CHAPTER I-4
CONTROVERSIES ABOUT THE CONCEPTS OF PROBABILITY AND CHANCE
Philosophers have wrestled long and hard with the nature of
probability and its proper interpretation. The two main
contending schools of thought - a argument which has been going
on for centuries if not for millenia - are most commonly labeled
as "objectivists" or "frequentists" on one side, and
"personalists" or "subjectivists" or "Bayesians" on the other
side.
Among the frequentists, the probability of an event is said
to be the proportion of times that the event has taken place in
the past, usually based on a long series of trials. Insurance
companies use this type of definition when they estimate the
probability that a thirty-five-year-old postman will die during a
period for which he wants to buy an insurance policy. (Notice
this shortcoming: Sometimes an insurance company must bet upon
events that have taken place only infrequently or never before -
damage to a female movie star's legs or a male football
quarterback's arm - and therefore one cannot reasonably reckon
the proportion of times the event occurred one way or the other
in the past.)
At the core of frequentists' thinking is the notion of
properties. For example, the analytic table of contents for
Hacking's influential Logic of Statistical Inference begins
with
Statistical inference is chiefly concerned with a
physical property, which may be indicated by the name
long run frequency. The property has never been well
defined. Because there are some reasons for denying
that it is a physical property at all, its definition
is one of the hardest of conceptual problems about
statistical inference - and it is taken as the central
problem of this book...The long run frequency of an
outcome on trials of some kind is a property of the
chance set-up on which the trials are
conducted...Chance is a dispositional property...(1965,
p. iii)
Another sort of property notion defined probability for
Charles Peirce. Isaac Levi tells us that "Probability, according
to Peirce, is a property of rules of inference" (1980, p. 130).
As noted in the previous chapter, the concept of the
properties of time was a stumbling block for physics until
Einstein. The concept of the properties of probability is a
similar stumbling block for statistics. An example of the
difficulty with properties: The theorist asserts that the
probability of a die falling with the "5" turned up is 1/6, on
the basis of the physics of equally-weighted sides. But if one
rolls a particular die a million times, and it turns up "5" less
than 1/6 of the time, one clearly would use the observed
proportion as a practical estimate. And the likelihood of a
given number turning up with cheap dice depends upon the number
of pips drilled out. We should think of the a priori engineering
estimate (the theoretical, or ideal) and the observed (empirical)
estimate as two alternative ways to proceed, as we do in other
applied situations (for example, when making estimates of a cost;
see my 1975 book). Such alternative estimating processes
constitute alternative operational proxies for the conceptual
probability. One must use additional knowledge to decide which
such proxy to use. I suggest that as a general rule, an
empirical proxy is better when it is available.
The great mathematician Littlewood asserts that the
fundamental axiom underlying any philosophical (frequentist)
theory of probability that would give the concept of probability
meaning in the real world is "inherently incapable of deductive
proof...also incapable of inductive proof" (1986, p. 72). The
mathematical theory of probability has no such problems. But
If he is consistent a man of the mathematical school
washes his hands of applications. To someone who wants
them he would say that the ideal system runs parallel
to the usual (mathematical) theory: `If this is what
you want, try it: it is not my business to justify
application of the system; that can only be done by
philosophizing; I am a mathematician'. In practice he
is apt to say: `try this; if it works that will justify
it' (p. 73)
For the pragmatist, that it does work is justification for
using it. And from the point of view of operational definitions,
not justification is needed.
Littlewood's argument is consistent with what is said about
the role of judgment in Chapter 00.
There is nothing new about the conclusion that one must
accommodate various ways of estimating and thinking about
probability; anyone who tries to work in a practical setting is
forced to this conclusion. In the 17th century, founding father
of probability James Bernoulli distinguished between a priori
(deductive) and a posteriori (empirical) probabilities (Hald,
1990, p. 193).
A conclusion: One should not think of what the probability "is"
in any given situation, or in general. Indeed, the word "is" is
(sic) one of the most confusion-inducing words in the English
language Rather, one think consider how best to estimate the
probability. The objectivists insist that any estimation not based on a
series of observations is so much at risk of potential bias that
we should never think of probability that way. They are worried
about the perversion of science, the substitution of arbitrary
assessments for value-free data-gathering. This is certainly a
legitimate concern, and one to be guarded against as much as
possible, especially in scientific work which inherently should
aim at objectivity.
