and efficiency. The topic deserves this fuller exposition.

I have argued that Richard Dawkins does not understand Darwinian evolution because he claims that the role of replacing a single stage of random mutation and natural selection with a series of sub-stages increases the probability of evolutionary success. In The God Delusion (p 121) he titles this ‘solving the problem of improbability’, i.e. the problem of low probability. My claim is that replacing the single stage with the series of sub-stages increases the efficiency of mutation while having no effect upon the probability of success.

Using Dawkins’ example of three mutation sites of six mutations each, I have illustrated the efficiency at a level of probability of 85.15%, where the series requires only 54 random mutations, while the single stage requires 478.

It may be noted that at a given number of mutations, the probability of success is greater for the series than for the single stage. A numerical example would be at 54 total mutations. For the series the probability of success in 85.15%, whereas at 54 total mutations, the probability of the single stage is only 22.17%. The series has the greater probability of success at a total of 54 mutations.

This would appear to be a mortal blow to my argument. It would seem that Richard Dawkins correctly identifies the role of the series of sub-stages as increasing the probability of success, while not denying its role of increasing the efficiency of mutation. It would seem that Bob Drury errs, not in identifying the role of the series as increasing the efficiency of mutation, but in denying its role in increasing the probability of evolutionary success.

Hereby, I address this apparently valid criticism of my position.

The Two Equations of Probability as a Function of Random Mutations

The probability of evolutionary success for the single stage, PSS, as a function of the total number of random mutations, MSS, is:
PSS = 1 – (215/216)^MSS

The probability of evolutionary success for the series of three sub-stages, PSR, as a function of the total number of random mutations per sub-stage, MSUB, is:
PSR = (1 – (5/6)^MSUB)^3.

For the series, the total number of mutations is 3 x MSUB.

Comparison of Probability at the Initial Condition

At zero mutations, both probabilities are zero. Initially, the probability of both processes, namely the single stage and the series of sub-stages is the same.

For the single stage at one random mutation, which is the minimum for a positive value of probability, the probability of success is 1/216 = 0.46%.

For the series of three stages, at one random mutation per stage, which is the minimum for a positive value of probability, the probability of success is (1/6)^3 = 1/216 = 0.46%. At this level of probability, the single stage has the greater mutational efficiency. It takes the series three random mutations to achieve the same probability of success as the single stage achieves in one random mutation.

Comparison of the Limit of Probability

For both the single stage and for the series of three stages, the limit of probability with the increasing number of mutations is the asymptotic value of 100% probability.

Comparison of the Method of Increasing Probability

For both the single stage and for the series of three stages, the method of increasing the probability is the same, namely increasing the number of random mutations. For both, probability is a function of the number of random mutations.

Comparison of the Intermediate Values between the Initial Condition and the Limit

For both the single stage and for the series of three stages, probability varies, but continually increases from the initial condition toward the limit.

Excepting for values of total mutations less than six, i.e. two per sub-stage, at every level of probability, the series requires fewer mutations than does the single stage. Correspondingly, at any number of mutations greater than six, the series has a higher value of probability than the single stage. Thus, if the comparison is at a constant value of probability, the series requires fewer mutations. If the comparison is at a constant value of mutations, the series has a higher value of probability.

Apparent Conclusion

Richard Dawkins is right in that the series increases the probability of success, without denying that it also increases the efficiency of mutation. Bob Drury is wrong in denying the increase in probability.

The Apparent Conclusion Is False, in Consideration of the Concept of Efficiency

Both the single stage and the series of sub-stages are able to achieve any value of probability over the range from zero toward the asymptotic limit.

Efficiency is the ratio of output to input. One system or process is more efficient than another, if its efficiency is numerically greater. There is no difficulty in comparing two processes where the efficiency of both systems is constant. In such a case, output starts at zero at input equals zero. Output is a linear function of input, having a constant positive slope. The process with the higher positive slope is more efficient than the other. However, in cases where the efficiencies vary, the comparison of efficiencies must be determined at the same value for the numerator of the ratio of efficiency, i.e. the output, or at the same value for the denominator, the input.

In this comparison of the single stage vs. the series of sub-stages, the output is probability and the input is the number of random mutations. Remember both processes increase probability by the same means, namely by increasing the number of random mutations. That is, output increases with increasing input. Also, remember that both processes do not differ in that they both approach the same limit of probability asymptotically.

Dawkins’ comparison of replacing the single stage with a series of sub-stages is the comparison of two processes.

In the numerical examples above we can calculate and compare the efficiencies of the two processes at a constant output, e. g. of 85.15% probability and at a constant input, e.g. of 54 mutations.

At the constant output of 85.15%, the efficiency for the single stage 85.15/478 = 0.18. For the series of sub-stages the efficiency is 85.15/54 = 1.57. The mutational efficiency is greater for the series than for the single stage at the constant output of 85.15% for both processes.

At the constant input of 54 mutations, the probability for the single stage is P = 1 – (215/216)^54 = 22.17%. Therefore, the efficiency is 22.17/54 = 0.41. At this constant input, efficiency for the series is 85.15/54 = 1.57. The mutational efficiency is greater for the series than for the single stage at the constant input of 54 mutations for both processes.

At the 85.15% probability level, the series is greater in mutational efficiency than the single stage by a factor of 478/54 = 8.8

Further evidence that Dawkins is illustrating an increase in efficiency and not an increase in probability is that he compares the temporal efficiencies of two computer programs. For both programs, the input of the number of random mutations is equated with the time of operation from initiation to termination. Termination is upon the random inclusion of one specific mutation. The sub-stages based program typically wins the race against the single stage based program. This demonstrates the greater mutational efficiency of the sub-series, not the greater probability of success.

