AT THIS stage of the story nobody should take the existence of extrasensory
perception on faith. What the two preceding chapters have done is to describe
certain experiments with cards, tell about the scores which some subjects were
able to make with them, and quote the odds against guesswork or chance as the
explanation. The tests as a whole, or in their natural subdivisions, gave
results that are clearly beyond the best that could be expected from the
socalled laws of luck. A figure of over a thousand digits would be required to
express the unlikelihood of that solution. Some other factor was causing our
results, and so far it has been called "extrasensory perception" in a
noncommittal way.
In explaining the results of these tests a mere abstract citation of
mathematical odds against chance as the explanation of our findings is not
sufficient. People want every question and doubt answered before they are
convinced, and indeed it is sensible to look over again and again every possible
alternative before accepting such a revolutionary explanation as our hypothesis
of extrasensory perception. Then, too, the tests on which it is based are so
incompatible with certain widely accepted points of view that even though we
have found many subjects who can produce high scores there is perhaps a flaw in
the method itself or in the way of handling the results. Is chance really
excluded by the mathematics used? However conclusive the figures appear, is
there not a possibility that the mathematics used in deriving them is faulty or
is wrongly applied?
Now, of course, my associates  particularly Mr. Stuart, who is mathematically
trained  and I have been asking ourselves these questions from the very
beginning. We have had others doing it in other places. We have been in touch
with mathematical people all along the way, but most readers have not. For their
sake I shall take up these questions here and see if I can show whether we have
adequately excluded chance as a possible explanation. This is important because,
if there is doubt at this crucial point, the rest of this book will not be even
interesting.
2
To begin with, let us take a simple, commonsense view of the results of the
tests. On that basis there are two things to be compared: (1) the scores a
subject gets by actually calling the cards through a long series of runs and (2)
the scores secured in tests where no subjects called the cards at all simply
crossmatching one deck against another. As we have already seen, thousands of
these tests have worked out to an average of almost exactly 5.0, and when you
reflect that mechanical shufflers have been employed in some of these control
tests, as they are called, it is impossible to believe that they did not exclude
the faculty or condition, or whatever it was, that made the higher scores of the
human subjects possible. But that is not all the precaution we have taken on
this point. Thousands and thousands of other matchings have been made to
crosscheck calls by the subjects themselves. This is easily done by using a
different pack from the one against which he made his calls  say, the order of
the deck in the run before or the run after. Thus, the records of the
subject's calls in his second run are compared with the actual order of the
cards in his first or his third run. This, too, ought to exclude extrasensory
perception because the calls were never intended for the cards against which
they are checked.
One of the very first things that was done in the evaluation of Pearce's scoring
was to take his first thousand ESP trials and compare them with a thousand card
records taken from the same pack of cards when no extrasensory perception was
involved. In other words, a thousand of his calls were checked against cards
that he did not intend them to match. The crosscheck series of 1,000
approximated 5 very closely, giving 5.1 as a result, while Pearce's first
thousand trials averaged 9.6 hits per 25, almost double the amount. Various
other kinds of crosschecks have been made, as well as the simple matching of
one pack of cards against another, and in no case has there been any
important departure from the theoretical chance average of 5.0. It is just as
easy, then, for one to judge by common sense that something is shown
where the results average 9.6 as it would be to look into one's account book and
find that there was a profit if the average sale amounted to $9.60 and the cost
was $5.10.
For a long time one of my friends, who did not understand the mathematics of
probability very well, kept repeating, "But sometime you may find your subjects
going just as far in the other direction as now they are going above chance."
In reply I used to appeal to his common sense and say, "For two years now
Pearce, to say nothing of the others, has been coming in here several days a
week and has been leaving every day a positive deviation. He never goes
below chance unless we ask him to. When we do ask him to go below chance and
deliberately try to miss the cards, he can do so, sometimes scoring zero. The
fact that he can go low at will and can regularly go high for so long a period
must be the answer to your claim that we are having just a run of luck. Such
voluntary scoring is the very opposite of chance. This man can get a score of 9
or 10 if I ask him for a high score; if I ask him to run low, he can get a 1 or
0; can go back up on the next run if I say 'high' and down on the run succeeding
that if I say 'low.' If this is a matter of chance performance, then the rise
and fall of that steam shovel I see out the window is a chance performance. And
even if the subject does reverse later, and go regularly below for two years? A
man may make money on his sales every day for two years; then he may turn round
and sell at a loss for the next two years and lose it all again. Is that to say
that the whole performance was merely chance?"
3
Most striking to the people who want the point made as simple as possible are
the long unbroken stretches of successive hits. Even 5 successive hits
represents odds of more than 3,000 to 1 against a chance occurrence. But when
one gets up into 9's, 15's, and finally 25's, one need only know the
multiplication table to follow through and find out what the chances are of such
an event's being due to nothing but random factors. Or you can even dispense
with the multiplication table. Actually, all we have to convince us of the
occurrence of things in life is simple repetition in unbroken succession. Apply
the simple, everyday rules of common sense to these long stretches, and few
people would be likely to say they were accidental.
Fortunately for the skeptics of common sense, the mathematics which applies to
these cases has been in use for many years and has been recognized over and over
again by the authorities in the field of special determinations of probability.