In Bayesian estimation, there is the serious problem of the
scientist estimating prior probabilities after seeing the sample
evidence instead of before. (Vulgarly, this might be called
"pulling priors from your posterior", but I would not be so
vulgar, of course.)
The Bayesians and other personalists argue, however, that
some judgment is unavoidable - about which observations will be
used, which topics will be examined in depth, and so on. And
they are certainly right in that.
For subjectivists, the probability that an event will take
place or that a statement is true can be said to correspond to
the odds at which you would bet that the event will take place.
But there is an important shortcoming of this definition: You
might be willing to accept a five-dollar bet at 2-1 odds that
your team will win the game, but you might be unwilling to bet a
hundred dollars at the same odds. A more important example is
the case of life insurance. (Here I follow Bertrand Russell,
1948, pp. 341-342.) The insurance firm has little difficulty in
estimating the probability that I will die in the next 12 months,
and there follows straightforwardly the expected loss if I die
and the minimum price at which they would sell me a policy. But
for my own decisions the problem is more complex. First, I can
adduce various factors about which the insurance company has no
knowledge, such as the probability that I will go scuba diving
and drown, but I cannot bring frequency series to bear on this
and many other factors. Second, my evaluation of the "utility"
to those I leave behind of the payments if I die is very complex;
indeed, the first "rational" approach to the problem has only
recently been done (Levy, Simon, and Doherty, 1988).
The subjectivists are surely right that we must make guesses
about likelihoods, using various common-sense devices, when we
lack long series of experiences. But as will be discussed in
Chapter 00 on statistical inference, this does not imply that we
should always work within a Bayesian framework and apply prior
probability distributions.
Consider this example of the difficulty of finding an
"inherent" difference between a frequency series and a degree of
belief. You want to estimate the chances of the Blue basketball
team beating the Green team. Perhaps you count the number of
times each has won in the past year, according to the scores in
the newspaper; that's a frequency series and you base a
probability estimate on the proportion... Or, you simply remember
the general numbers of wins, without checking the papers. Is
that still a frequency series?. Probably...Now, how about if you
have only seen them play once without knowing what the score is,
but you observe the players and the plays closely, and you run
through a good many scenarios in your mind, each one leading to a
game score, and overall to a proportion one by Blue. Is that a
frequency series? If not, why not?
As is so often the case, the various sides in the argument
have different cases in mind. And unfortunately the disputants
(especially the philosophers) argue on without citing particular
examples (even as good a philosopher as Braithwaite, 1953). The
arguments disappear if one thinks operationally with respect to
the purpose of the work, rather than in terms of properties. But
the difficulties of deciding how best to estimate probabilities
can never disappear, because estimates always will require wisdom
and balanced judgment - qualities which will always be rare among
us very-human human beings.
Different ideas about concepts of probability are
appropriate for business and engineering decisions, for
biostatistics, and for scientific explanation (the case the
philosophers mostly focus upon).
It is instructive that the probability estimate "prob value"
that emerges from the complex results of an empirical study, and
the statistical inference based on that empirical study, do not
seem to be easily classifiable as either a frequentist idea or as
a "degree of belief" idea; they seem to be neither, and a very
different sort of concept entirely.
Again, one should not think about what probability "is", or
"really is", but rather, how best to estimate it in a particular
circumstance. It is quite amazing how much confusion can follow
from the notion of "really". It is not rare to find a person who
will agree that the probability of a head on a forthcoming toss
of a fair coin is 1/2. But after the coin has been tossed and is
on the back of your hand yet still covered and has not yet been
examined, the person will say that the probability is "really"
either 0 or 1. Ask that person: What if the coin toss will be
seen on video and has already been taped but the toss has not
been shown? If that example does not alter the person's point of
view, next ask: What about if you know that the toss is to be
shown on television, but you do not know if it has been photo-
graphed yet? And then: How about if you are told that the chance
is 1/2 that it has or has not been photographed? Trying to hew
to a position of "really", rather than to the idea that
probability refers to our knowledge of an event, quickly leads to
absurdities.