In the numerical example of three sites of six mutations each, the specific mutation would be one of 216. Let us modify the computer program races slightly. This will give us a greater insight into the meaning of probability and the meaning of efficiency.

Let each program be terminated after 54 and 478 mutations for the series and the single stage, respectively. If the comparison is performed 10,000 times, one would anticipate that on the average, both programs would contain at least one copy of the specific mutation in 8,515 of the trials and no copies in 1,485 of the trials. The series program would be more efficient because it took only 54 mutations or units of time, compared to 478 mutations or units of time for the single stage program to achieve a probability of 85.15%.

For the numerical illustration of three mutation sites of three mutations each, both the single stage and the series of sub-stages have the same initial probability of success greater than zero, namely, 0.46%. Both can achieve any value of probability short of the asymptotic value of 100%. They do not differ in the probability of success attainable.

It doesn’t matter whether we compare the relative efficiencies of the series vs. the single stage at a constant output or a constant input, the series has the greater mutational efficiency for total mutations greater than six.

For the numerical illustration of three mutation sites of three mutations each, at a probability of 85.15%, the series is greater in mutational efficiency by a factor of 8.8. At 90% probability, the factor of efficiency is 8.9 in favor of the series. At a probability of 99.9999%, the factor of efficiency is 12.1 in favor of the series.

Analogy to a Different Set of Two Equations

Let the distance traveled by two autos be plotted as a function of fuel consumption. Distance increases with the amount of fuel consumed. Let the distance traveled at every value of fuel consumption be greater for auto two than auto one. Similarly, at every value of distance traveled, auto two would have used less fuel than auto one. My understanding would be inadequate and lacking comprehension, if I said that replacing auto one with auto two increases the distance traveled. It would be equally inane to say that auto two solves the problem of too low a distance. My understanding would be complete and lucid, if I said that replacing auto one with auto two increases fuel efficiency.

There is a distinction between distance and fuel efficiency. Understanding the comparison between the two autos is recognizing it as a comparison of fuel efficiency. Believing it to be a comparison of distances is a failure to understand the comparison.

For both the single stage and the series of sub-stages of evolution, probability increases with the number of random mutations. Except for the minimum number for the sub-series, at every greater number of random mutations, the probability is greater for the series of sub-stages than for the single stage of evolution. Similarly, except for the minimum positive value, at every value of probability, the series requires fewer random mutations. My understanding would be inadequate and lacking comprehension, if I said that replacing the single stage with the series increases the probability attained. It would be equally inane to say that the series solves the problem of too low a probability. My understanding would be complete and lucid, if I said that replacing the single stage with the series increases mutational efficiency.

The role of a series of sub-stages in replacing a single stage of random mutation and natural selection is to increase the efficiency of random mutation while having no effect on the probability of evolutionary success. This is evident by comparing the equations of probability for the series and for the single stage as functions of the number of random mutations. This is the very comparison proposed by Richard Dawkins for the sake of understanding evolution. He misunderstood it as “a solution to the problem of improbability” (The God Delusion, page 121), i.e. as solving the problem of too low a probability.

There is a distinction between probability and mutational efficiency. Understanding the comparison between the series of sub-stages and the single stage is recognizing it as a comparison of mutational efficiency. Believing it to be a comparison of probabilities is a failure to understand the comparison.

In order to understand Darwinian evolution, every high school student must know the basic arithmetic involved. The following example illustrates the mathematical explanation of Darwinian evolution by evolutionary biologist, Richard Dawkins (The God Delusion, page 121).

The Example

The improbability of the result of the simultaneous flip of a coin, the roll of a die and the random selection of a card from a deck is 99.84%, i.e. 1 – (0.5 x 0.166 x 0.0185). Sequentially flipping a coin, rolling a die and randomly selecting a card breaks up the improbability of 99.84% into three smaller pieces, namely, 50%, which is 1 – 0.5; 83.33%, which is 1 – 0.166 and 98.08%, which is 1 – 0.0185. This is how natural selection works.

The Explanation

“(N)atural selection is a cumulative process which breaks the problem of improbability up into small pieces. Each of the small pieces is slightly improbable, but not prohibitively so. When large numbers of these slightly improbable events are stacked up in series, the end product is very very improbable indeed, improbable enough to be far beyond the reach of chance. It is these end products that form the subjects of the creationist’s wearisomely recycled argument. The creationist completely misses the point, because he (women should once not mind being excluded by the pronoun) insists on treating the genesis of statistical improbability as a single, one-off event. He doesn’t understand the power of accumulation.”

Dawkins illustrated this with his own numerical example of three mutation sites of six mutations each. Taking random mutation as a one-off event affecting all three sites simultaneously, the improbability of the outcome is 99.54% for one random mutation, i.e. 1 – (1/216). Subjecting each site individually to random mutation, the improbability of each of the three stages is 83.33%, i.e. 1 – (1/6), for one random mutation in each sub-stage. In biological evolution a sub-stage is terminated by natural selection. Thereby, natural selection breaks the improbability of 99.54% into three smaller pieces each of which is 83.33%.

The creationist, would claim that there is no change in the improbability. The overall improbability of the series equals the improbability of the single one-off stage. The improbability would be 99.54% in both the single stage affecting all three sites and in the series of three sub-stages, each affecting only one site. The creationist mistakenly thinks that the probabilities of a series multiply, thereby yielding the same improbability for the one-off event and for the series. The creationist doesn’t understand the power of accumulation, which applied to a single, one-off event, breaks up the improbability into smaller pieces of improbability.

