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."