The economist Frank Knight wrote a famous book
distinguishing between risk and uncertainty in business -
uncertainty as akin to frequency series in insurance, risk as
akin to investment decisions. The distinction may be a useful
one in understanding business behavior, just as it can be useful
to an insurance executive to know whether shipping insurance
rates are being based on extensive data or on crude guesses. But
there need be no conceptual distinction made in this regard in
our discussion of the general concept of probability.
The most important general point: One cannot sensibly talk
about a given probability in the abstract, without reference to
some set of actual facts. The notion then loses meaning and
invites confusion and argument. This also is a reason why
general formalizations of a probability concept make little or no
sense. One cannot formalize an empirical probability, but only a
theoretical probability.
There is simply no need for the type of discussions of
probability engaged in by Keynes, von Mises, and Reichenbach,
among many others. The frequentist-subjectivist argument is of
no interest to the pure mathematician; the probabilist and the
mathematical statistician simply do not need it. They pick up
the matter at the point at which the sample space is defined, and
do not inquire as to the meaning of the sample space or the
points within it, but simply focuses on the manipulations of that
space. As the pre-eminent probabilist William Feller put it, "A
salient feature of mathematics is that it is concerned solely
with relations among undefined things" (1950, p. 1). The great
mathematician J. E. Littlewood put it this way: "Mathematics (by
which I shall mean pure mathematics) has no grip on the real
world; if probability is to deal with the real world it must
contain elements outside mathematics" (1986, p. 71). And the
practitioners in various fields do not need it either; it would
be nothing but a distracting, confusing, irrelevancy.
The fact of the matter is that the philosophers are
literally not in touch with the relevant reality - that is, the
use of probabilities in everyday life. This can be seen in the
complete absence of actual examples in their writing.
Consider again the estimation of the probability of failure
for the tragic flight of the Challenger shuttle discussed in the
previous chapter. Little of the philosophical discussion seems
relevant to this case. It seems doubtful that philosophers who
are doctrinaire as frequentists or for other positions could
throw helpful light on space shuttle flight discussions.
THE CONCEPT OF CHANCE
The related concepts of randomness and chance are
fundamental in statistical inference, so we must discuss them.
The concept of risk also is intertwined with statistics, but this
book assumes that decision-makers are risk-neutral with respect
to all matters touched on here, or that risk will be allowed for
at a later stage of analysis. (See Simon, 1975, concerning risk
in decision-making.)
The study of probability focuses on randomly generated
events - that is, events about which there is uncertainty whether
or not they will occur. The older view is illustrated by the
sentence "So what is chance?" written by Mark Kac (1985, p. 76),
one of the founders of modern probability theory. I urge the
point of view (which seems to be the emerging consensus view)
that the uncertainty refers to your knowledge of a future event
rather than to a property of the event itself.[fn] Avoiding
discussion of the properties of chance is quite consistent with
avoiding discussion of the properties of probability, as
discussed earlier in this chapter.
The incomparable David Hume wrote a very clear and useful
statement about chance: "[C]hance, when strictly examined, is a
mere negative word, and means not any real power which has
anywhere a being in nature" (An Enquiry, p. 104, Open Court Ed.)
But this insight was left to molder by generations of
philosophers, physicists, statisticians, and sports
enthusiasts.<1>
For example, consider this lecture illustration with a salad
spoon. I spin the salad spoon like a baton twirler. If I hold
it at the handle and attempt to flip it so that it turns only
half a revolution, I can be almost sure that I will correctly get
the spoon end and not the handle. And if I attempt to flip it a
full revolution, again I can almost surely get the handle
successfully. It is not a random event whether I catch the handle
or the head (here ignoring those throws when I catch neither end)
when doing only half a revolution or one revolution. The result
is quite predictable in both these simple maneuverers so far.
When I say the result is "predictable", I mean that you
would not bet with me about whether this time I'll get the spoon
or the handle end. So we say that the outcome of my flip aiming
at half a revolution is not "random".
When I twirl the spoon so little, I control (almost
completely) whether it comes down the handle or the spoon end;
this is the same as saying that the outcome does not occur by
chance.