Coordinately, the power of accumulation must break up the small piece of probability of the single, one-off event into three larger pieces of probability forming the series. The small piece of probability of the single, one-off event is 0.46%, i.e. 1/216, while each of the three larger pieces of probability, into which the small piece is broken, is 16.67%, i.e. 1/6.

According to Dawkins, the creationist thinks the probability of the single, one-off event is equal to the product, not the sum, of a sub-series of probabilities. Thereby, according to Dawkins, the creationist is oblivious to the power of accumulation, i.e. to the power of addition in arithmetic.

Evaluation

In fact it is Dawkins who has the arithmetic wrong. The probabilities of a series are the factors, whose multiplication product is the overall probability of the series. The overall probability of a series is not the sum of the probabilities of a series, nor is the overall improbability of a series the sum of the improbabilities of the series. The overall improbability is not broken up into smaller pieces, which are united by accumulation, i.e. by summation. Neither is the overall probability broken up into larger pieces, which are united by accumulation, i.e. by summation.

Dawkins’ confusion of the arithmetical operations of addition and multiplication, leads him to the false belief that sub-staging in Darwinian evolution increases the probability of evolutionary success. It also blinds Dawkins to what natural selection truly accomplishes through sub-stages. Through sub-staging, natural selection increases the efficiency of random mutation.

In the case of one random mutation per sub-stage the probability of evolutionary success per sub-stage is 16.67%. Overall the probability of success is 0.46%, while the total number of mutations for the three stages is three. To achieve this same probability of success, i.e. 0.46%, the single stage requires only one mutation. At the 0.46% level of success, the single stage is more efficient in the number of mutations by a factor of three. However, at higher levels of evolutionary success, this quickly changes, resulting in greater mutational efficiency for the series of sub-stages.

At two random mutations per sub-stage for a total of six, the probability of success overall is 2.85%, while for that level of success, the single stage also requires six mutations by rounding down from a calculated 6.23 mutations. At three random mutations per sub-stage for a total of nine, the probability of success overall is 7.48%, while for that level of success, the single stage requires sixteen random mutations by rounding down from a calculated 16.75 mutations.

Mutational efficiency in favor of sub-stages increases the higher the level of evolutionary success. Eighteen random mutations per sub-stage for a total of fifty-four random mutations, yield an overall probability of evolutionary success of 89.15%. To achieve the 89.15% level of probability of success, the single, one-off stage requires 478 random mutations.

For levels of 0.46%, 2.85%, 7.48% and 89.15% of the probability of Darwinian evolutionary success, the efficiency factor for random mutations in favor of the series of sub-stages goes from less than one, namely 1/3, to 6/6 to 16/9 to 478/54. That last ratio is an efficiency factor of 8.85.

By confusing multiplication and addition, Dawkins fails to understand the role of sub-stages in Darwinian evolution. Sub-staging has no effect on the probability of evolutionary success. Rather, it increases the efficiency of random mutation in Darwinian evolution.

Note

The probability, P, of Darwinian evolutionary success for a stage of random mutation and natural selection is a function of n, the total number of different mutations defined by a stage, and of x, the number of random mutations occurring in that stage. P = 1 – ((n – 1)/n)^x. The probability of a series is the product of the probabilities of the stages in the series.

This essay is presented from the perspective of the philosophy of probability.

It is of the nature of a horse to have two eyes. Such is the ancient and naive ‘explanation’. However, that is not an explanation or even an answer. At best, it is simply a statement of the observation, which prompts the question. At worst, it is a copout, which evades the question.

More recent observations are that predatory animals have two eyes in a plane enabling binary vision. In contrast, animals, like the horse, which are prey to others, have eye placement approximately in two planes, which are the two sides of their heads. Such placement affords nearly a complete view of the sphere from which attack by a predator may occur.

In prey, the placement of the eyes approximately forms the axis of a globe of visual sensation, over which the eyes are uniformly distributed. In the case of fewest eyes, namely two, the eyes are at the axis of the globe. The axis may be viewed as the diameter of one longitudinal circumference.

The more eyes distributed uniformly over the globe of visual sensation, the more complete and uniform would be the monitoring of the sphere of predatory attack. Let the algorithm of adding uniformly distributed eyes to the globe of visual sensation be by adding equally spaced longitudinal circumferences, where each circumference has a number of eyes equaling twice the number of circumferences, while maintaining the location of the two original eyes at the axis of the globe.

By this algorithm, the relationship between the number of equally spaced eyes, N, and the number of longitudinal circumferences, n, would be: N = (n^2 – (n -1)) × 2. For n = 1, 2, 3 and 4, the number of uniformly distributed eyes, N, is 2, 6, 14 and 26. The practical upper limit would be determined by the size of the eyes and the size of a horse’s head. Also, the practical number of eyes would be diminished by those of the virtual globe of visual sensation which would be eliminated due to the neck of the horse.

Why do horses have two eyes? Horses have two eyes as a uniform distribution of visual sensors forming a global base for monitoring an attack from any point of the sphere of predation.

It should be apparent that the fact that animals of prey in the scope of human observation, such as the horse, have just two eyes is one possibility among several. It happens that in our universe the number is two. However, the fact that our observation is limited to our earth in our universe voids the rationale, which claims that it is of some fundamental character of a horse that it has two eyes. Indeed, it has two eyes in our universe, but horses may have any range of a number of eyes, notably greater than two, in other regions of the multiverse. It is the multiverse which explains the fact that we observe within our universe just one of many possibilities, where in accord with probability, the number of eyes of earth horses is two.

It is evident that the existence of the multiverse in cosmology is not a consequence solely of the science of physics and the numerical values of the physical constants in our universe. The multiverse is harmonious with biology as well and with what seems like a simple question, ‘Why do horses have two eyes?’

