Figures 1-2, table 1
CHAPTER IV-I
WHY STATISTICS IS SO DIFFICULT, AND WHY RESAMPLING IS EASIER
Julian L. Simon
A successful statistical inference is as difficult a feat of
the intellect as one commonly meets. This is not because of
mathematical difficulties. Rather, it is due to the long chain
of reasoning connecting the original question with a sound
conclusion; indeed, the mathematical operations involved in
estimating probabilities once the problem has been correctly
specified can be straightforward, especially when one estimates
with the experimental resampling method rather than with
formulaic sample-space methods.
Indeed, perhaps the greatest benefit of the resampling
approach is that it clears away the mathematical difficulties so
that the difficult philosophical and procedural issues can be
seen clearly, and hence may be tackled head on. Ironically,
however, this characteristic of reducing the computational
difficulties has also been a drawback of the resampling method;
by making the necessary mathematical operations so simple as to
be accessible to any clear-thinking layperson, resampling has
made the assistance of professional statisticians seem less
necessary, which naturally educes resistance by the
traditionalists in that profession.
To restate the point: The great challenge presented by the
ideas of statistics does not stem from the need for a large body
of prerequisite knowledge, or for mathematical sophistication and
inclination. The challenge stems, rather, from the inherent
difficulty of making sense of a complicated situation. Indeed, if
any particular situation is not hard to understand, statistical
inference is not needed. The difficulty lies in there being a
very long sequence of decisions that must be made correctly about
such matters as the nature of the correct hypothetical
population, the correct sampling procedure, and so on. This
essential difficulty will be seen more clearly in Chapters 00
where canonical procedures for confidence intervals and
hypothesis tests are set forth.
THE ESSENCE OF THE DIFFICULTY OF MATHEMATICS\statlect\xintro for
use in first day's lecture and maybe improvbed
there\statbook\yintro_c. Equated on December 28, 1995 What is
the area of a table that is 6 feet long and 4 feet wide?
Twenty-four square feet, you say. But how do you know that
the answer is 24?
Multiply length times width, you say.
But are the 6 and 4 units in the lengths you are multiplying
the same as the 24 units in the answer? Notice that a length is
in feet and the answer is in square feet. So how can you start
talking about one entity and end up talking with another entity?
Never mind; that's just a minor complication.
Why do you multiply length times width?
Because that's the formula I learned in school, you say.
But can you prove that that is the correct formula? Most of us
can't. Simple as it is, such a deductive proof is not easy. It
includes a chain of definitions and substitutions which is not
easy to follow, let along to produce. So it should not be
surprising to us that more complex problems in mathematics are
hard to follow.
Do we have to do such difficult intellectual work? No.
Let's instead take a piece of graph paper, mark off a length of 6
units and a width of 4 units, and count the number of squares
that are included. The formula does the same thing as we do in
the graph but more abstractly, and the proof of the formula
probably only deduces why the formula is the same as what we have
done. So we can do without the formula. And we can use the same
device for more complex problems - say, a trapezoid.
One day I sat down to read an essay about the famous
Goedel's proof which shook mathematics in the 1930s by showing
that the foundations of mathematics are less intellectually solid
that mathematicians had believed. I had no hope of ever reading
Goedel's paper itself, but I tackled an essay by the philosopher
Ernest Nagel and the mathematician-journalist James R. Newman
(1956) in hopes that they could relate to me enough of the
central idea so that I could grasp its spirit.
The paper contains almost no formulae, and very little
mathematical manipulation. Yet as I tackled it weekend after
weekend, I found that I could not follow its logic in a
satisfactory manner. So I tried to figure out why I was having
such difficulty.
The difficulty for me seems to be that in the series of
steps in the logic there is continual substitution of one set of
symbols - either a formula, or part of a formula - for another
formula. This means that at each substitution one must remember
the meaning of both the symbols in the previous step as well as
in the next step - that is, one must keep in one's mind the
meaning of all the many symbols being used. Nagel and Newman
reduce the number of symbols and sets of them very sharply - they
note that Goedel's paper contains "Forty-six preliminary
definitions together with several important lemmas must be
mastered" before even beginning the main work (p. 1688);
nevertheless, I realized that my memory could not hold the
necessary meanings of even the smaller number of symbols Nagel
and Newman use as they make one substitution after another.