The terms "random" and "chance" implicitly mean that you
believe that I cannot control or cannot know in advance what will
happen.
Whether this twirl will be the rare time I miss, however,
should be considered chance. Though you would not bet at even
odds on my catching the handle versus the spoon end if there is
to be only a half or one full revolution, you might bet - at
(say) odds of 50 to 1 - whether I'll make a mistake and get it
wrong, or drop it. So the very same flip can be seen as random
or determined depending on what aspect of it we are looking at.
Of course you would not bet against me about my not making a
mistake, because the bet might cause me to make a mistake
purposely. This "moral hazard" is a problem that emerges when a
person buys life insurance and may commit suicide, or when a
boxer may lose a fight purposely. The people who stake money on
those events say that such an outcome is "fixed" (a very
appropriate word) and not random.
Now I attempt more difficult maneuvers with the ladle. I
can do 1 1\2 flips pretty well, and two full revolutions with
some success - maybe even 2 1/2 flips on a good day. But when I
get much beyond that, I cannot determine very well whether I'll
get handle or spoon. The outcome gradually becomes less and less
predictable - that is, more and more random.
If I flip the spoon so that it revolves three or more times,
I can hardly control the process at all, and hence I cannot
predict well whether I'll get the handle or the head. With 5
revolutions I have absolutely no control over the outcome; I
cannot predict the outcome better than 50-50. At that point,
getting the handle or the spoon end has become a very random
event for our purposes, just like flipping a coin high in the
air. So at that point we say that "chance" controls the outcome,
though that word is just a synonym for my lack of ability to
control and predict the outcome. "Chance" can be thought to
stand for the myriad small factors that influence the outcome.
We see the same gradual increase in randomness with
increasing numbers of shuffles of cards. After one shuffle, a
skilled magician can know where every card is, and after two
shuffles there is still much order that s/he can work with. But
after (say) five shuffles, the magician no longer has any power
to predict and control, and the outcome of any draw can then be
thought of as random chance.
At what point do we say that the outcome is "random" or
"pure chance" as to whether my hand will grasp the spoon end, the
handle, or at some other spot? There is no sharp boundary to
this transition. Rather, the transition is gradual; this is the
crucial idea, and one that I have not seen stated before.
Whether or not we refer to the outcome as random depends upon the
twirler's skill, which influences how predictable the event is.
A baton twirler or juggler might be able to do ten flips with a
non-random outcome; if the twirler is an expert and the outcome
is highly predictable, we say it is not random but rather is
determined.
A baton twirler or juggler might be able to do ten flips
with a non-random outcome. Again this shows that the randomness
is not a property of the physical event, but rather refers to a
person's capacity to predict or explain or control a
phenomenon.[1]
So we define "chance" as the absence of predictive power
and/or explanation and/or control.
As to those persons who wish to inquire into what the
situation "really" is: I hope they agree that we do not need to
do so to proceed with our work. I hope all will agree that the
outcome of flipping the spoon gradually becomes unpredictable
(random) though still subject to similar physical processes as
when predictable. I do not deny in principle that these processes
can be "understood"; certainly one can develop a machine (or a
baton twirler) that will make the outcome predictable for many
turns. But this has nothing to do with whether the mechanism is
"really" something one wants to say is influenced by "chance".
This is the point of the cooking-spoon demonstration. The
outcome traverses from non-chance (determined) to chance (not
determined) in a smooth way even though the physical mechanism
that produces the revolutions remains much the same over the
traverse[1].
Consider, too, a set of fake dice that I roll. Before you
know they are fake, you assume that the probabilities of various
outcomes is a matter of chance. But after you know that the dice
are loaded, you no longer assume that the outcome is chance. This
illustrates how the probabilities you work with are influenced by
your knowledge of the facts of the situation.
Admittedly, this way of thinking about probability takes
some getting used to. For example, suppose a magician does a
simple trick with dice such as this one:
The magician turns his back while a spectator throws
three dice on the table. He is instructed to add the
faces. He then picks up any one die, adding the number
on the bottom to the previous total. This same die is
rolled again. The number it now shows is also added to
the total. The magician turns around. He calls
attention to the fact that he has no way of knowing
which of the three cubes was used for the second roll.
