home | what's new | other sitescontact | about



Word Gems 

exploring self-realization, sacred personhood, and full humanity


Not Enough Time 



return to "Evolution Controversy" contents page


Editor's note:

Evolutionists speak in terms of “given enough time” probability and chance will explain the appearance of life on Earth. There is a major problem, however, with this notion once we look carefully at the mathematics involved.

There is not enough time in the entire history of the universe, nor in 10 universes, or 100, or 1000 universes, to “make pigs fly” in evolutionary theory. Probability and chance are dead in the water for those who understand the math involved.

Below, you will find many quotations and excerpts from articles addressing this issue.




Dr. Amit Goswami, Creative Evolution: “Can geographical isolation explain the fossil gaps in all cases of speciation [the change of one species into another]? No, it cannot. It certainly is a viable explanation of speciation for what we call microevolution, the evolution of simple organisms involving only a few genetic changes. However, it cannot explain macroevolution, or evolution of new species involving a large number of genetic changes. The reason is subtle… Probability calculations alone preclude Darwinism’s ability to explain all evolution, whether micro or macro. I have cited biologist Robert Shapiro’s (1986) work… Shapiro showed that the maximum number of chance events available over a billion years of evolution is 2.5 x 1051. The astrophysicist Arne Wyller (2003), on the basis of very conservative assumptions, deduced that to create the billion multicellular species that have ever existed on Earth until now (according to Harvard biologist Richard Lewontin) requires more than 101000000000000 chance events. This figure is obviously far, far, far greater than the number of chance events available as calculated by Shapiro… Mere chance and necessity cannot do it all. Biologists must face up to this.”

Dr. Amit Goswami, Creative Evolution: “An argument based on probability calculations shows that the manufacture of even a relatively small protein enzyme would take a time much longer than the age of the universe.”

"...life cannot have had a random beginning...The trouble is that there are about two thousand enzymes, and the chance of obtaining them all in a random trial is only one part in 10 to the 40,000 power, an outrageously small probability that could not be faced even if the whole universe consisted of organic soup. If one is not prejudiced either by social beliefs or by a scientific training into the conviction that life originated on the Earth, this simple calculation wipes the idea entirely out of court....The enormous information content of even the simplest living systems...cannot in our view be generated by what are often called "natural" processes...For life to have originated on the Earth it would be necessary that quite explicit instruction should have been provided for its assembly...There is no way in which we can expect to avoid the need for information, no way in which we can simply get by with a bigger and better organic soup, as we ourselves hoped might be possible a year or two ago." Fred Hoyle and N. Chandra Wickramasinghe, Evolution From Space

“How can one gain some conception of the size of such a huge number? [10 to the 40,000 power] According to most Evolutionists, the universe is less than 30 billion years old -- and there are fewer than 10 to the 18th power seconds in 30 billion years. So, even if nature could somehow have produced trillions of genetic code combinations every second for 30 billion years, the probabilities against producing the simplest one-celled animal by trial and error would still be inconceivably immense!” What are the Odds of Life evolving by chance alone? Alan McDougall

“The 1/10 40,000th probability actually refers to the chance of obtaining the required set of enzymes for the simplest living cell. How was this probability derived? We know that amino acids are the building blocks of enzymes, so what part of the probability that you give is the chance of obtaining the required amino acids for those enzymes? The researchers probably calculated the odds for a limited set of molecules to combine to the desired configurations. What they fail to take into account is that there are an exponential number of molecules performing exponential random combinations over billions of years...some of which will combine to the desired configuration. Combine those together and the odds shorten to the extent that it obviously becomes a certainty judging by the fact that DNA, enzymes et al are extant.” Alan McDougall

Chemist Dr. Grebe: "That organic evolution could account for the complex forms of life in the past and the present has long since been abandoned by men who grasp the importance of the DNA genetic code."

the whole evolution debate should have come to a screeching halt; but, it didn't - because with some the facts don't matter

Mathematician I.L Cohen: "At that moment, when the [complexity of the] DNA/RNA system became understood, the debate between Evolutionists and Creationists should have come to a screeching halt...the implications of the DNA/RNA were obvious and clear....Mathematically speaking, based on probability concepts, there is no possibility that [Darwinian] Evolution [was] the mechanism that created the approximately 6,000,000 species of plants and animals we recognize today."