You hear and read mathematicians saying that the ability to
hold a long chain of reasoning in mind is a characteristic of
their field. Let's agree with them about that. But we need not
accept the apparent implication of the statement - that being
able to do so is the mark of superior intelligence. Rather,
handling long chains of reasoning is just one special ability,
and one that can be counter-productive.
Consider an analogy: A juggler shows you a pile of glass
balls atop a tool that you need to do a job. S/he says: "Watch
how I get this tool for you", and proceeds to throw all the glass
balls in the air and keep them there, remove the tool and give it
to you, then take the balls out of the air and replace them one
by one in the pile without breaking any. Then s/he puts the tool
back and says, "Here is the pile. Now you lift the balls, remove
the tool, and put the balls back in the original spot".
The implication is that we, too, should learn how to juggle
the balls so as to extract the tool. But there is another way:
Take all the balls off the pile and put them in another pile
nearby, take the tool away, and again pile the balls on the
original spot. That way you do not need to keep the balls in the
air at all.
And so it is with statistics and probability. You do not
need to obtain the tool you need by expending great amounts of
time and effort to learn how to keep all the intellectual balls
in your mind - which you probably could not learn to do
satisfactorily no matter how you try, if you are like me.
Instead, you can learn how to get what you need without keeping
anything in memory by instead using concrete elements - such as
the use of a picture for solving the fathers-and-sons puzzle (see
Chapter 00, page 000).
Many of us find it difficult to turn away from the
mathematician's demand that we carry out the task using his/her
deductive method. In contrast, if the juggler brags about
his/her skill with hand and eye, we reject the claim that that is
a sign that s/he is just plain a better athlete than the rest of
us. For example, I doubt my ability to learn juggling well, but
my hand-eye coordination hitting a a ball to a tiny spot with a
squash or racquetball racket was at one time quite good for my
age and experience; and juggling is just one among many hand-eye
skills.
But with formulaic mathematics it is different. We all have
to play that game at least a little bit in school and college.
And hence it is harder for us to turn away from the claim that
skill in mathematical manipulation is just one special talent,
and not a higher one.
It is interesting to note the discussion by Friedrich Hayek
- a Nobel-prize winner in economics, and in my view the greatest
social scientist of the 20th Century - of two kinds of minds
among eminent scientists (1978, Chapter 4). One type calls
"master of the subject", and the other he calls "the puzzler" and
sometimes "the muddler"; it is the latter that he said he himself
possessed. I would suggest that a "master of the subject" is the
type that (among other characteristics) handles formulaic methods
with dexterity, and operates comfortably at high levels of
abstraction. In contrast, a puzzler constantly feels the need to
go back to first principles, if only because s/he cannot well
remember what underlies propositions at a high level of
abstraction. The puzzler is likely to more drawn to resampling
and experimental methods generally that is a master of the
subject.
WHY SIMULATION MAKES STATISTICS EASIER
The intellectual advantage of simulation in general, and of
the resampling method in statistics, is that though it takes
repeated samples from the sample space, it does not require that
one know the size of the sample space, or the number of points in
a particular partition of it. To calculate the likelihood of
getting (say) 26 "4"s in 130 dice throws with the binomial
formula requires that one calculate the total number of possible
permutations of 130 dice, and the number of those permutations
that include 26 or more "4"s. Gaussian distribution-based
methods often are used to approximate such processes when sample
sizes become large, which introduces another escalation in the
level of intellectual difficulty and obscurity for persons who
are not skilled professional mathematicians. In contrast, with a
simulation approach one needs to know only the conditions of
producing a single "4" in one roll.
