Page 47
Chapter 2
The Heart of Modern Probability Theory
It is truth very certain that, when it is not in our power to
determine what is true, we ought to follow what is most probable.
–René Descartes^{1}
SCIENTISTS CANNOT SAY how the world came to be or how life
began. There was no human observer on the scene to record
with technical data whether God created or things just evolved.
Until one arrives at faith in the Bible record, the only logical
course is to use inductive reasoning and follow what is most
probable. Let us begin now to learn the main rules of probability.
We will need only two central principles. Here is the first, sometimes
called the “law of averages.”
The Law of Large Numbers
Probability theory applies mainly to “long runs.” If you toss
a coin just a few times, the results may vary a lot from the
average. As you continue the experiment, however, it levels out
to almost absolute predictability. This is called the “law of large
numbers.” Here is how physicist George Gamow stated it:
Thus whereas for 2 or 3, or even 4 tosses, the chances to
have heads each time or tails each time are still quite appreciable,
in 10 tosses even 90 per cent of heads or tails is
very improbable. For a still larger number of tosses, say 190
or 1000, the probability curve becomes as sharp as a needle,
48 Evolution: Possible or Impossible?
and the chances of getting even a small deviation from fiftyfifty
distribution becomes practically nil.^{2}
The long run serves to average out the fluctuations that you
may get in a short series. These variations are “swamped” by
the longhaul average. When a large number of tries is involved,
the law of averages can be depended upon quite closely.
This rule, once called the “law of great numbers,” is of central
importance in this field of probability. By the way, in the
popular sense, probability theory, the laws of chance, and the
science of probability can be considered to be simply different
expressions for the same general subject.
Make Your Experiments Scientific
To be exact, the theory of probability deals not with material
objects, but with ideal theoretical models or mental pictures.
If we use objects that are reasonably identical, however, the
results of our experiments will be close to the same as with
the abstract mathematical models on which the laws are based.
When we do experiments such as coin tosses or drawings of
numbered objects, it is important to insure that there is equal
likelihood of the different outcomes or “events” as they are called.
If one of the objects to be drawn is heavier than the others, it
may tend to settle to the bottom of the group, thus giving results
that are inaccurate. Different rules might be involved if the
various possible results are not made equally probable.
In selecting coins or letters at random, they must, of course,
be thoroughly mixed before each drawing. If they are not shaken
sufficiently, the same one that was just drawn might remain near
the top to be more easily drawn again. Objects also should be
drawn without looking, to avoid the possibility that the choice
is influenced by sight of the various objects. The purpose is to
find out what chance can do, and chance is blind.
If other articles are used instead of coins, they should as nearly
as possible be the same size and shape and weight. This makes
the experiment more scientific and assures more accurate results.
Experimenting may mean more to you if first you read on a few
pages farther.
The Multiplication Rule (Learn It Well!)
We now come to the most important rule of all for the
The Heart of Modern Probability Theory 49
purposes of our study. It is the second of the two principles.
Let’s go back to the ten numbered coins. Why is there only
one chance in one hundred that we will get the number one
coin on the first draw followed by the number two coin on
the next draw?
Here is the principle involved, as described clearly by Adler:
“Break the experiment down into a sequence of small steps.
Count the number of possible outcomes of each step. Then
multiply these numbers.”^{3} This important “multiplication rule”
is most often used where the various outcomes of a particular
step are all equally probable and the steps are independent.
In the experiment with ten similar coins numbered one through
ten, we want to know the probability of getting the number
one coin on the first try followed by the number two coin on
the second try. Divide this into steps as Adler suggested. Our
first step will be to draw one coin. There are ten different outcomes
we could get on that first draw. There are also ten different
possible results when we get to the second step. Multiplying,
as Adler said, we have 10 x 10 = 100. So, the chance
is 1 in 100 of getting the two desired coins in order. The probability
is 1/100, on the average.