It was first applied to these problems back in the [eighteen] eighties and nineties by the
physiologist, Professor Richet, and it was then used essentially as today*. It
was used again by Coover (who, you will recall, mistook his evidence to be
against nonsensory perception), by Estabrooks, and by several others, including
the experts called in to evaluate the results of the widely publicized
Scientific American tests for telepathy. It has had the endorsement of the
leading authorities of Britain and America. To my knowledge no question of its
validity has been raised by any professional statistician or mathematician of
probability.
* This article was written in 1937.
Granted, then, that the mathematics is sound and appropriate to these results.
have we somehow made a mistake in the way we have applied it? There is a good
test for this too: we may know reliably that we have not made such a mistake
because we get only figures appropriate to chance when we apply the mathematical
tests in the same way to experiments carried out under conditions identical in
every point with the test experiments, except that, since no human mind has made
any of the calls in the series, ESP has been so positively excluded that only
chance factors can possibly be operative. From these nonESP experiments we get
the same results to be anticipated from mere chance data. At the moment of
writing, a group of papers is going to press for the Journal of
Parapsychology reporting such parallel experiments. In every case chance
conditions give figures that would be expected. In every case the ESP tests,
differing only in that extrasensory perception was allowed to operate if it
could, show that something beyond chance is at work. There is logically no
criticism left to level at the use of the mathematics in the case.
So much for chance. We have had it as our everpresent competitor. We have
always been alert to its claims. But as a theory for these results it "hasn't a
chance"!
But suppose the subject has personal preferences and calls twice as many circles
as other symbols. Might this not favor him? The answer is "no" since, even if he
called all 25 of the cards circles, the most he could get would be 5. The more
circles he calls the greater chance he has, of course, of getting a good
proportionate score among the five circles in the pack, but a proportionately
small chance is left for his getting the other twenty cards right. Preference
cannot help him on his total score.

A shuffling box to insure mechanical
shuffling. The lid is to put on and the box slowly tipped, one end
up and then down, not less than five times. Five ESP cards are
displayed against the lid of the box. 

Can any method of shuffling the cards or any natural sequence of cuts give
peculiar upcurves or downcurves in the scoring of these control series? The many
practical test checks that have been made on just this point furnish the best
answer. They average close to 5, with no longdrawnout stretches of runs that
would yield significant deviations.
For years one of the most common objections that we encountered was: "But might
not the subject use reasoning, as in card games? Suppose he has called all the
symbols five times over except one let us say, star, and he has two calls to
make. Will he not reason that these now must be stars because he has called all
the others?" Obviously, as I have suggested already, he has no way of knowing
whether or not the other calls have been correct, so it would be most unfounded
reasoning to conclude that the last two must be stars. Only if he knew the
correctness of the cards already called could reasoning help him, and this he
does not know. Therefore, the chances remain the same on the twentyfifth call
as on the first, since he is just as ignorant about what that card is.
"Might not a subject use some system of advantage to him?" How can he, if he has
nothing to go on? If he does not know whether his calls are correct or
incorrect, no system could work. A system without a basis in fact would be
nothing but a delusion.
A curious question has been raised and vehemently urged in one or two places. It
is supposed that all our investigators in this research might be stopping at
some strategic moment  say, after some high scores have been made and just
before a series of low ones might be made. The very essence of this question is
to assume that we can tell somehow by previous runs what the next ones are going
to be. If our results are due to chance, this could not be done. What we mean by
the term "chance" is the very absence of a fixed order and predictability.
However, to settle the matter, one of my critical colleagues, who believed this
was a weakness in our work, tested out the supposed principle in actual
experimentation and found no evidence of it.
At times we have been told that perhaps something is wrong with our using a pack
of 25 cards, and we have been urged to try packs of 100 or 1,000. There are no
adequate mathematical grounds available for such insistence, and even from a
commonsense point of view it is difficult to see what difference it would make.
However, some of our best work has been done without adhering strictly to a pack
of 25. It will be recalled that Pearce's twentyfive straight successes were
made by calling one card, checking it at once, returning it to the pack, and
cutting. In this way the pack was an unending one. It might have been a hundred
or a thousand or any other number. Considerable later work has been done with
packs of 50, and on some occasions of even larger size.
After weighing all the criticism we have been able to get in seven years' time,
I have come to feel as much security in the general soundness of the research as
is good for an investigator in science to have. Reflecting upon the enormous
amount of work that has been done here and elsewhere, it seems to me that no
inferential scientific conclusion has ever had so much evidence in its support;
that is, in excluding a chance hypothesis. The mathematics has been
questioned, yes, but not by a single mathematician. Two psychologists have
written a total of four articles criticizing it, but the author of three of them
has become satisfied that his criticisms do no apply now that he has what he
feels is sufficient further information. A third psychologist has more recently
published a review of the criticisms, and he asserts that the statistics used in
this research are substantially correct.
Among mathematicians the best authority is with us. Confirmatory mathematical
checks have mounted by tens of thousands, not only in this laboratory but in a
number of other places. It is difficult to see what further mathematical
criteria can be applied to evaluate the results of our tests.
Thus far, it would appear, we have been on sound territory. Whatever we have
claimed to be beyond chance has stood the tests and is safe. But our experiments
are still going on. They are going on into yet more meaningful, more
revolutionary lines. The strain upon this mathematics of probability will be
increasingly great with every advancing step along the lines we are at present
following. With the enormously greater burden anticipated for this technique of
evaluation, it is high time that we secure the last word, both in criticism and
in support. We shall need it.
Note:
The article above was taken from J. B. Rhine's "New Frontiers of the Mind"
(1937, Farrar & Rhinehart).