Indeed, it is the much lesser degree of intellectual
difficulty which is the secret of simulation's success, because
it improves the likelihood that the user will arrive at a sound
solution to the problem at hand - which I hope that you will
agree is the ultimate criterion.
One may wonder how simulation performs this economical trick
of avoiding the complex abstraction of sample-space calculations.
The explanation is that it substitutes the particular information
about how elements in the sample are generated in that specific
case, as derived from the facts of the actual circumstance; the
analytic method does not use this information.
Recall that Galileo solved the problem of why three dice
yield "10" and "11" more frequently than "9" and "12" by
inventing the concept of the sample space. He listed all 256
possible permutations, and found that there are 27 permutations
that yield "10" and "11", but only 25 that yield "9" and "12".
But it took a Galileo to perform this intellectual feat, simple
as it seems to professionals now; lesser theorists had made the
error of assuming that the probabilities of all four numbers are
the same because the same number of combinations yields all four
numbers.
That is, for the gamblers before Galileo - who from long
experience had correctly determined that "10" and "11" had the
higher chances of success than "9" and "12" - simulation used the
assumed facts that three fair dice are thrown with an equal
chance of any outcome. They then took advantage of the results
of a series of such events, performed one at a time
stochastically; in contrast, Galileo made no use of the actual
stochastic element of the situation, and did not gain information
from a sequence of such trials; instead, he replaced all
possible sequences by computation of their number (actually, a
complete enumeration).
Simulation is not "just" a stochastic shortcut to the
results of the formulaic method. Rather, it is a quite different
route to the same endpoint, using different intellectual
processes and utilizing different sorts of inputs. As a partial
analogy, it is like fixing a particular fault in an automobile
with the aid of a how-to-do-it manual's checklist, compared to
writing a book about the engineering principles of the auto; the
author may be no better at fixing cars than the hobbyist is at
writing engineering books, but the hobbyist requires less time to
learn the task to which he addresses him\herself -- fixing the
car's fault -- than the author needs to learn to write learned
works.
COMPLEXITY, NOT BUILT-IN PERVERSITY, MAKES PUZZLES DIFFICULT
A key implication of the deservedly-famous research on
errors in probabilistic judgments of Daniel Kahnemann and Amos
Tversky (interchangeably, Tversky and Kahnemann) is that human
thinking is often unsound. And some writers in their school of
thought assert that the unsoundness of thinking is hard-wired
into our brains; this point of view is expressed vividly in the
title of Massimo Piattelli-Palmarini's book Inevitable Illusions;
he calls the unsoundness "bias", and says that "we are
instinctively very poor evaluators of probability" (1994, p. 3,
italics in original).
Another possibility - not necessarily inconsistent with
genetic explanation - is that the reason we arrive at unsound
answers to certain types of problems is that the problems are
inherently very difficult, especially when they are tackled
without the assistance of tools, because the problems require
many steps and also because the steps often involve reversals in
the path. Without the aid of memory aids such as paper and
pencil, and the skill of using them well, the problems are just
too difficult for most persons.
One piece of evidence against the genetic-bias explanation
is that the wrong answers to problems are not all the same; they
do not even concentrate at one end of the probability spectrum.
As the work of Kahnemann and Tversky amply shows, the errors
often are widely distributed among most or all of the simple
arithmetical combinations of the numbers involved in the
problems. The outstanding characteristic of the answers is that
they are wrong, and not the nature of the errors. In following
long chains of logic and assessing complex assortments of
information, our brains may be weaker than we would like, but we
need not think of our brains as twisted.
The two explanations have quite different implications for
remediation, and two different remedies are offered; I suggest
resorting to simulation whereas others suggest additional
training (especially in probability theory) to improve people's
logic. The different remedies are not necessarily connected to
the two explanations, however; I believe that the remedy I
suggest is implied by the bias explanation as well as by the
weakness explanation.
Martin Gardner paraphrases the great American philosopher
Charles Sanders Peirce as saying that "in no other branch of
mathematics is it so easy for experts to blunder as in
probability theory" (1961, p. 220).Even great mathematicians have
blundered on simple problems, including D'Alembert and Leibniz.