Before the first draw, we know intuitively that there is a
loutof10 chance of success in getting the number one coin.^{4}
Therefore, whatever chance the second step will have must be
multiplied by 1/10, because there is only that 1/10 chance
of success on the first step. But the second step also has 1/10
probability of success. As we have just seen, that will have to
be multiplied by the 1/10 probability from step one. This will
give the answer for both steps together, which is 1/100. If such
an experiment is continued long enough, about once in every
hundred draws the number one coin will be followed by the
number two. Remember, however, the law of large numbers.
There will be deviations unless you do several hundred and
average them.
The principle is: If you seek first “this outcome” and then
“that outcome,” the probability of getting both is the product
of their separate probabilities, in cases where one outcome does
not affect the other. George Gamow said it in these words:
50 Evolution: Possible or Impossible?
Here we have the rule of “multiplication of probabilities,”
which states that if you want several different things, you
may determine the mathematical probability of getting them
by multiplying the mathematical probabilities of getting the
several individual ones.^{5}
Perhaps this may seem to be much ado about a minor point.
Some who are mathematically minded or knew the principle
beforehand may have gotten it easily. For most people, however,
it is hard to believe that the chances are that slim–just
one in one hundred. This is the average outcome one can expect.
It will be worthwhile to stay with this matter until thoroughly
convinced that it is true. One’s mind may be slow to accept
the idea. Darrell Huff wrote that “even intelligent adults confuse
addition with multiplication of probabilities.” That is why
actual experimenting may be such a help. Much depends on
becoming certain in one’s own thinking that this is correct. A
little later, we will suggest quicker methods for experimenting
that will lead to the certainty of the truth of this rule.
This one point is absolutely vital to the whole process of this
approach to certainty. It may be mastered by rereading and
by experimenting as described a little farther on, and by pondering
the matter until one’s mind will accept its truth. All
probability theory used in science and industry builds from this
multiplication rule.
Can Chance Count to Ten?
What is the probability of drawing all ten coins in order?
Remember the multiplication rule. For each of these steps,
there are ten possible outcomes. For all ten steps, we must
multiply ten by itself until the figure is used ten times: 10 x 10 x
10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 = 10,000,000,000. So, the
chances are quite small of getting all ten in a row. Once in ten
billion selections we will get the number one followed in order
by all the rest. Chance will succeed on the average only once
in ten billion attempts.
To absorb the meaning of that fully is to be well on the
way to the assurance that we seek. Chance requires ten billion
tries on the average in order to count to ten!
Shorten Your Experiment Time
The reader has doubtless already realized that the experiment
The Heart of Modern Probability Theory 51
with ten coins is too long for any reasonable chance of success
if done properly. If a person could draw and record one coin
every five seconds day and night, it would take over 1,500 years
to complete the time in which one success could be expected!
In all that time, the outlook is for chance, on the average, to
succeed just once in counting to ten.
Perhaps we get the gist of the idea that chance is not very
capable when we need an ordered result. Consider the difference
intelligence makes–even a limited intelligence. Give an eightyearold
the coins, and ask the child to arrange and pick up
each one in order and return it. Chance is blind, and has no
intelligence. The child is not thus limited. The child can do it
in a few moments. Chance takes 1,500 years–just to count to
ten once.
The same principle can be learned with shorter experiments,
using fewer coins. If you try it with three or four or five numbered
coins long enough to average out any shortrun fluctuations,
you will see that the rules hold true. With five coins,
the probability of getting the number one and the number two
in order on the first two draws is naturally 1 in 5 x 5 = 1 in 25.
In tossing a coin, the probability of four heads in a row is
1/2 x 1/2 x 1/2 x 1/2 = 1/16. What would be the probability
of ten heads in a row?
To Spell “Evolution” by Chance
Suppose, instead of numbers, we use the letters of the alphabet.
As a substitute for coins, any small, similarly shaped objects
may be used if they are practically identical in size, weight, and
shape. (The party game called “Scrabble” has small letters on
wooden squares quite suitable for this.)