Gradualism in Darwinian evolution is identified as the replacement of a conceptual overall cycle of random mutation and natural selection with an actual, gradual series of sub-cycles. It is assumed that the series of sub-cycles randomly generates all the mutations defined by the conceptual overall cycle, but in stages rather than in one gigantic evolutionary cycle. The gigantic set of mutations is itself a graded set, not of sub-cycles, but of mutations.

The mutations defined in each sub-cycle is a subset of the graded set of mutations defined by the overall cycle. Each sub-cycle consists of subjecting its subset of mutations to random generation and the resulting pool of random mutations to natural selection.

The gradualism of sub-cycles is often taken to be synonymous with the gradualism represented by the entire graduated sequence of mutations defined by the associated conceptual overall cycle of evolution.

Everyone agrees that replacing a single cycle of Darwinian evolution by a series of sub-cycles yields a series of sub-cycles, each of which has a probability of evolutionary success greater than the probability of success overall. This is simple arithmetic. The product of a series of factors, each a fraction of 1, is less than any of its factors.

However, proponents of Intelligent Design claim that there are some biological structures that cannot be assembled gradually in a series of subsets because survival in the face of natural selection requires the full functionality of the surviving mutant in each subset.

There are in fact two distinct Intelligent Design arguments against Neo-Darwinism. One argument is entitled, irreducible complexity. The other argument is the argument of gradualism presented by Stephen Meyer (Ref. 1). Both of the Intelligent Design arguments cite complex biological structures such as the ‘motor assembly’ of the bacterial flagellum. In opposition to Neo-Darwinism, the Intelligent Design arguments claim that it is the integrity of the assembled unit that confers functionality and thereby survivability when subjected to natural selection. Those mutants, which have partial assemblies, have no functionality and therefore no survivability based on functionality.

Intelligent Design’s Irreducible Complexity Argument

This argument acknowledges that the gradualism of sub-cycles increases the probability of evolutionary success in terms of the probability of the individual sub-cycle. However, it is argued that the integrity of a cited biological assembly requires a single cycle of such a size that it thereby has a level of probability too low to be acceptable. The assembly is not reducible, by a series of sub-cycles, to a lower level of complexity without sacrificing survivability. The lower the complexity, the greater the probability of evolutionary success. Thus, the level of complexity is irreducible by means of sub-cycles to a low level of complexity, which would raise the probability of each sub-cycle to an acceptably high level of probability. According to this argument, Darwinian evolution fails on the basis of probability.

Stephen Meyer’s Intelligent Design argument (Ref. 1) ignores the numeric values of probability and an alleged value of probability above which probability becomes large enough to serve as an explanation. Rather, the argument concentrates on the proposition that the gradualism of Darwinian evolution requires the actual generation of the entire, graduated spectrum of mutations. If there is a sub-sequence of graduated mutations, which have no functionality and therefore have no survivable utility, then the terminal of this sub-sequence could never be generated. The ‘motor assembly’ of the bacterial flagellum is cited as one example. Consequently, the evolution of such a terminal mutation is incompatible with gradualism, which is a key characteristic of Darwinian evolution.

According to this argument Darwinian evolution fails on two grounds with respect to gradualism. (1) The complete biological assembly, in the cited instances, cannot be assembled gradually because the necessary precursors, namely incomplete biological assemblies, would not pass the test of natural selection to which each precursor would be subjected in its sub-stage of gradualism. (2) The fossil record has gaps in the spectrum of mutations, which gaps are incompatible with gradualism.

Critique of Both Irreducible Complexity and Darwinian Evolution with Respect to Probability

Probability is defined over the range of 0 to 1. There is no logical basis for dividing this continuous range of definition into two segments: one segment from 0 up to an arbitrary point, where probability is too small to serve as an explanation; a second segment from that point through 1, where probability is numerically large enough to serve as an explanation.

The Irreducible Complexity Argument claims there are cycles of evolution for which the probability is in the segment near zero. Because these cycles cannot be replaced by sub-cycles, the gradualism required by evolution cannot be achieved. The Neo-Darwinian response is that there are no such cycles. Any cycle, which due to its size would have too low a probability, superficially may appear to be indivisible into sub-cycles, but is not so in fact.

Both Irreducible Complexity and Darwinian evolution claim that it is the replacement of a cycle by a series of sub-cycles which solves the ‘problem of improbability’ (Ref. 2).

Granted that each probability of a series of probabilities is larger than the overall or net probability, the net probability remains constant. The net probability is the product of the probabilities in the series. Consequently, it is nonsensical to say ‘natural selection is a cumulative process, which breaks the problem of improbability up into small pieces. Each of the small pieces is slightly improbable, but not prohibitively so’ (Ref. 2). The individual probabilities of the sub-cycles in the series do not change the net probability of evolution.

Critique of Both the Intelligent Design and the Darwinian Evolution Claims of Gradualism

Meyer’s Intelligent Design argument agrees with Neo-Darwinism that the gradualism of sub-cycles ensures the generation of the entire spectrum of graded mutations defined by the overall cycle of evolution (Refs. 1 and 3). To what they agree is false. The fact is that role of sub-cycles is to increase the efficiency of mutation by eliminating the possibility of the generation of most of the graded mutations in the defined spectrum. Although he misinterpreted what he was demonstrating, Dawkins did an admirable job of demonstrating this efficiency (Ref. 4).