But when you tackle problems in probability with experimental
simulation methods rather than with logic, neither simple nor
complex problems need be difficult for experts or beginners.
THE LOGICAL STEPS IN THE "THAT MAN'S FATHER..." PUZZLE
Though the main issue is problems in probability,
it is illuminating to begin with a famous deterministic
(non-probabilistic) problem:
A man points to the image of a person and says,
"Brothers or sisters have I none. That man's father is my
father's son."
This puzzle is difficult for most of us to solve in our heads.
In his book of puzzles, Raymond Smullyan says (and only about
this puzzle),
The remarkable thing about this problem is that most people get
the wrong answer but insist (despite all argument) that they are
right. I recall one occasion about 50 years ago when we had some
company and had an argument about this problem which seemed to
last hours, and in which those who had the right answer just
could not convince the others that they were right (1978, p. 7).
What is it that causes puzzles to be difficult? There seem
to be at least two elements that cause trouble: 1) A large
number of logical steps, as will soon be documented. This
requires a large memory. 2) Often one must switch directions of
thought back and forth several times. This requires that you
hold all of the relevant information in your mind, which makes it
hard to remember where you are - just as when moving around in a
city and making many lefts and rights.
Both of these elements are found in the "That man's father
is my father's son" puzzle. When we spell out and number the
separate logical steps in the problem, as in Figure 1, there are
fully 11 of them (13, including the sub-steps), and they go
backwards and forwards. It should not be surprising that one
cannot effectively store and sort out all this material in short-
run memory.
Figure 1 here
Here is the list of the steps. (The steps are not perfectly
denominable, and what I list as a single step might be considered
two steps by someone else, and vice versa.)
1. Point to a picture of a man and say "That man".
2. Follow the logic backwards a generation from "That man"
to "That man's father".
3. Draw a picture frame at a generation earlier than the
generation of "That man", and label it "That man's father".
3. Show the equation (taken from the original language of
the puzzle) of "That man's father" with "My father's son", by
drawing another box at the same generational level as "That man's
father", and connecting the two boxes with a sign of equality.
4. From "My father's son"--a complex label for a person who
is not yet specifically identified--we pass to the previous
generation and to the completely identified framed personage of
"My father".
5. With "My [historical] father" identified, we now move
again to the next generation and identify the historical person
"My father's son".
6. Now equate the two-element concept of "My father's son"
with the one-element "I".
7. We can now make clear exactly who "That man's father" is
by working back through the two equalities on the generational
level of "I", and then equating "I" directly with "That man's
father".
8. Now we pass from "I", who is "That man's father", to
"The son of I" ("son of me", to grammarians), and picture that
concept on the generational level below "I".
9. We can now pass from the logical concept "The son of I"
to the same (because logically equal) personage, "That man".
10. Last step: Re-label (because they are equal) the
concept "That man" as "My son", the latter a clearly-defined
historical person. This is the answer to the riddle.
The lightning thinker with a laser-like mentality may
contemptuously dismiss the piecemeal, step-wise reasoning above
as obvious and unnecessary, and write "It can be shown that...".
But as such great mathematicians and quantifiers as Alfred North
Whitehead and Wassily Leontief have noted, there can be an
enormous psychological difference between two equivalent logical
constructions. And the task at hand is to find safe and
efficient psychological routes to sound logical endpoints, even
if they are extended rather than compact and therefore
unesthetic.
When you show the pictorial analysis in Figure 1, people
quickly understand and agree about the right answer. This is
unlike Smullyan's experience (cited above) that he could not
persuade some people of the right answer. This should be strong
evidence of the power of concrete illustration to ease logical
difficulties and help obtain the correct answer. Simulation's
strength is its concreteness. (This is also the power of
mathematical notation, of course.)
If you use paper and pencil, or a simulation, some people
will say, "But you didn't do the problem in your head". True.