With one set of the twentysix letters of the alphabet, you
have 1/26 probability of getting the “A” on the first draw. To
get “A” followed by “B” (replacing the letter after each draw,
as before) your probability by the multiplication rule is: 1/26 x
1/26 = 1/676. To get ABC in order, the chance is 1 in 17,576,
by the same rule.^{6}
To spell the word “evolution,” obtaining the nine letters in
order, each having a 1/26 probability, you have a probability
52 Evolution: Possible or Impossible?
of 1 in 5,429,503,678,976. This, as you will realize, comes from
multiplying 26 by itself, using the figure 9 times. If every five
seconds day and night a person drew out one letter, he could
expect to succeed in spelling the word “evolution” about once in
800,000 years!
Further Tests for Chance
Suppose we put chance to a test which is less simple, yet
something that would be quite easy for any school child. Let
it spell this phrase: “the theory of evolution.” Drawing from a
set of twentysix small letters and one blank for the space between
letters, what is the probability expectance?
All that is needed is simply to get those twentythree letters
and spaces in proper order, selecting them at random from the
set of twentyseven objects (twentysix letters and one space).
By the multiplication rule we learned, it will be 27 x 27 x 27 . . .
x 27 using the figure twentythree times.
The probability when computed is 1 in approximately 834,390,000,000,000,000,000,000,000,000,000;
that is, one success in
over 8 hundred million trillion trillion draws.
To get an idea of the size of that number, let us imagine that
chance is employing an imaginary machine which will draw,
record, and replace the letters at the speed of light, a BILLION
draws PER SECOND! Working at that unbelievable rate, chance
could spell “the theory of evolution” once in something over
26,000,000,000,000,000 years on the average!
Again, a child could do it in a few minutes. Chance would
take more than five million times as long as the earth has existed
(if we use the fivebillionyear rounded figure which some evolutionists
now estimate as the age of the earth).
If we are drawing from a set which contains both small
letters and capital letters and one blank for the space between
words to spell “The Theory of Evolution,” the probability is
1 in 4,553,500,000,000,000,000,000,000,000,000,000,000,000. Our
machine drawing at the speed of light, a billion draws per second,
would require 140,000,000,000,000,000,000,000 years. That
is 28,000,000,000,000 times the assumed age of the earth!
Chance Is Moronic
So chance requires twentyeight trillion times the age of the
earth to write merely the phrase: “The Theory of Evolution,”
drawing from a set of small letters and capitals as described,
The Heart of Modern Probability Theory 53
drawing at the speed of light, a billion draws per second!^{7} Only
once in that time could the letters be expected in proper order.
Again, a child can do this, using sight and intelligence, in a
few minutes at most. Mind makes the difference in the two
methods. Chance really “doesn’t have a chance” when compared
with the intelligent purpose of even a child.
“In the beginning, God . . . ” begins to appear more scientific,
as we see how limited are the abilities of mindless chance.
Perhaps the alphabet experiments just described may help to
emphasize how important it is fully to understand the multiplication
rule we studied earlier. It’s hard to believe at first.
Try drawing alphabet letters for a few hours to become really
convinced! Remember in doing so that chance has no intelligence,
no purpose. It cannot purposefully choose one correct
letter and discard unwanted ones until it finds the next one
needed.
In the next two chapters, we will make some actual use of
what we have learned. We are to apply probability theory to
the strange phenomenon of the “lefthanded” molecules which
are used in proteins. We will use that as a practice field in applying
the laws of chance. It is ideal for this, because only two
possible outcomes are involved for each step. It is similar, therefore,
to the experiment of tossing a coin.
Special Note to the Reader
Most of this book is in plain, easytounderstand language. In
a few places, however, we must go far enough into certain areas
of biology to apply the laws of chance in logical manner. This
will require the use of a small amount of mathematics, but not
much–mostly just arithmetic. It is a necessary part of the process
in gaining certainty by the approach which we are following.