In Ref. 4, Dawkins used an example of three mutation sites of six mutations each to illustrate the efficiency of sub-cycles, where efficiency is achieved by eliminating the possibility of the generation of most intermediate evolutionary forms. The excellence of his illustration of increased mutational efficiency is not vitiated by the fact that Dawkins mistakenly thought he was illustrating an increase in the probability of evolutionary success by natural selection. The net probability of success is unaffected by the introduction of sub-cycles (Ref. 5).

Three mutation sites of six mutations each defines 216 different graded mutations, i.e. 6 x 6 x 6 = 216. These mutations are the two end points and 214 intermediates. Let the 216 mutations of the graded spectrum be designated 000 to 555.

In a single cycle of Darwinian evolution, all 216 different mutations are liable to be generated randomly. In Dawkins’ illustration of gradualism, a series of three sub-cycles replaces the single overall cycle. Each of the three sub-cycles in the series subjects only one site out of the three to random generation and natural selection, independently of the other two sites. This entails only six different mutations per sub-cycle. In the first sub-cycle, the six mutations are between 00’0′ and 00’5′ inclusively. The six mutations of the second sub-cycle are between 0’0’5 and 0’5’5 inclusively. The six mutations of the third sub-cycle are between ‘0’55 and ‘5’55 inclusively.

Although there are six possible different mutations per sub-cycle, in the second sub-cycle mutation 005 is a duplicate of 005 of the first sub-cycle. In the third sub-cycle mutation 055 is a duplicate of 055 of the second sub-cycle. That yields only 16 different mutations in total, which are liable to random generation, not 18.

In the single overall cycle there are no missing links or missing gaps in the spectrum of 216 mutations which are liable to random mutation. These are 000 to 555, for a total of 216.

In the first sub-cycle of the illustration, all six graded mutations are liable to be randomly generated, i.e. 00’0′ to 00’5’. In the second sub-cycle the six mutations liable to be randomly generated are separated by gaps in the graded spectrum. The gaps are of 5 mutations each. The six mutations which can be generated in the second sub-cycle are 0’0’5, 0’1’5, 0’2’5, 0’3’5, 0’4’5 and 0’5’5. The first gap comprises the five mutations between 0’0’5 and 0’1’5. These are 010, 011, 012, 013 and 014. There are 5 gaps of 5 mutations each, for a total of 25 mutations of the overall spectrum which cannot be generated in the second sub-cycle due to the gradualism of sub-cycles.

In the third sub-cycle of the illustration, the six mutations which are liable to be randomly generated are ‘0’55, ‘1’55, ‘2’55, ‘3’55, ‘4’55 and ‘5’55. Between each of these mutations there is a gap of 35 graded mutations which cannot be generated due to the gradualism of sub-cycles. For the first gap, the 35 are 100 to 154, inclusive. The total of different graded mutations, which cannot be generated in the third sub-cycle, is 35 x 5 = 175.

The totals for the three sub-cycles of different mutations are: Sub-cycle one: 6 mutations possibly generated, 0 mutations in non-generated gaps; Sub-cycle two: 5 mutations possibly generated, 25 mutations in non-generated gaps; Sub-cycle three: 5 mutations possibly generated, 175 mutations in non-generated gaps. Totals for the three sub-cycles: 16 mutations possibly generated, 200 mutations in non-generated gaps.

For a critique of gradualism from the perspective of probabilities see Refs. 5 and 6.

Conclusion

Both its proponents (Ref. 3) and its critics (Ref. 1) assume that a key characteristic of Darwinian evolution is the generation of a complete spectrum of graded mutations. This shared view assumes that the generation of all mutations in this spectrum is facilitated by the gradualism of a series of sub-cycles of random mutation and natural selection. This is false. The Darwinian algorithm of random mutation and natural selection, applied in series, ensures that most of the mutations, defined by the overall graded spectrum, cannot be generated. The role of sub-staging in Darwinian evolution is the increased efficiency of mutation due to the non-generation of most of the mutations comprising the defined graded spectrum. This results in huge gaps in the spectrum of mutations actually generated.

To the typical bystander (Ref. 7), the debate between Intelligent Design and Neo-Darwinism appears to be one of science vs. science or, as the Dover Court ruled, faith vs. science. In fact, the arguments of both sides are based on their mutual misunderstanding of the arithmetical algorithm, which is Darwinian evolution.

References

Note: The single quote marks are used simply to highlight the mutation site in question.

It is perfectly acceptable in thought to characterize material processes as mathematically random. For example, the roll of two dice is characterized as random such that their sum of seven is said to have a probability of 1/6. The equation of radioactive decay may be characterized as the probability function of the decay of a radioactive element. Wave equations in quantum mechanics may be characterized as probability functions. However, such valid mathematical characterizations do not attest to randomness and probability as being characteristics of material reality. Rather, such characterizations attest to mathematical randomness and probability as being characteristic of human knowledge in its limitations.

If randomness and probability were characteristic of material reality at any level, including the atomic and sub-atomic level, material reality would be inherently unintelligible in itself. Material reality would be inexplicable and as such inherently mysterious. Yet, to view material reality as inherently mysterious is superstition. Superstition denies causality by claiming that material results are a mystery in themselves, e.g. that they are materially random.

It is an erroneous interpretation to hold that quantum mechanics requires material reality to be random and probable in itself. Wave equations may be viewed as probability functions only in the same sense that the result of rolling dice is mathematically probable. That sense is in the suspension of human knowledge of material causality at the level of physical forces for the sake of a mathematical utility without the denial of material causality.

A commenter on a recent post at catholicstand.com (ref. 1), was so enamored with the validity, utility and beauty of the mathematics of quantum mechanics that he declared, “This randomness is inherent in nature.” Indeed it is inherent in nature, i.e. in human nature in the limitations of the human knowledge of the measurable properties of material reality.