One might well be amused and impressed by someone who can do
incredible mathematical/logical feats in his/her head. But let's
keep separate the sphere of amusement-cum-awesome-spectacle and
the sphere of useful tools. If you want to perform great works
in science, business, and the rest of life, it is dexterity with
functional instruments you want, not the ability to impress other
people with feats akin to sword-swallowing and lifting small
autos with your bare hands.
THE LOGICAL STEPS IN BERTRAND'S PUZZLE
Next, let us consider a problem that Piattelli-Palmarini
considers a canonical "illusion", the three-chests problem
discussed in Chapter III-4. Here it is again:
A Spanish treasure fleet of three ships was sunk at sea off
Mexico in the 1500s. One ship had a trunk of gold forward and
another aft, another ship had a trunk of gold forward and a trunk
of silver aft, while a third ship had a trunk of silver forward
and another trunk of silver aft. A scuba diver just found one of
the ships and a trunk of gold in it, but she ran out of air
before she could check the other trunk. On deck, they are now
taking bets about whether the other trunk found on the same ship
will contain silver or gold. What are fair odds that the trunk
will contain gold?
These are the logical steps I distinguish in arriving at a
correct answer with deductive logic (portrayed in Figure 2):
Figure 2
1. Postulate three ships, one (call it "I") with two gold
chests (G-G), II with one gold and one silver chest (G-S), and
III with S-S. (Choosing notation might well be considered one or
more additional steps.)
2. Assert equal probabilities of each ship being found.
3. Step 2 implies equal probabilities of being found for
each of the six chests.
4. Fact: Diver finds a chest of gold.
5. Step 4 implies that S-S ship III was not found; hence
remove it from subsequent analysis.
6. Three possibilities: 6a) Diver found chest I-Ga, 6b)
diver found I-Gb, 6c) diver found II-Gc.
From step 2, the cases a, b, and c in step 6 have equal
probabilities.
7. If possibility 6a is the case, then the other trunk is
I-Gb; the comparable statements for cases 6b and 6c are I-Ga and
II-S.
8. From steps 6 and 7: From equal probabilities of the
three cases, and no other possible outcome, p (6a) = 1/3, p (6b)
= 1/3, p (6c) = 1/3,
9. So p(G) = p(6a) + p(6b) = 1/3 + 1/3 = 2/3.
Now let us list the steps in a simulation that would answer
the question:
1. Create three urns each containing two balls labeled
"0,0", "0,1", and "1,1" respectively.
2. Choose an urn at random, and shuffle its contents.
3. Choose the first element in the chosen urn's vector. If
"1", stop trial and make no further record. If "0", continue.
4. Record the second element in the chosen urn's vector on
the scoreboard.
5. Repeat (2 - 5), and calculate the proportion "0's" on
scoreboard.
AN APPLIED BAYESIAN PROBLEM
Now consider this classic Bayesian problem that Tversky and
Kahnemann quote from Cascells, Schoenberger, and Grayboys (1978,
p. 999):
If a test to detect a disease whose prevalence is 1/1000 has a
false positive rate of 5%, what is the chance that a person found
to have a positive result actually has the disease, assuming you
know nothing about the persons's symptoms or signs?
Tversky and Kahnemann note that among the respondents -
students and staff at Harvard Medical School - "The most common
response, given by almost half of the participants, was 95%",
very much the wrong answer.
To obtain an answer by simulation, rephrase the question
above with hypothetical numbers as follows:
If a test to detect a disease whose prevalence has been estimated
to be about 100,000 in the population of 100 million persons over
age 40 (that is, about 1 in a thousand) has been observed to have
a false positive rate of 60 in 1200, and never gives a negative
result if a person really has the disease, what is the chance
that a person found to have a positive result actually has the
disease, assuming you know nothing about the persons's symptoms
or signs?
(Please note in passing that the use of percentages rather
than raw numbers is an unnecessary abstraction and is often
misleading, in addition to being a barrier to simulation. If the
raw numbers are not available, the problem can be phrased in
terms of "about 1 case in 1000" or "about 5 cases in 100".)