Evolutionist Michael Denton: "The complexity of the simplest known type of cell is so great that it is impossible to accept that such an object could have been thrown together suddenly by some kind of freakish, vastly improbable, event. Such an occurrence would be indistinguishable from a miracle."




God and the Laws of Science: The Laws of Probability


Jeff Miller, Ph.D.


A typical misconception about science is that it can tell us what will definitely happen now or in the future given enough time, or what would certainly have happened in the past, given enough time. The truth is, science is limited in that it does not grant absolute truth, but only yields degrees of probability or likelihood. Science observes the Universe, records evidence, and strives to draw conclusions about what has happened in the past, is happening now, and what will potentially happen in the future, given the current state of scientific knowledge—which is often times woefully incomplete, and even inaccurate. The late, prominent evolutionist George Gaylord Simpson discussed the nature of science and probability several years ago in the classic textbook, Life: An Introduction to Biology, stating:

We speak in terms of “acceptance,” “confidence,” and “probability,” not “proof.” If by proof is meant the establishment of eternal and absolute truth, open to no possible exception or modification, then proof has no place in the natural sciences. Alternatively, proof in a natural science, such as biology, must be defined as the attainment of a high degree of confidence (Simpson and Beck, 1965, p. 16, emp. added).

In other words, science observes and attempts to answer for mankind such things as: what could have happened in the past; what most likely happened; what is probably happening now; what could happen in the future; or what will likely happen in the future. Science does not necessarily tell us what will certainly always be or has always been the case. Rather, it tells us what has always been observed to be the case and what will almost certainly always be the case, without exception, and which coincides with logic, intuition, and mathematics. When enough evidence is gathered and all that evidence points to some truth and therefore yields an extremely high level of confidence in that truth (i.e., the probability of the same truth always being the case is considered so high that it is beyond doubt), the truth is made a law. Such a step is not taken lightly. Extensive observation must be conducted before doing so. Therefore, the laws of science are highly respected and considered to be essentially beyond doubt. However, there is always the slightest potential that a law could be broken in the future by some unknown event. Thus, probability is intimately intertwined with science. Mark Kac, famous mathematician and professor at Cornell and Rockefeller Universities, said, “Probability is a cornerstone of all the sciences, and its daughter, the science of statistics, enters into all human activities” (as quoted in Smith, 1975, p. 111, emp. added).

Many evolutionists understand the significance of probability in science and yet go too far in their use of the laws of probability, presumptuously claiming that they can do more than they profess to do. These assert that anything—no matter how far-fetched—will inevitably happen, given enough time, as long as it does not have a probability of zero. Supposedly, objects will pop into existence, and eventually, those things will come to life and transform into humans. Many evolutionists have long cited the principles of probability in an effort to support such unscientific dogmas (e.g., Erwin, 2000). As far back as 1954, George Wald, writing in Scientific American concerning the origin of life on Earth, penned the words:

However improbable we regard this event, or any of the steps it involves, given enough time, it will almost certainly happen at least once. And for life as we know it, once may be enough. Time is the hero of the plot…. Given so much time, the “impossible” becomes possible, the possible becomes probable, and the probable becomes virtually certain. One has only to wait; time itself performs miracles (Wald, p. 48, emp. added).


The second problem with the assertion of evolutionary inevitability is implied by the work of the renowned French mathematician, Emile Borel, for whom the lunar crater, Borel, is named (O’Connor and Robertson, 2008). In 1962, Borel discussed in depth the law of probability known as the Single Law of Chance—a law that he said “is extremely simple and intuitively evident, though rationally undemonstrable” (1962, p. 2). This principle states that “events whose probability is extremely small never occur” (1965, p. 57). He further stated that we “at least…must act, in all circumstances, as if they were impossible” (1962, p. 3, italics in orig.). The law, he said, applies to

the sort of event, which, though its impossibility may not be rationally demonstrable, is, however, so unlikely that no sensible person will hesitate to declare it actually impossible. If someone affirmed having observed such an event we would be sure that he is deceiving us or has himself been the victim of a fraud (1962, p. 3, italics in orig., emp. added).

To clarify the meaning of “extremely small” probabilities, he defined different categories of events in which the probabilities are so small that they are “practically negligible,” including events from the human, terrestrial, and cosmic perspectives (1965, p. 57).