Material reality is not random in its nature. The nature of material reality, in light of the utility of application of the mathematics of probability or in light of perceiving a mathematical function as one of probability, is not a question within the scope of science or mathematics. The nature of material reality is always a question within philosophy. In contrast, the mathematical and scientific question is the suitability of specific mathematics in representing the relationships among the measurable properties of material reality including those properties, which can only be detected and measured with instruments.

Let it be noted that scientific knowledge cannot demonstrate the fundamental invalidity of human sensory knowledge and human intellectual knowledge because the validity of scientific knowledge depends on the fundamental validity of these.

It has been recognized since the time of Aristotle that the human intellect is extrinsically dependent in its activity upon a sensual phantasm, i.e. a composite of sense knowledge. This and all visualizations or imaginative representations are necessarily restricted to the scope of the senses, although the intellect is not. Consequently, science at the atomic and sub-atomic level cannot consist in an analysis of visual or imaginative simulations, which are confined to the scope of human sensation. Rather, the science consists in the mathematics, which identifies quantitative relationships among instrumental measurements. It would be a fool’s quest to attempt to determine a one to one correspondence between science and an imaginative representation of the atomic and sub-atomic level or to constrain the understanding of the science to such a representation (Ref. 2).

Remarkably, in an analogy of a wave function in quantum mechanics as a probability function which collapses into a quantum result, the physicist, Stephan M. Barr, did not choose an example of mathematical probability (Ref. 3). He could have proposed an analogy of mathematical probability simulated by flipping a coin. When the coin is rotating in the air due to being flipped it could be viewed as a probability function of heads of 50%, which collapses into a quantum result of heads, namely one, or tails, namely zero, upon coming to rest on the ground.

Instead, he chose an example where the meaning of probability is not mathematical, but qualitative.

Mathematical probability is the fractional concentration of an element in a logical set, e.g. heads has a fractional concentration of 50% in the logical set of two logical elements with the nominal identities of heads and tails. A coin is material simile.

A completely unrelated meaning of the word, probability, is an individual’s personal lack of certitude of the truth of a statement. Examples: ‘I probably had eggs for breakfast in the past two weeks’ or ‘Jane will probably pass the French exam.’ These statements identify no set of elements or anything quantitative. Personal human certitude is qualitative. Yet, we are bent upon quantitatively rating the certitude with which we hold our personal judgments.

Barr succumbs to this penchant for quantifying personal certitude. He illustrates the collapse of a wave function in quantum mechanics with the seemingly objective quantitative statement:
“This is where the problem begins. It is a paradoxical (but entirely logical) fact that a probability only makes sense if it is the probability of something definite. For example, to say that Jane has a 70% chance of passing the French exam only means something if at some point she takes the exam and gets a definite grade. At that point, the probability of her passing no longer remains 70%, but suddenly jumps to 100% (if she passes) or 0% (if she fails). In other words, probabilities of events that lie in between 0 and 100% must at some point jump to 0 or 100% or else they meant nothing in the first place.”
Barr mistakenly thinks that probability, whether referring either to mathematics or to human certitude, refers to coming into existence, to happening. In fact, both meanings are purely static. The one refers to the composition of mathematical sets, although its jargon may imply occurrence or outcome. The other refers to one’s opinion of the truth of a statement, which may be predictive. That Jane has a 70% chance or will probably pass the French exam obviously expresses the certitude of some human’s opinion, which has no objective measurement even if arrived at by some arbitrary algorithm.

Probability in mathematics is quantitative, but static. It is the fractional composition of logical sets. Probability in the sense of human certitude, like justice, is a quality. It cannot be measured because it is not material. This, however, does not diminish our penchant for quantifying everything (Ref. 4).

Barr’s identification of probability, as potential prior to its transition to actuality in an outcome, is due to taking the jargon of the mathematics of sets for the mathematics of sets itself. We say that the outcome of flipping a coin had a probability of 50% heads prior to flipping, which results in an outcome or actuality of 100% or 0%. What we mean to illustrate by such a material simulation is a purely static relationship involving the fractional concentration of the elements of logical sets. The result of the coin flip illustrates the formation or definition of a population of new sets of elements based on a source set of elements. In this case the source set is a set of two elements of different IDs. The newly defined population of sets consists of one set identical to the original set, or, if you wish, a population of any multiple of such sets.

Another illustration is defining a population of sets of three elements each, based on the probabilities of a source set of two elements of different nominal IDs, such as A and B. The population is identified by eight sets. One set is a set of three elements, A, at a probability (fractional concentration) of 12.5% in the newly defined population of sets. One set is a set of three elements, B, at a probability of 12.5%. Three sets are of two elements A and one element B, at a probability of 37.5%. Three sets are of two elements B and one element A, at a probability of 37.5%. The relationships are purely static. We may imagine the sets as being built by flipping a coin. Indeed, we use such jargon in discussing the mathematics of the relationship of sets. The flipping of a coin in the ‘building’ of the population of eight sets, or multiples thereof, is a material simulation of the purely logical concept of random selection. Random selection is the algorithm for defining the fractional concentrations of the population of eight new sets based on the probabilities of the source set. It is only jargon, satisfying to our sensual imagination, in which the definitions of the eight new sets in terms of the fractional concentration of their elements are viewed as involving a transition from potency to act or probability to outcome. The mathematics, in contrast to the analogical imaginative aid, is the logic of static, quantitative relationships among the source set and the defined population of eight new sets.

Random selection, or random mutation, is not a material process. It is a logical concept within an algorithm, which defines a logical population of sets based on the probabilities of a logical source set.