He picks up the dice, shakes them in his hand a moment,
then correctly announces the final sum.
Method: Before the magician picks up the dice he
totals their faces. Seven [the opposite sides of the
dice always add to seven] added to this number gives
the total obtained by the spectator. (Gardner, 1956,
pp. 42-44).
Can the dice's sum really be random if the magician knows
exactly what it is - as you also could, if you knew the trick?
Forget about "really", I suggest, and accept that this simply is
a useful way of thinking.
The concept of many small, independent influences also is
the explanation for the shape of the Normal distribution (for
"normalized", rather than being the "normal" shape of many
distributions found in nature). Chapter IV-1 explains how it is
that many sets of data are observed in the pattern of the Normal
distribution. To preview that discussion with respect to the
topic of heights: When we consider all living things taken
together, the shape of the overall distribution - many
individuals at the tiny end where the viruses are found, and very
few individuals at the tall end where the giraffes are - is
determined mostly by the distribution of species that have
different mean heights. Hence we can explain the shape of that
distribution, and we do not say that is determined by "chance".
But with a homogenous cohort of a single species - say, all 25-
year-old human females in the U.S. - our best description of the
shape of the distribution is "chance". With situations in
between, the shape is partly due to identifiable factors - e.g.
age - and partly due to "chance".
Or consider the case of a basketball shooter: What causes
her or him to make (or not make) a basket this shot, after a
string of successes? Only chance, because the "hot hand" does
not exist. But what causes a given shooter to be very good or
very poor relative to other players? For that explanation we can
point to such factors as the amount of practice or natural
talent.
Again, all this has nothing to do with whether the mechanism
is "really" chance, unlike the arguments that have been raging in
physics for a century. That is the point of the ladle
demonstration. Our knowledge and our power to predict the
outcome gradually transits from non-chance (that is,
"determined") to chance ("not determined") in a gradual way even
though the same sort of physical mechanism produces each throw of
the ladle.
There is a long-running argument in physics, with Einstein
and Bohr embroiled on opposite sides of the issue, about whether
one must "ultimately" arrive at a point at which in principle
further investigation cannot elucidate the factors that determine
an event. Ekeland, based on Poincare's work, says yes. And
this is the point of the famous Uncertainty Principle of
Heisenberg. Perhaps the situation is easier for non-physicists
to comprehend if we simply recognize that as one seeks finer and
finer measurement, one inevitably arrives at a point at which
there is measurement error - whether the measurement is of a
person's height or the speed of an electron. Indeed, dealing
with such measurement error in astronomy is one of the main
historical roots of modern statistics.
Hayek (1980, Chapter 2?) argues that this is also the case
for individual human behavior in the context of a market or other
social system. The desires and knowledge inside a human mind can
never be specified fully in advance, in considerable part because
they depend upon the behavior of other persons who also are
influenced by unknown desires and knowledge.
Earlier I mentioned that when we say that chance controls
the outcome with the spoon after (say) five revolutions, we mean
that there are many small forces that affect the outcome - each
of whose effect is not known and each of which is independent of
the other. None of these forces is large enough for me (as the
spoon twirler) to deal with, or else I would deal with it and be
able to improve my control and my ability to predict the outcome.
This concept of many small influences - "small" meaning in
practice those influences whose effects cannot be identified and
allowed for - which affect the outcome and whose effects are not
knowable and which are independent of each other is fundamental
in statistical inference. This concept is the basis of the
Theory of Errors and the Central Limit Theorem, which enable us
to predict how the mean of a distribution will behave in repeated
sampling from the distribution, as will be discussed later.
That is, the assumptions of the Central Limit Theorem and of
the Normal distribution are the conditions that produce an event
that we say is chance-like.
It is interesting to consider the relationship of this
concept to the quincunx: Therein, any one ball's fate seems
chance-like, but the overall distribution is determined.
A definition of the complexity of a series of numbers is the
number of lines of computer code required to produce it. One
might then think of a series as random if representing it in the
shortest fashion with a computer program requires as many lines
of computer code as the series has entries (Barrow, 1992, pp.
135-136).