In his discussion on the probabilities of certain cosmic events, he argues convincingly from mathematical calculations and intuition that reasonable human beings consider probabilities of chance cosmic events that fall below one in 1045 to be negligible (1965, p. 59). In other words, if the probability of a certain event happening in the Universe is less than one in 1045 (i.e., a one with 45 zeros after it), human beings intuitively categorize that event as so unlikely that we consider it to be an impossible event.

Several years ago, evolutionist Harold Morowitz of Yale, and currently professor of biology and natural philosophy at George Mason University, estimated the probability of the formation of the smallest and simplest living organism to be one in 10340,000,000 (1970, p. 99). A few years following Morowitz’s calculations, the late, renowned evolutionist Carl Sagan made his own estimation of the chance that life could evolve on any given single planet: one in 102,000,000,000 (1973, p. 46)!

Note also that these calculations were made before the last several decades have revealed with even more clarity the complexity of life (cf. Deweese, 2010). These probability estimations for the formation of life, made by the evolutionists themselves, are, of course, so far beyond the limit articulated for cosmic events by the Single Law of Chance that we must respond in shock, rather than humor, at the big lie that has been perpetrated on the world at large by so many in the scientific community in thrusting macroevolution on the masses. The distinguished British astronomer, Sir Fred Hoyle once said regarding evolution, “the chance that higher forms have emerged in this way is comparable with the chance that a tornado sweeping through a junk-yard might assemble a Boeing 747 from the materials therein” (1981b, 294:105). He further stated:

At all events, anyone with even a nodding acquaintance with the Rubik cube will concede the near-impossibility of a solution being obtained by a blind person moving the cubic faces at random. Now imagine 1050 blind persons each with a scrambled Rubik cube, and try to conceive of the chance of them all simultaneously arriving at the solved form. You then have the chance of arriving by random shuffling at just one of the many biopolymers on which life depends. The notion that not only biopolymers but the operating programme of a living cell could be arrived at by chance in a primordial organic soup here on the Earth is evidently nonsense of a high order (1981a, 92:527, emp. in orig.).


Further, even if there were not a probability of zero when it comes to macroevolution, it is important to note as was discussed earlier that probabilities do not guarantee that an event will or will not happen, regardless of how much time is allotted. Sproul, Gerstner, and Lendsley correctly observed:

The fact is, however, we have a no-chance chance creation. We must erase the “1” which appears above the line of the “1” followed by a large number of zeroes. What are the real chances of a universe created by chance? Not a chance. Chance is incapable of creating a single molecule, let alone an entire universe. Why not? Chance is no thing. It is not an entity. It has no being, no power, no force. It can effect nothing for it has no causal power within it, it has no itness to be within. Chance…is a word which describes mathematical possibilities which, by a curious slip of the fallacy of ambiguity, slips into discussion as if it were a real entity with real power, indeed, supreme power, the power of creativity (1984, p. 118, emp. in orig.).

We certainly agree. There is only one causal Power capable of creating the Universe, and there is certainly nothing random about Him.


Recall what Borel said of events prohibited under the Single Law of Chance—that sensible humans “must act, in all circumstances, as if they were impossible” (1962, p. 3, italics in orig.). Unfortunately, so many scientists today do not act sensibly. They do not follow this simple and intuitive truth when it comes to the matter of origins. Rather, they hold to the impossible, pouring thousands of hours and billions of dollars into researching it, writing on it, speaking on it, thrusting it into the minds of people of all ages, and attacking anyone who contradicts them. They, themselves, admit that the spontaneous generation of life from non-life has never been observed and that the odds are shockingly against it, and yet, since they start with the presumptuous assumption that there is no God, they believe the existence of life is proof enough that spontaneous generation occurred. But if the scientific evidence is so strongly against it, how can it be considered scientific? Even if there was a 0.0000…1% chance that macroevolution could happen, why would a scientist stake his/her name and entire career on such astronomical, outrageous odds when, if biased assumptions are dropped, there is a much more plausible explanation for the origin of this Universe? Prominent evolutionist, Richard Dawkins, himself admitted, “The more statistically improbable a thing is, the less we can believe that it just happened by blind chance. Superficially the obvious alternative to chance is an intelligent Designer” (1982, p. 130, emp. added). We certainly agree, and sadly, the implication of that alternative is the very reason so many people irrationally hold onto impossibilities—the intelligent Designer has expectations to which this rebellious generation refuses to submit.