It is a serious error to conflate mathematical probability with the certitude of human judgment. It is also a serious error to believe that either refers to coming into existence or to the transition from potency to act, which are subjects of philosophical inquiry.

Ref. 1 “When Randomness Becomes Superstition” http://catholicstand.com/randomness-becomes-superstition/

Ref. 2 “Random or Non-random, Math Option or Natural Principle?” https://theyhavenowine.wordpress.com/2014/08/24/random-or-non-random-math-option-or-natural-principle/

Ref. 3 “Does Quantum Physics Make It Easier to Believe in God?” https://www.bigquestionsonline.com/content/does-quantum-physics-make-it-easier-believe-god

Ref. 4 “The Love of Quantification” https://theyhavenowine.wordpress.com/2013/08/11/the-love-of-quantification-2/

Richard Dawkins has extensively discussed arithmetic. The theme of The God Delusion is that there is an arithmetical solution to the improbability of evolution in a one-off event, namely gradualism, whereas there is no arithmetical solution to the improbability of God. Obviously, the ‘improbability’ of God cannot be solved by gradualism.

It is encouraging that Richard Dawkins is interested in mathematics. If he were to learn to correct his mistakes in math, he might do very well in re-educating those whom he has deceived in mathematics, science and philosophy due to his errors in arithmetic.

The following is a math quiz based on problems in arithmetic addressed by Richard Dawkins and his answers, whether implicit or explicit, in his public work. I present this as a helpful perspective in the delineation of Dawkins’ public errors in arithmetic.

1) What is the opposite of +1?
Student Dawkins: Zero. Let us then take the idea of a spectrum of probabilities seriously between extremes of opposite certainty. The spectrum is continuous, but can be represented by the following seven milestones along the way:
Strong positive, 100%; Short of 100%; Higher than 50%; 50%; Lower than 50%; Short of zero; Strong negative, 0% (p 50-51. Ref 1).
Those, who aver that we cannot say anything about the truth of the statement, should refuse to place themselves anywhere in the spectrum of probabilities, i.e. of certitude. (p 51, Ref 1)
On page 51 of The God Delusion, Dawkins devotes a paragraph to discussing the fact that his spectrum of certitude, from positive to its negative opposite, does not accommodate ‘no opinion’. Yet, he fails to recognize what went so wrong that there is no place in his spectrum for ‘no opinion’. The reason, that there is no place, is that he has identified a negative opinion as zero, rather than as -1. If he had identified a negative opinion as -1, which is the opposite of his +1 for a positive opinion, then ‘no opinion’ would have had a place in his spectrum of certitude at its midpoint of zero. Instead Dawkins discusses the distinction between temporarily true zeros in practice and permanently true zeros in principle, neither of which are accommodated by his spectrum ‘between two extremes of opposite certainty’ in which the opposite extreme of positive is not negative, but a false zero.

2) Is probability in the sense of rating one’s personal certitude of the truth of a statement a synonym for mathematical probability, which is the fractional concentration of an element in a logical set? For example, is probability used univocally in these two concepts: (a) The probability of a hurricane’s assembling a Boeing 747 while sweeping through a scrapyard. (b) The probability of a multiple of three in the set of integers, one through six?
Correct Answer: No. The probability of a hurricane’s assembling a Boeing 747 does not identify, even implicitly, a set of elements, one of which is the assembling of a Boeing 747. Probability as a numerical rating of one’s personal certitude of the truth of such a proposition has nothing to do with mathematical probability. In contrast to the certitude of one’s opinion about the capacity of hurricanes to assemble 747’s, the probability of a multiple of 3 in the set of integers, one through six, namely 1/3, is entirely objective.
Student Dawkins: Probability as the rating of one’s personal certitude of the truth of a proposition, has the same spectrum of definition as mathematical probability, namely 0 to +1 (p 50, Ref. 1). The odds against assembling a fully functioning horse, beetle or ostrich by randomly shuffling its parts are up there in the 747 territory of the chance that a hurricane, sweeping through a scrapyard would have the luck to assemble a Boeing 747. (p 113, Ref. 1) There is only one meaning of probability, whether it is the probability of the existence of God, the probability of a hurricane’s assembling a Boeing 747, the probability of success of Darwinian evolution based on the random generation of mutations or the probability of seven among the mutations of the sum of the paired output of two generators of random numbers to the base, six.

3) In arithmetic, is there a distinction between the factors of a product and the parts of a sum? Is the probability of a series of probabilities, the product or the sum of the probabilities of the series?
Correct Answers: Yes; The product. The individual probabilities of the series are its factors.
Student Dawkins: Yes; The relationship of a series of probabilities is more easily assessed from the perspective of improbability. An improbability can be broken up into smaller pieces of improbability. Those, who insist that the probability of a series is the product of the probabilities of the series, don’t understand the power of accumulation. Only an obscurantist would point out that if a large piece of improbability can be broken up into smaller pieces of improbability as the parts of a sum, i.e. as parts of an accumulation, then it must be true that its complement, the corresponding small piece of probability, is concomitantly broken up into larger pieces of probability, where the larger pieces of probability are the parts, whose sum (accumulation) equals the small piece of probability. (p 121, Ref. 1).