Ekeland makes the point that "throwing a die may be seen as
both a deterministic and a random event. Celestial mechanics
also has this double character. The laws of motion are purely
deterministic. But certain trajectories are so irregular...that
they seem to arise from some kind of random motion. They
certainly become unpredictable very quickly" (1988, p. 49) This
inherent unpredictability of celestial systems was the discovery
of the great mathematician Henri Poincare'(Ekeland, p. 35). In
technical terms, "the equations of dynamics are not completely
integrable" (p. 35), which appears in connection with the famous
three-body problem.
Ekeland explains that "Randomness appears because the
available information, though accurate, is incomplete" (p. 62).
And he continues, "So, if determinism means that the past
determines the future, it can only be a property of reality as a
whole, of the total cosmos. As soon as one isolates, from this
global reality, a sequence of observations to be described and
analyzed, one runs the risk of finding only randomness in that
particular projection of the deterministic whole" (p. 62). All
this leads directly into chaos theory.
Ekeland then asks:
[I]f we cannot predict individual trajectories, what is
there left to study? What can science say about an
unpredictable system?
For dice throwing the answer has been known for a
long time. One should forget about individual throws,
and consider instead the set of all possible
throws...We attribute to each of them a probability
1/6, and this is the beginning of probability theory
(p. 70).
Einstein's famous comment that God does not play dice with
the universe is, upon consideration, a red herring. It is rather
sure that Einstein would also have denied the apparent opposite,
that God does not operate a set of levers which control the
outcomes of all natural and human events. Just what "God" does
or does not do is not a relevant question in this context. OUr
aim should be to get about our business of understanding the
world and acting in it as well as we can.
There must be interesting psychological dimensions to the
concepts of randomness and determinism, else why would it matter
to someone if a series of numbers used in statistical inference
is "truly" random or only "quasi-random". Does it matter if the
series used are the digits of pi, which are predictable once you
know that that is the series being used and where the start is,
if in fact you do not know those things? From any practical
standpoint I think not; yet many people care. Perhaps there is a link to other matters of "natural" versus
"artificial". I remember as a youth seeing the famous glass
flowers in a Harvard University museum. With the naked eye these
artifacts" (which are "determined" by human activity) cannot be
distinguished from the real thing (which are "random" in their
characteristics) - which is what makes you marvel at the
artificial flowers, whereas we do not marvel at real flowers
which look like them. Why? ...On the other hand, nature lovers
enjoy "pristine" wilderness but reject "artificial" wilderness
that has the same characteristics. Why?...There are deep
connections between pattern and order and meaning which are
interesting but beyond our scope here.<2>
SUMMARY AND CONCLUSIONS
We define "chance" as the absence of predictive power and/or
explanation and/or control. The gradualness of the transition
from determined to chance is the key new element in the
discussion of chance.
When the spoon rotates more than three or four turns I
cannot control the outcome - whether spoon or ladle end - with
any accuracy. That is to say, I cannot predict much better than
50-50 with more than four rotations. So we then say that the
outcome is determined by "chance".
As for those people who wish to inquire whether a given
series of numbers are "really" random: In principle, there can
never be a test that shows a series to be "truly" random<3>. The
best that can be done is to subject a series of numbers to tests
that show that is a series is not random; a series can be said
to be random if it has passed all tests invented until now. (The
essence of the tests is to ask whether one can predict a given
digit with probability higher than chance.) Even from a
classical formal point of view, one of Kac's basic theorems "even
raise[s] serious doubts as to whether randomness is an
operationally viable concept" (Kac, 1985, p. 765). And it
follows from Godel's work that "whether a sequence of numbers is
random or not is logically undecidable" (Barrow, 1992, p. 138).
"There is no general algorithmic criterion which would enable us
to determine whether any given system is chaotically random or
not" (p. 241).
**FOOTNOTES**
[1]: This discussion suggests an irony in the curious fact
that probability theory is a conceptually-exact and therefore
deterministic replica of an particular situation that can never
be specified completely (a la Poincare and Ekeland in Ekeland,
1988). Yet probability theory claims to be a better replica than
the resampling method, which mirrors a particular situation's
indeterminism (though of course it is not a perfect replica,
either).