Nevertheless, in the words of Emile Borel:

When we calculated the probability of reproducing by mere chance a work of literature, in one or more volumes, we certainly observed that, if this work was printed, it must originally have emanated from a human brain. Now the complexity of that brain must therefore have been even richer than the particular work to which it gave birth (1963, p. 125, emp. added).

And if we might add another line to Borel’s statement: “And further, the complexity of the Mind that gave birth to that brain must be truly incomprehensible!”




Evolution: Rationality vs. Randomness, by Dr. Gerald Schroeder

An M.I.T. trained scientist takes a look at Darwin, the fossil record, and the likelihood of random evolution.

At the basis of the theory of neo-Darwinian evolution lie two basic assumptions: that changes in morphologies are induced by random mutations on the genome, and that these changes in the morphology of plant or animal make the life form either more or less successful in the competition to survive.

With nature doing the selection, evolutionists claim to remove the theory of evolution from that of a random process. We are told that the selection is in no way random. It is a function of the environment. The randomness, however, remains as the basic driving force that produces the varied mutations from among which the selection by survival takes place.

The question is: Can random mutations produce the evolution of life?

Because evolution is primarily a study of the history of life, statistical analyses of evolution are plagued by having to assume the many conditions that were extant during those long gone eras. Rates of mutations, the contents of the "original DNA," and environmental conditions – all these affect the rate and direction of the changes in morphology. And these are all unknowns.

From a secular view, one must never ask what the likelihood is that a specific set of mutations will occur to produce a specific animal. This would imply a direction to evolution, and basic to all Darwinian theories of evolution is the assumption that evolution has no direction. The induced changes, and hence the new morphologies, are totally random. The challenges presented by the environment determine which will survive to produce the new generations and which will perish.

Protein Combinations

With this background, let's look at the process of evolution. Life is in essence a symbiotic combination of proteins (and other structures, but here I'll discuss only the proteins). The history of life teaches us that not all combinations of proteins are viable. At an event recorded in the fossil record and known as the Cambrian explosion of animal life, some 50 phyla (basic body plans) suddenly and simultaneously appeared in the fossil record. This is the first appearance of complex animal life. Only 30 to 34 of the phyla survived. The rest perished.

Since then the fossil record and modern existing biota reveal that no new phyla have evolved. At a later stage in the flow of life, a catastrophic event (possibly the collision of the earth with a massive comet or meteor) eliminated 90% of all life forms. The ecology was wide open for new phyla to develop. Again, no new phyla appear. The implication is that only a limited number of life forms (phyla) are viable.

It is no wonder that the most widely read science journal, Scientific American, asked "has the mechanism of evolution altered in ways that prevent fundamental changes in body plans of animals" (November 1992). It is not that the mechanism of evolution has changed; it is our understanding of how evolution functions that must change to fit the data presented by the fossil record and by the discoveries of molecular biology.

Pure randomness as the source of the mutations no longer stands against mounting evidence of scientific data.

It is difficult and painful to discard entrenched notions of what is actually true, even when scientific data demand such an abandonment. Pure randomness as the source of the mutations that neo-Darwinian concepts demand to drive the evolution of life no longer stands against the mounting evidence of scientific data. Unfortunately, the emotional commitment to a totally materialist view of life makes discarding this notion problematic.

Let's look at the likelihood that random mutations could have produced viable forms of life. Life as we know it is built largely of combinations of proteins working in symbiotic harmony. But as we've seen, only certain combinations produce viable life. Other combinations fail.

Humans and all mammals have some 50,000 genes. That implies, as an order of magnitude estimate, some 50,000 to 100,000 proteins active in mammalian bodies. It is estimated that there are some 30 animal phyla on Earth. If the genomes of each animal phylum produced 100,000 proteins, and no proteins were common among any of the phyla (a fact we know to be false, but an assumption that makes our calculations favor the random evolutionary assumption), there would be (30 x 100,000) 3 million proteins in all life. (The actual number is vastly lower.)

Now let's consider the likelihood of these 3 million viable combinations of proteins forming by chance, recalling that the events following the Cambrian explosion of animal life and the later decimation of 90% of life taught us that only certain combinations of proteins are viable.