If one has the habit of saying to oneself, "Let's simulate
it", one may then get an answer as follows:
1. Construct urn A with 999 white beads and 1 black bead,
and urn B with 95 green beads and 5 red beads. A more complete
problem that also has false negatives would need a third urn.
2. Pick a bead from urn A. If black, record "T", replace
the bead, and end the trial. If white, continue to step 3.
3. If a white bead is drawn from urn A, select a bead from
urn B. If red, record "F" and replace the bead, and if green
record "N" and replace the bead.
4. Repeat steps 2-4 perhaps 10,000 times, and count the
proportion of "T"s to "T"s plus "F"s (ignore the "N"s) in the
results.
Of course 10,000 draws would be tedious, but even after a
few hundred draws a person would be likely to draw the correct
conclusion that the proportion of "T"s to ("T"s plus "F"s) would
be small. And it is easy with a computer to do 10,000 trials
very quickly.
Note that the respondents in the Cascells et al. study were
not naive; the staff members were supposed to understand
statistics. Yet most produced wrong answers. If simulation can
do better, then simulation would seem to be the method of choice.
And only one piece of training for simulation is required:
Mastery of the self-reminder "Try it".
THE THREE-DOOR PROBLEM
The now-famous problem of the three doors, discussed in
Chapter III-v, is Piattelli-Palmarini's piece de resistance and
"Grand Finale" - his "Super-Tunnel" (p. 161) of an "inevitable
illusion" due to a "super blind spot" (p. 7). But it is indeed
if complex problem, as one sees if one diagrams it. But as we
have seen, hands-on simulation with physical symbols, rather than
computer simulation is a surefire way of obtaining and
displaying the correct solution. Not only does the best choice
become obvious, but one is likely to understand quickly why
switching is better. No other mode of explanation or solution
brings out this intuition so well. And it is much the same with
other problems in probability and statistics. Simulation can
provide not only answers but also insight into why the process
works as it does. In contrast, formulas produce obfuscation and
confusion for most non-mathematicians.
One may attempt to elucidate the three-box problem by
listing the elements of the sample set (all of which have the
same probability). But though this sample space is easy to
understand after it is placed before you, constructing it
requires much more depth of understanding of the problem's logic
than does simulation - which does not require that one count the
possible outcomes or the number of "successes"; this may explain
why it is much easier to err with sample-space analysis in this
and other problems.
DISCUSSION
Kahnemann and Tversky and others of their school (e.g.,
Nisbett et al.) infer from their findings the need for better
instruction in the logic of probability and statistics. The role
of logic may be seen in these statements by Piattelli-Palmarini:
The ultimate measuring instruments are those offered by
pure logic, probability theory, economics, and decision
theory. (Piattelli-Palmarini, 1994, p. 5.)
Yet, the mere fact that we intuitively come to see the
situation as anomalous is not sufficient to set us
right. It requires thought, and thought based on real
data (such as those offered by the cases in this book)
and on well-constructed theories that can ultimately
and persuasively gain our assent. That is how
rationality is fostered.
The very fact that we turn our backs on such abstract concepts
is, however, an unpardonable resistance to the progress of
reason. The game is worth it, because these are important
matters for all of us; they are fundamental to ourselves and to
those whom we love. (Piattelli-Palmarini, 1994, p. 14.)
Through thinking enough about these matters, one day we
may come to a certain rationality. (Piattelli-
Palmarini, 1994, p. 88.)
After presenting a problem asking the probabilities of
RGRRR (5 outcomes), GRGRRR (6 outcomes), and GRRRRR (six
outcomes), he asks the subject: "On which do you bet? Think."
(Piattelli-Palmarini, 1994, p. 50.). In contrast, I'd say only
"On which do you bet?", and hope that people would try out the
possibilities.
But this inference that more instruction in logic is needed
flows from the way the cognitive psychologists frame their test
questions, forcing a choice among a narrow and specific range of
alternatives and thereby excluding the possibility of simulating
the problem. If instead one widens the range of alternative
answers, we may see that a better choice is to teach people the
habit of not relying on the sort of logic that Kahnemann and
Tversky find to be often defective, but instead resorting to some
kind of experimentation - simulation - to seek an answer.