[1]: The idea that our aim is to advance our work in
improving our knowledge and our decisions, rather than to answer
"ultimate" questions about what is "really" true is in the same
spirit as some writers about quantum theory. In 1930 Ruarck and
Urey wrote: "The reader who feels disappointed that the
information sought in solving a dynamical problem on the quantum
theory is [only] statistical...should console himwelf with the
thought that we seldom need any information other than that which
is given by the quantum theory" (quoted by Cartright, 1987, p.
420).
This does not mean that I think that people should confine
their learning to what they need in their daily work. Having a
deeper philosophical knowledge than you ordinarily need can help
you deal with extraordinary problems when they arise.
ENDNOTES
**ENDNOTES**
<1>: A wonderfully clear statement of this point of view was
given almost a century ago by Whitworth (1897/1965, pp. xxiv-xxx.
(Indeed, this is one of the most penetrating writings on
probability and statistics that I have found, and perfectly
contemporary now.
...all chance is a function of limited knowledge...
(DCC Exercises, by William Allen Whitworth, Hafner
Publishing Company, New York and London, 1897/1965, p.
xxi.)
Chance implies a limited knowledge, and in every ques-
tion of chance there are certain data known and certain
conditions unknown. (DCC Exercises, by William Allen
Whitworth, Hafner Publishing Company, New York and
London, 1897/1965, p. xxi.)
...Chance has to do altogether with what we have reason
to expect. It therefore depends upon our knowledge or
upon our ignorance. It is a function of our knowledge,
but that necessarily a limited and imperfect knowledge.
(DCC Exercises, by William Allen Whitworth, Hafner
Publishing Company, New York and London, 1897/1965, p.
xxii.)
...It is because they have not grasped the idea of
Probability as a science concerned with limited knowl-
edge that some writers feel a repugnance to speaking of
chance in regard to an event already decided. (DCC
Exercises, by William Allen Whitworth, Hafner Publish-
ing Company, New York and London, 1897/1965, p. xxiv.)
...a friend sets out for a voyage on a vessel which is
carrying 50 passengers, of whom 30, including our
friend, are Englishmen and the rest are foreigners. A
telegram is published that a passenger fell overboard
and was lost. Apart from any knowledge I may have of
our friend's ability to take care of himself the chance
that the lost passenger is our friend if 1/50. The
event is actually decided. To the captain of the
vessel there is no probability whatever; he knows for
certain which passenger is lost, but in my limited
knowledge the odds are 49 to 1 against it being our
friend.
But a second telegram is received which describes the
lost passenger as an Englishman. The accession of new
knowledge at once raises the chance to 1/30, or reduces
the odds to 19 to 1 against the loss of our friend.
(DCC Exercises, by William Allen Whitworth, Hafner
Publishing Company, New York and London, 1897/1965, p.
xxv.)
...the theory that the chance of an event is something
inherent in the event itself, independent of the condi-
tions of the observer. I assert that the mathematical
probability of the same event may be different to the
same observer as he becomes possessed of greater infor-
mation. (DCC Exercises, by William Allen Whitworth,
Hafner Publishing Company, New York and London,
1897/1965, p. xxvii.)
...in practice, when we get beyond the region of cards
and dice, probabilities are generally of a mixed kind,
that is, they involve elements which can only be esti-
mated by an exercise of judgment as well as others
which lend themselves to arithmetical calculation. The
probability of a man of 20 living to be 60 has been
stated as 59/97, but if you have reason to think the
particular man to be stronger or less strong than the
average, if you know him to belong to a long-lived or a
short-lived family, the price at which you would buy a
remainder contingent on his reaching the age of 60 will
be modified, and the modifying element will be one
which can only be estimated, not calculated. (DCC
Exerises, by William Allen Whitworth, Hafner Publishing
Company, New York and London, 1897/1965, p. xxix.)
<2>: On a personal note, it is a great gift for me to
be able to work on a subject at age 61 that allows me
to connect up threads of thought that first came into
my hands when I was perhaps 18, and that have been
loose in my mind since then. Perhaps this very feeling
of comfort from this order is related to the topic at
hand.
<3>: Gardner (1977, p. 163) puts it this way: "Is there
any objective, mathematical way to define a completely
disordered series? Apparently there is not."