Proteins are complex coils of several hundred amino acids. Take a typical protein to be a chain of 200 amino acids. The observed range is from less than 100 amino acids per protein to greater than 1000. There are 20 commonly occurring amino acids that join in varying combinations to produce the proteins of life. This means that the number of possible combinations of the amino acids in our model protein of 200 amino acids is 20 to the power of 200 (i.e. 20 multiplied by itself 200 times), or in the more usual 10-based system of numbers, approximately 10 to the power of 260 (i.e. the number one, followed by 260 zeros!). Nature has the option of choosing among the 10 to power of 260 possible proteins, the 3 million proteins of which all viable life is composed. In other words, for each one correct choice, there are 10 to power of 254 wrong choices!

Simon Conway Morris, professor of evolutionary paleontology at the University of Cambridge and fellow of the Royal Society of England, is the scientist who revealed the significance of the Cambrian explosion of animal life. He refers to this vast biological waste land of failed life forms as the "multidimensional hyperspace of biological reality."

Can this have happened by random mutations of the genome? Not if our understanding of statistics is correct. It would be as if nature reached into a grab bag containing a billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion non-viable proteins – and pulled out the one that worked.

And then repeated this trick a million times.

With odds like that, it is amazing that nature and our bodies ever got it or get it right.

But perhaps not every amino acid can join with every other amino acid. If this is the case, then the number of possible combinations will be reduced. To get even hint for what this would do to the hyperspace of failed choices, I looked at combinations of amino acids that actually exist in just six proteins. Among the proteins I used were bovine insulin and bovine ribonuclease. The number of potential amino acid combinations just from this modest sampling of proteins was 10 to the power of 20. Again, nature would have had to select the one viable combination from among 100 billion billion wrong choices. Either our knowledge of statistical probability is skewed or something other than randomness is operating.

The late Harvard professor, Stephen Jay Gould, suggested that the flow of life is "channeled" along these basic animal phyla.

Nobel laureate, organic chemist and a leader in origin of life studies, Professor deDuve writes in his excellent book, Tour of a Living Cell, "If you equate the probability of the birth of a bacteria cell to chance assembly of its atoms, eternity will not suffice to produce one... Faced with the enormous sum of lucky draws behind the success of the evolutionary game, one may legitimately wonder to what extent this success is actually written into the fabric of the universe." Life written into the fabric of the universe sounds a bit metaphysical.

Morris, in his book Life's Solutions (Cambridge University Press, 2003), writes: "Life is simply too complex to be assembled on any believable time scale... evolution's uncanny ability to find the short cuts across the multidimensional hyperspace of biological reality. It is my suspicion that research might reveal a deeper fabric to biology..." Elsewhere Morris identifies this "deeper fabric" as having "metaphysical implications."

This impossibility of randomness producing order is not different from the attempt to produce Shakespeare or any meaningful string of letters more than a few words in length by a random letter generator. Gibberish is always the result. This is simply because the number of meaningless letter combinations vastly exceeds the number of meaningful combinations.

With life, such gibberish was and is lethal.

In brief, randomness cannot have been the driving force behind the success of life. Our understanding of statistics and molecular biology clearly supports the notion that there must have been a direction and a Director behind the success of life.




  • Editor's note: The mathematics of probability inform us that randomness cannot account for the complexity we find in nature. Even a "simple" production of "monkeys typing the works of Shakespeare" would require far, far more time than the age of the universe. The forces of evolution are still in play but do not work according to the rules taught in high-school biology textbooks. (See the following, original article online at https://plus.maths.org/content/infinite-monkey-businesst)


Understanding uncertainty: Infinite monkey business

David Spiegelhalter and Owen Smith

The idea that an infinite number of monkeys typing at random on an infinite number of typewriters will eventually produce the complete works of Shakespeare apparently dates from 1913, and has appeared repeatedly in popular culture ever since. When the BBC Horizon team decided to make a programme about infinity, they contacted us about simulating the monkeys. We said we needed a program to churn out random letters and match them to Shakespeare, and so they commissioned a Monkey Simulator program from Aaron Russell, which is available from this website.

The Monkey Simulator program generates random symbols from a list of 31 options: 26 lower-case letters, a space, a comma, a full stop, a semicolon and a hyphen. After a sequence of four symbols has been generated, the program searches for a match in a stored plain-text version of the Complete Works of Shakespeare, ignoring whether a letter is capital or lower-case. If the procedure finds one or more matches, it generates a further character and again checks if there is a match for five symbols, and so on, until no matches are found. Then it starts again with a new sequence of four characters. Characters are generated at a rate of 50 per second. (We note that this procedure cannot match sequences that stretch over the 36,357 line returns in the text file, and also cannot match all the other punctuation in the text, such as the 10,475 question marks and 8,827 exclamation marks.)