For the reader who happens to have an absorbing interest in
biology, it is unlikely to involve any strain or confusion as a rule.
Perhaps, on the other hand, you have only a casual interest
in the details of science. Does that rule out the value to you of
this method of seeking assurance on evolution? Not at all. A
great number of people may not have any engrossing interest
in biology, and yet may attain that valuable certainty.
54 Evolution: Possible or Impossible?
If you plow on through any places that seem somewhat technical,
you will at least get the general idea and you will soon
be back into easier reading. In the process, you will realize
that the actual facts and figures are there in print for anyone
who wishes to dig into the subject more thoroughly. The conclusions,
moreover, are always in easily grasped speech. Without
the actual reasoning and figures, and without the references, the
reader would have little to depend on except an author’s words,
and that is a poor basis for certainty. Don’t worry, then, if you
strike sections that you do not quickly comprehend completely.
Just read on through. You can return later to those sections
that you may wish to reread.
Before going on, we will confess that (to the horror of mathematicians) we have oversimplified a bit, to make the ideas
accessible to people not trained in mathematics. The recurrent
phrase, “on the average,” needs more explaining when it is used
with experiments which are repeated. The footnote below goes
into this, for the noncasual reader.^{8} Now, let’s look at lefthanded molecules.
^{1}
In Darrel Huff and Irving Geis, How To Take a Chance (New York: W. W.
Norton & Go., 1959), p. 7.
^{2}
George Gamow, One, Two, Three–Infinity (New York: Viking Press, 1961), p. 209.
^{3}Irving Adler, Probability and Statistics for Everyman (New York: John Day Co., 1963), pp. 58, 59.
^{4}
This probability arises partly because of equivalence or symmetry, and we
sense its logic.
^{5}
Gamow, One, Two, Three–Infinity, p. 208.
^{6}
We tried such an experiment at the Center for Probability Research in Biology.
In 30,000 alphabet letters drawn, only once did we get ABC in order!
(Of course, there were other reasons for the experiment. The main
purpose is explained in chapter 6 where it provides an analogy for usable an nonsense
chains of amino acids.)
^{7}
The imaginary machine is considered as moving slightly less than a foot per
draw, round trip. The letter is recorded during the return trip so that no time
is taken up except the actual travel, round trip, of .98 foot at the speed of light,
to allow 1,000,000,000 draws per second.
^{8}
Our figuring thus far has been the kind where “success” was getting a certain result once, on the average. in a series of trials. Now, consider the different
concept of at least once: the desired event may happen once or more than once,
but the main thing is that it happens at all.
If we draw from ten coins (with replacement), what is the probability that
the No.1 coin will show up at least once if we make two draws? Here is what
can happen: (1) We may obtain the No.1 coin just once from the two draws;
(2) we may get it both times; or (3) not at all. Either the first or the second
of these results would be a “success,” because in each the event occurs: “No.1
coin at least once.”
We see that success can happen in more than one way, but failure can
happen just one way. We therefore first figure the chance of failure. It is 9/10
on anyone draw, and we can use the multiplication rule for two draws, because
we need failure both times–“this and that.” 9/10 x 9/10 = 81/100. Now to
find the chance of success:
Always, if one adds the probability of success and the probability of failure,
the total is exactly one. We can obtain the probability of success by subtracting
81/100 from 100/100 (which is the same as one). The answer is 19/100, the
chance of getting the No.1 coin at least once. A mathematician might write
the formula thus: where n is the number of draws, and p is the probability of
success in one draw: p_{n} = 1 – (1 – p)^{n}.
With the large figures we will encounter, it would make virtually no difference
if we used this more exact method, so we will save confusion by figuring the
much simpler probability on the average. Chapter 10 will give more details on
this. (The difference between the two methods is less than just adding one to
an exponent of ten. The exact method would be even harder on evolution.)