4) Jack and Jill go to a carnival. They view one gambling stand where for one dollar, the gambler can punch out one dot of a 100 dot card, where each dot is a hidden number from 00 to 99. The 100 numbers are randomly distributed among the dots of each card. If the gambler punches out the dot containing 00, he wins a kewpie doll. Later they view another stand where for one dollar, the gambler gets one red and one blue card, each with 10 dots. The hidden numbers 0 to 9 are randomly distributed among the dots of each card. If in one punch of each card, the gambler punches out red 0 and blue 0, he wins a kewpie doll. This second stand has an interesting twist, lacking in the first stand. A gambler, of course, may buy as many sets of one red card and one blue card as he pleases at one dollar per set. However, he need not pair up the cards to win a kewpie doll until after he punches all of the cards and examines the results.
(a) If a gambler buys one card from stand one and one pair of cards from stand two, what are his respective probabilities of winning a kewpie doll?
Correct Answer: The probability is 1/100 for both.
Student Dawkins: The probability of winning in one try at the first stand is 1/100. At the second stand the probability of winning is smeared out into probabilities of 1/10 for each of the two tries.
(b) How many dollars’ worth of cards must a gambler buy from each stand to reach a level of probability of roughly 50% that he will win at least one kewpie doll?
Correct Answers: \$69 worth or 69 cards from stand one yields a probability of 50.0%. \$12 worth or 24 cards (12 sets) from stand two yields a probability of 51.5%. (A probability closer to 50% for the second stand is not conveniently defined.)
Student Dawkins: A maximum of \$50 and 50 cards from stand one yields a probability of 50%. A maximum of \$49 and 49 cards from stand one yields a probability of 49%.A maximum of \$14 and 28 cards yields a probability of 49%. (A probability closer to 50% for the second stand is not conveniently defined.)
(c) In the case described in (b), is the probability of winning greater at stand two?
Student Dawkins: Yes
(d) In the case described in (b), is winning more efficient or less efficient in terms of dollars and in terms of total cards at the second carnival stand?
Correct Answer: More efficient. The second stand is based on two sub-stages of Darwinian evolution compared to the first stand, which is based on one overall stage of Darwinian evolution. The gradualism of sub-stages is more efficient in the number of random mutations while having no effect on the probability of evolutionary success. Efficiency is seen in the lower input of \$12 or 24 random mutations compared to \$69 or 69 random mutations to produce the same output, namely the probability of success of roughly 50%.
Student Dawkins: Efficiency is irrelevant. It’s all about probability. The gradualism of stand two breaks up the improbability of stand one into smaller pieces of improbability. (p 121, Ref. 1)
This problem is an illustration of two mutation sites of ten mutations each. I analyzed these relationships in Ref. 2, using an illustration of three mutation sites of six mutations each. In that illustration, I introduced two other modifications. One modification was that the winning number was unknown to the gambler. The other was that the gambler could choose the specific numbers on which to bet, so his tries or mutations were non-random. With the latter deviation from the Darwinian algorithm, the probability of winning a kewpie doll required a maximum of 216 non-random tries for the first stand and a maximum of 18 non-random tries for the second stand. The gradualism of the second stand smears out the luck required by the first stand. The increased probability of winning a kewpie doll at the second stand is due to the fact that one need not get his luck in one big dollop, as one does at the first stand. He can get it in dribs and drabs. It takes, respectively at the two stands, maxima of 216 and 18 tries for a probability of 100% of winning a kewpie doll. Consequently and respectively, it would take maxima of 125 and 15 tries to achieve a probability of 57.9% of winning a kewpie doll. Whether one compares 216 tries for the first stand to 18 tries for the second stand or 125 tries to 15 tries, the probability of winning a kewpie doll is greater at the second stand because it takes fewer tries. (See also the Wikipedia explanation, which is in agreement with Student Dawkins, Ref. 3)
Another example of extreme improbability is the combination lock of a bank vault. A bank robber could get lucky and hit upon the combination by chance. In practice the lock is designed with enough improbability to make this tantamount to impossible. But imagine a poorly designed lock. When each dial approaches its correct setting the vault door opens another chink. The burglar would home in on the jackpot in no time. ‘In no time’ indicates greater probability than that of his opening the well-designed lock. Any distinction between probability of success and efficiency in time is irrelevant. Also, any distinction between the probability of success and efficiency in tries, whether the tries are random mutations or non-random mutations is irrelevant. (p 122. Ref. 1)

5) If packages of 4 marbles each are based on a density of 1 blue marble per 2 marbles, how many blue marbles will a package selected at random contain?
Student Dawkins: 2 blue marbles.

6) If packages of 4 marbles each are based on a probability of blue marbles of 1/2, how many blue marbles will a package selected at random contain?
Correct Answer: Any number from 0 to 4 blue marbles.
Student Dawkins: 2 blue marbles. This conclusion is so surprising, I’ll say it again: 2 blue marbles. My calculation would predict that with the odds of success at 1 to 1, each package of 4 marbles would contain 2 blue marbles. (p 138, Ref. 1)

References:
1. The God Delusion
3. http://en.wikipedia.org/wiki/Growing_Up_in_the_Universe#Part_3:_Climbing_Mount_Improbable

Calculations:
Where the total number of generated mutations, x, is random, and n is the number of different mutations, the Probability of success, P, equals 1- ((n – 1)/n)^x.
For n = 100 and x = 69, P = 50.0%
For n = 10 and x = 12, P = 71.7%. For P^2 = 51.5%, the sum of x = 24
Where the total number of generated mutations, x, is non-random, and n is the number of different mutations, the Probability of success, P, equals x/n
For n = 100 and x = 50, P = 50%
For n = 100 and x = 49, P = 49%
For n = 10 and x = 7, P = 70.0%. For P^2 = 49%, the sum of x = 14
For n = 216 and x = 216, P = 100%
For n = 6 and x = 6, P = 100%. For P^3 = 100%, the sum of x = 18
For n = 216 and x = 125, P = 57.9%
For n = 6 and x = 5, P = 83.3%. For P ^3 = 57.9%, the sum of x = 15