A screenshot from the Monkey Simulator programe

The image to the right shows the situation after the program has generated the sequence "y sak" and matched it with the phrase "and for my sake even so doth she abuse me", which comes from Sonnet 42. In fact, this was only one of 37 matches of this particular sequence in the whole of the Complete Works.

The image shows that after 113 million monkey seconds (or around 26 days for 50 monkeys typing one character a second) the longest match was "we lover", which matched with the speech by Boyet, "With that which we lovers entitle affected", in Love's Labours Lost, Act 2 scene 1.

The claim is that, given enough time, such a program would generate the entire Complete Works. What is the chance of this happening for any particular generated sequence? The sequence would have to start with the first character (the first "T" in "The complete works...") and continue through around 5,000,000 characters (including spaces) until it reaches the end.

Let's be generous to the poor monkey and ignore the fact that there are lower and upper case characters and various extra bits of punctuation, so there is one of only 31 characters in each position. Each character typed therefore has a 1 in 31 chance of being the right one, so the chance that the first 2 are correct is 1 in 31 × 31, which equates to 1 in 961. The chance that they are all right, from beginning to end, is 1 in 315,000,000. This is approximately equal to 1 in (101.5)5,000,000 = 107,500,000. So each time the monkey starts typing, there is a chance of p=1/107,500,000 of completing Shakespeare. This probability is rather small but it is finite. Since 107,500,000 225,000,000, it is about the same chance as flipping a fair coin 25,000,000 times and it coming up heads every time.

Alternatively we can think of the National Lottery, where the chance of winning the jackpot is around 1 in 14,000,000 107.1. So the tiny chance p is equivalent to winning around 1,000,000 lotteries in a row, since 107,500,000 is around 107.1 to the power 1,000,000. If you bought one lottery ticket a week, this is like winning every week for 20,000 years. Of course in real life people might start getting suspicious.

It's just a matter of time...

Even though the event of one monkey producing Shakespeare when randomly typing 5,000,000 characters has such a microscopic probability, we are "certain" of it eventually occurring in exactly the same sense that we are "certain" that if we repeatedly flip a fair coin, it will eventually come up heads. This will take on average two flips, just as the time to get the first double-six when throwing two dice is on average 36 throws. So we expect to see Shakespeare after an average of 1/p tries by the monkey, which of course is a very long time. Even with our program, which generates 50 characters a second, corresponding to 50 monkeys typing slowly or one typing incredibly fast, we would not expect to get Shakespeare until well after the Universe has come to an end.

This is an average, but how long might we actually have to wait? Waiting times have what is known as a geometric distribution. If p is the probability of the event of interest, and there are an unlimited number of independent opportunities for that event to occur, then the chance that the event happens at exactly the nth attempt is the chance that it does not happen for (n-1) attempts (that is, (1-p)n-1) times the chance that it finally does happen at the nth try (which is p), to give the total probability as:

Prob (First event happens at attempt n) = (1-p)n-1p.

The chance that the event does not happen at attempt n or before is the probability of n failures in a row, or (1-p)n. This means that the chance that the event happens by attempt n, that is, at n or beforehand, is

Prob (First event happens at or before attempt n) = 1 - (1-p)n.

As long as p is not zero, this number can be made as small as you like by increasing the number n of attempts. This is essentially a special case of the law of large numbers, which (very) roughly says that things tend to average out in the end. After the Horizon programme was aired I had an email discussion with a philosopher who argued that it was not logically certain that the monkeys would eventually type Shakespeare, which is true, but it is probabilistically certain, which is good enough for me.

We can easily work out how long we expect to wait to have, say, a 99% chance of generating Shakespeare. Suppose we want a chance F of being finished by attempt n. Then F = 1 - (1-p)n, and so rearranging terms we get





where the logarithms can be to any base, but we shall assume natural logarithms to base e. For small p we can get a convenient approximation, since and so





We can express this in an even simpler way. Suppose p = 1/T, that is there is a 1 in T chance of the event occurring. And suppose we want to be really sure of observing the event, so that F = 1 - 1/K where K is large. That is, we want there to be only a 1 in K chance of still being waiting after time n. Then





So if you have an event with a 1 in 1000 chance of occurring, and you want to have only a 1 in 100 chance of still waiting by time n, then set n to be 1,000 log(100) = 4,600. This means that there is a 99% chance that the event will have occurred before 4,600 attempts.