Indeed, Kahneman and Tversky sometimes find that even
persons trained in statistics do little or not at all better than
untrained respondents -- and this means doing badly. "[T]he
study of research psychologists...reveals that a strong tendency
to underestimate the impact of sample size lingers on despite
knowledge of the correct rule and extensive statistical training"
(1972/1982, pp. 45-46). So training in statistical logic does
not teach people what they need to know. Yet the response of
Kahnemann and Tversky - and even more so, the recommendation of
such researchers as Nisbett et al. - is that the appropriate
remedy is more conventional statistical training.
This remedy of giving more instruction in statistical logic
has a bit of the flavor of: If beating does not produce
improvement in the child's behavior, beat the child three times
as hard. Or, if drilling for oil produces no results at 1,000
feet and then at 2,000 feet, resolve to dig another 2,000 feet.
A better choice of remedy than the instruction that has
already failed might to instruct people in a different fashion --
that is, by simulation.
It is, of course, an empirical question whether people will
with higher probability arrive at a correct answer with
simulation or with deductive logic. Studies with other sorts of
questions clearly give the palm to simulation (Simon, Atkinson,
Shevokas, 1976).
Though I do not think that a genetic bias is responsible for
our inability to deductively calculate probabilities well, I do
believe that another (and even more dangerous) genetic bias is
implicated in the situation, to wit: The powerful urge to use
our brains deductively rather than by producing and assessing
experimental data. Even among scientists whose everyday business
is experimentation, it is extremely difficult to get a person to
address problems like those alluded to here with actual trials;
people generally resist the suggestion of simulation as if it
threatens an end to pleasure - which indeed it does.
There is one major shortcoming of the simulation approach
that we must be mentioned: the inadequacy of problem-solution
work done by simulation to satisfy the powerful human desire to
meet and solve challenges to one's power of reasoning. This is
what David Hume had in mind when he compared the excitement of
difficult intellectual activity to the hunt and the chase.
Further discussion of this crucial topic must be left to another
place, however.
REFERENCES
Gardner, Martin, The Second Scientific American Book of
Mathematical Puzzles & Diversions (New York: Simon and Schuster,
1961).
Kahneman, Daniel, and Amos Tversky, "Subjective
probability: A judgment of representativeness," abbreviated
version of a paper originally appearing in Cognitive Psychology,
1972, 3, 430-454, reprinted in Judgment under uncertainty:
Heuristics and biases, edited by Kahneman, Daniel, Paul Slovic,
and Amos Tversky (Cambridge: Cambridge University Press, 1982),
pp. 32-47.
Nisbett, Richard E., David H. Krantz, Christopher Jepson,
and Geoffrey T. Fong, "Improving inductive inference," Judgment
under uncertainty: Heuristics and biases, edited by Kahneman,
Daniel, Paul Slovic, and Amos Tversky (Cambridge: Cambridge
University Press, 1982), pp. 445-459.
Piattelli-Palmarini, Massimo, Inevitable Illusions (New
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Smullyan, Raymond, What is the Name of This Book? (Englewood
Cliffs: Prentice-Hall, 1978)
ENDNOTES
SUMMARY
Simulation - and its sub-class of resampling problems in
statistics - is a much simpler task intellectually than the
formulaic method of probabilistic calculations because it does
not require that one calculate a) the number of points in the
entire sample space, and b) the number of points in some sub-set,
so as to estimate the ratio of the latter to the former.
Instead, one directly estimates the ratio from a sample. Even
slightly difficult problems involving permutations and
combinations are sufficiently difficult as to require advanced
training. Similarly, it is much easier to sample the proportion
of black grains of sand on a beach than it is to take a census of
the total number of grains of each color. The latter is a task
both of great intellectual difficulty as well as great practical
difficulty.
page # teachbk IV-Idiff May 9, 1996