We can use the exact and approximate formulas to generate the following table for different events:


Coin coming up heads
p=0.5000, T=1/p=2

Two 6s from 2 dice
p=0.028, T=1/p=36

Desired probability F of success


Number of attempts n

Number of attempts n






















Any 17 character sequence from Shakespeare being produced by the monkeys
p=2.2×10-19, T=1/p=4.5×1018

Desired probability F of success


Number of attempts n

Billion years



























The number of attempts required to have a specified probability of success when repeatedly trying to observe an event with probability p = 1/T. For small p and F near 100%, n≈T log K.


So to be 99% sure of observing at least one double-six when throwing two dice, you will have to plan to make 163 throws. And to get a particular sequence of 17 characters right, for example "to be or not to b", would require an event with chance 1 in 3117 2.3 × 1025. This is for a particular sequence: there are around five million 17-letter sequences in Shakespeare. So to get any 17-letter sequence right is an event with probability 1 in 4.5 × 1018. Our program types characters at 50 a second, so assuming the program generates a continuous sequence of characters, we would expect to wait around 2.9 billion years, or to be 99% sure, 13.2 billion years — as long as the time since the Big Bang signalled the start of our Universe.

Theory does not always meet practice. A wonderful arts project in Paignton Zoo put a computer in a monkeys' enclosure to see how they got on. The monkeys typed 5 pages, mainly consisting of the letter "s", and then used the keyboard as a toilet. So rather a limited output of classic English literature, which just shows the problems that turn up when maths meets the real world.

The image of monkeys and typewriters is powerful and will keep on being used whenever we need an image of an apparently impossible event made feasible by thinking on a large enough scale. I think it provides a fine incentive for analysing waiting times for rare events, but does not really give us any insight into what forms of infinity play out in the physical universe. But who cares about that anyway?




Dr. William Stenger

If you flip a coin, there are two equally likely outcomes:

The probability of "heads," is one out of two or l/2, and the probability of "tails" is one out of two or 1/2.

If you flip two coins, a penny and a nickel, there are four (2 x 2) equally likely outcomes:

The probability of two "heads" is one out of four or 1/4, the probability of two "tails" is 1/4, but the probability of one "head" and one "tail" is two out of four or 2/4= 1/2.

If you flip three coins, a penny, a nickel, and a dime, there are eight (2x2x2) equally likely outcomes:

The probability of three "heads" is l/8, the probability of three "tails" is 1/8, the probability of two "heads" and one "tail" is 3/8, and the probability of one "head" and two "tails" is 3/8.

Monkey Business

   "GIVEN a monkey, a typewriter and a stack of paper, by chance alone, words, sentences, even whole books could be written." This is a doctrine deemed holy by a not-insignificant number of educated men in the biological and geological sciences today.
   "Given enough time, chance alone will produce simple life forms... given more time, complex form of life will come into being... given enough time, even man... highly intelligent man could arise." But is this true? And if true, what span of time would be needed?
   The Origin of Species — by chance alone? Could life forms — species — arise by chance "given enough time"? Could Darwin's book also have come by blind chance?
   Given a chimpanzee, a typewriter and a stack of paper, how long would it take the chimp to peck out just the title of Darwin's controversial book The Origin of Species? Eighteen letters and three spaces, a total of twenty-one strokes with the typewriter. And we will simplify the problem by allowing him to use all capital letters.
   It should take but a few hundred attempts? Or should it?
   Begin with a chimp — of course you must provide him with an intelligent human assistant to note his progress and to insert a fresh sheet of paper each time the chimp makes an error. How long will it take to type Darwin's title? Twenty-one strokes might take perhaps twenty-one seconds! That is IF he hit the right keys. There is a chance, you will admit, that he might by chance hit the right keys, even the first time. And that given enough time he would certainly get the title right.
   How long then to type out Darwin's entire volume? How long to type out the Encyclopaedia Britannica with its 24 volumes, each with over a thousand pages? And given enough time, how long for all the books and publications in the world?
   Do we acknowledge that given enough time this chimp could type out all the books in the world? By mere random pecking at the keys?
   And by analogy, do we then admit that life forms and man too could originate by chance?... that the millions of species and billions of billions of different individual creatures alive today are the result of blind chance?

Prepare to Multiply

   Let us go back to our chimp and watch over his shoulder as he begins ARX BLMX... but wait, his human assistant removes the sheet of paper, turns in a fresh one and our chimp begins again.
   T — Success! He has hit a "T" by accident and we are on our way to words, then books... and even LIFE itself....
   There are 44 keys on the typewriter (26 letters, 9 digits, a space bar and numerous symbols and punctuation marks). For simplicity, the other controls are being handled by our human assistant. He watches for errors (and success), changes sheets and times the proceedings. And feeds the chimp.
   The odds of getting that first letter right were 1 in 44 (that's how many keys there are on a typewriter), but he did it. Now only 20 more successful strokes to go: T, TH, THE!
   What were the odds on that first word? One chance in 44 on the T, one in 44 on the H, one in 44 on the E. Prepare to multiply 44 times 44 times 44. The answer is 85,184.
   By blind chance done, our chimp has only one chance in 85,184 of getting these first three letters correct. But let him proceed. One chance in 44 that he will hit the space bar next, then one chance in 44 of hitting the 0, then the R, then I, G, I and N. The odds now become astronomical, for 44 times itself 10 times turns out to be about 27,720 million million.
   Remember that each time an error was made, his assistant removed the sheet from the typewriter and inserted a fresh one. How high a stack of paper can we expect to result from our chimp typing out these first two words? On the "average," 27,720 million million sheets' would be wasted to achieve a two-word goal — THE ORIGIN — typed neatly on the twelfth line, where the human assistant started our chimp off.
   And the stack of 27,720 million million wasted sheets? How high? A thousand sheets of paper might be about six inches thick; two million sheets, a thousand feet high; ten million, roughly a mile high.
   But 27,720 million million sheets would reach into the heavens — out past the moon — all of the 93 million miles to the sun — and over a thousand million miles further for a total of 2,772 million miles. All this waste by using the "random pecking" method of a chimp to type out the first two words!
   THE ORIGIN... just that much by chance. How many tons of paper? Eight pounds for a thousand sheets. Four tons for a million sheets. Thus our chimp has wasted 110,880 million tons of writing paper — tens of thousands of times the production of the U. S. in a year.

1The probability of success is p=4410 and of failure is q=1-p. The probability of the chimp succeeding the first time is p, of failing the first time and succeeding the second is qp, of failing the first two times and succeeding the third is q2p, etc. Therefore, the expected number of wasted sheets of paper is 0p+qp+ 2q2p+3q3p+... =(1/p )-1, which is approximately equal to 4410.

   How many trainloads of paper, and who would build the trains, and the railroads? How many square miles of timber would have to be cut to produce the pulpwood? How many lumberjacks?
   To inquire where we got the monkey (I mean chimpanzee) in the first place, and where the intelligent human assistant, and where the typewriter, would only cause further embarrassment at this point. And... whether any chimp would either submissively or mischievously peck keys at random? Finding the X, wouldn't he go XXXXXX, then XX, XXX, and then go off in search of something else to monkey with?
   But let us return to our chimpanzee. How long has this taken him? Not 21 seconds... not 21 years but...? Allow a second to insert the sheet, a few seconds to make an error and a bit more time to discover the error, stop the chimp and remove the paper.
   Five seconds per sheet would allow twelve sheets per minute, 720 sheets an hour and 5,760 for an 8-hour day. Our 27,720 million million sheets would then represent more than 4,800,000 million days. With 235 working days in a year (a month's vacation for other monkey business), our chimp is either 20,000 million years old... or we are four thousand million chimp generations later.

A Title for Darwin's Book

   But we began a task; let us complete it. Twenty-one correct typewriter strokes will complete the title — THE ORIGIN OF SPECIES. And the chance of being correct in each stroke is only 1 in 44. (The chances of error are 43 in 44.) The odds to type out 21 correct consecutive strokes are 44 times itself 21 times, or in the mathematician's mode of expression 4421.
   A few minutes with logarithms changes 4421 to about 1034.5, then back to layman's mathematics as the number 325 followed by 32 zeros. (Six zeros would produce a million.) But adding 32 zeros produces an answer of 32,500 million million million million million. That many sheets would produce a stack 3,250 million million million million miles high! But that distance is ten thousand times the assumed extent of the universe. The number of sheets of paper is millions of millions of times greater than the estimated total number of stars (1,000 million million million) in the universe.



Editor's last word: