CHAPTER I-3 PROBABILITY AND CHANCE: CHARACTERISTICS AND MEANING Uncertainty, in the presence of vivid hopes and fears, is painful, but must be endured if we wish to live without the support of comforting fairy tales. Bertrand Russell, A History of Western Philosophy (New York: Simon and Schuster, 1945, p. xiv) INTRODUCTION The central concept for dealing with uncertainty is probability. Hence we must inquire into the "meaning" of the term probability. (The term meaning is in quotes because it can be a confusing word.) After sketching the intuitive ground from which the concept of probability emerges, I shall suggest a theoretical concept of probability for applied work, and then argue that the appropriate empirical concept to apply this theoretical concept in statistics and elsewhere is an operational definition of probability. An operational definition at one stroke cuts away the difficulties that have arisen over the centuries in disputes among philosophers and statisticians about the appropriate concepts and definitions of probability. This chapter offers a way of dealing with the issue of probability that has been a bone of controversy for centuries. The following chapter discusses the history and nature of that controversy. THE INTUITIVE GROUND OF THE CONCEPT OF PROBABILITY You wonder: Will the footballer's kick from the 45 yard line go through the uprights? How much oil can you expect from the next well you drill, and what value should you assign to that prospect? Will you be the first one to discover a completely effective system for converting speech into computer-typed output? Will the next space shuttle end in disaster? Your answers to these questions rest on the probabilities you estimate. And you act on the basis of probabilities: You place your blanket on the beach where there is a low probability of someone's kicking sand on you. You bet heavily on a poker hand if there is a high probability that you have the best hand. A hospital decides not to buy another ambulance when the administrator judges that there is a low probability that all the other ambulances will ever be in use at once. NASA decides whether or not to send off the space shuttle this morning as scheduled. The common meaning of the term "probability" is as follows: Any particular stated probability is an assertion that indicates how likely you believe it is that an event will occur. THE THEORETICAL CONCEPT OF PROBABILITY FOR APPLIED WORK To say that an event has a high or low probability is the equivalent of making a statement that forecasts the future or predicts the outcome of some other event whose result is not yet known; an example of the latter is the result of a historical inquiry or the result of a coin flip when the coin already has been thrown and is in your palm. In practice, probability is stated as a decimal number between "0" and "1," such that "0" means you estimate that there is no chance of the event happening, and "1" means you are certain the event will happen. A probability estimate of .2 indicates that you think there is twice as great a chance of the events happening as if you had estimated a probability of .1. The probabilities associated with the possible outcomes in a given situation sum to one 1 by definition. The idea of probability arises when you are not sure about what will happen in an uncertain situation - that is, when you lack information and therefore can only make an estimate. For example, if someone asks you what your name is, you do not use the concept of probability to answer; you know the answer to a very high degree of surety. To be sure, there is some chance that you do not know your own name, but for all practical purposes you can be quite sure of the answer. If someone asks you who will win tomorrow's ball game, however, there is a considerable chance that you will be wrong no matter what you say. Whenever there is a reasonable chance that your prediction could be wrong, the concept of probability can help you. The concept of probability helps you to answer the question, "How likely is it that ...?" The purpose of the study of probability and statistics is to help you make sound appraisals of statements about the future, and good decisions based upon those appraisals. The concept of probability is especially useful when you have a sample from a larger set of data - a "universe" - and you want to know the probability of various degrees of likeness between the sample and the universe. (The universe of events you are sampling from is also called the "population", a concept to be discussed below.) Perhaps the universe of your study is all high school seniors in 1994. You might then want to know, for example, the probability that the universe's average SAT score will not differ from your sample's average SAT by more than some arbitrary number of SAT points - say, ten points. A probability statement is usually about the future, or about your future knowledge of past events. (Not always, though; historians put probabilities on the likelihoods that events occurred in the past, and the courts do, too, though the courts hesitate to say so explicitly.) But often one does not know the probability of a future event, except in the case of a gambler playing black on an honest roulette wheel, or an insurance company issuing a policy on an event with which it has had a lot of experience, such as a life insurance policy. Therefore, our concept must include situations where extensive data are not available. The conceptual probability in any specific situation is an interpretation of all the evidence that is then available. For example, a wise biomedical worker's estimate of the chance that a given therapy will have a positive effect on a sick patient should be an interpretation of the results of not just one study in isolation, but of the results of that study plus everything else that is known about the disease and the therapy. A wise policymaker in business, government, or the military will base a probability estimate on a wide variety of information and knowledge. The same is even true of an insurance underwriter who bases a life-insurance or shipping-insurance rate not only on extensive tables of long-time experience but also on recent knowledge of other kinds. The choice of a method of estimating a probability constitutes an operational definition of probability. Some writers (e. g. Hacking, 1975) suggest that the concept of probability is fairly new, emerging in recent centuries. But in my view the concept of probability is as old as the concept of uncertainty, which in turn is as old as the concept of certainty - that is, they have been with us forever. (The ancient Egyptians used gaming devices quite similar to our dice, and we can be sure that those players thought probabilistically.) What is new is our gradually acquiring better capacities to understand and deal with uncertainty. AN OPERATIONAL DEFINITION OF PROBABILITY As we shall see in the next chapter, there has long been controversy about what probability "is" - that is, about the supposed properties of probability. Typically, Parzen (1960, p. 2) asks, "What is it that is studied in probability that enables it to have such diverse applications?" He goes on to answer, "A random (or chance) phenomenon is an empirical phenonomenon characterized by the property..." But the search for the characterising property or properties is fraught with unavoidable confusion and contradiction. For example, is the distribution of a box of 3 one-inch and 3 two-inch nails to be considered a "random phenomenon"? What about if one draws a nail blindfolded from the box? And if you draw with your eyes open? How about drawing one from a box of only 3 one-inch nails? And from a box of 1000 one-inch and 1 two-inch nails? How about drawing 100 nails from a well-shuffled box of 1000 of each length? We can certainly make some useful distinctions among these cases, but the distinctions will depend upon what we know and think as we draw a nail rather than just upon the distribution of nails, and upon what we consider variation or stability (is picking between 45 and 55 one-inch nails among a 100 picks consider variation or sameness?), among other complicating factors. This dispute is reminiscent of the controversy about what time "is" that held up the progress of physics until Einstein swept away the difficulties by saying that time should simply be defined as what one reads on a clock. This operational definition became the keystone in a major advance in the philosophy of science which overreached itself for a while, but whose central idea - defining a difficult concept by the operations used to measure an empirical proxy for the concept - has cut threw many gordian knots of confusion in physics, psychology, and elsewhere in science. (For two examples in economics, the concepts of utility and product differentiation, see Simon, 1974 and 1969). Similarly, an operational definition of probability sidesteps the pitfalls into which probability has for too long been mired.[1] An operation definition is the all-important intellectual procedure that Einstein employed in his study of special relativity to sidestep the conceptual pitfalls into which discussions of such concepts as probability also often slip. An operational definition is to be distinguished from a property or attribute definition, in which something is defined by saying what it consists of. For example, a crude attribute definition of a college might be "An organization containing faculty and students, teaching a variety of subjects beyond the high-school level." An operational definition of university might be "An organization found in The World Almanac's listing of `Colleges and Universities.'" (Simon, 1969, p. 18.) P. W. Bridgman, the inventor of operational definitions, put it that "the proper definition of a concept is not in terms of its properties but in terms of actual operations." It was he who explained that definitions in terms of properties had held physics back until Albert Einstein and constituted the barrier that it took Einstein to crack (Bridgman, 1927, pp. 6-7). A formal operational definition of "operational definition" may be appropriate. "A definition is an operational definition to the extent that the definer (a) specifies the procedure (including materials used) for identifying or generating the definiendum and (b) finds high reliability for [consistency in application of] his definition" (Dodd, in Dictionary of Social Science, p. 476). A. J. Bachrach adds that "the operational definition of a dish ... is its recipe" (Bachrach, 1962, p. 74). The language of empirical scientific research is made up of instructions that are descriptions of sets of actions or operations (for instance, "turn right at the first street sign") that someone can follow accurately. Such instructions are called an "operational definition." An operational definition contains a specification of all operations necessary to achieve the same result. The language of science also contains theoretical terms (better called "hypothetical terms") that are not defined operationally. The clock-reading which operationally defines time may be a windup spring clock, an electric clock, an atomic clock, sunset on a given date in a given place, or the postman's delivery. Similarly, we should simply say that probability "is" ( or better, that we "define probability as") what you calculate from the data in a life table or the experience with a slot machine, or the number you assign to the chance that the competitor across the street will reduce her price tomorrow. And just as different sorts of clock proxies are used to measure time in various circumstances in physics, different sorts of data proxies are used to stand for probability - even for the same probability. Back to Proxies Example of a proxy: The "probability risk assessments" (PRAs) that are made for the chances of failures of nuclear power plants are based, not on long experience or even on laboratory experiment, but rather on theorizing of various kinds - using pieces of prior experience wherever possible, of course. A PRA can cost a nuclear facility $5 million. Another example: If a manager looks at the sales of radios in the last two Decembers, and on that basis guesses how likely it is that he will run out of stock if he orders 200 radios, then the last two years' experience is serving as a proxy for future experience. If a sales manager just "intuits" that the odds are 3 to 1 (a probability of .75) that the main competitor will not meet a price cut, then all his past experience summed into his intuition is a proxy for the probability that it will really happen. Whether any proxy is a good or bad one depends on the wisdom of the person choosing the proxy and making the probability estimates. THE VARIOUS WAYS OF ESTIMATING PROBABILITIES How does one estimate a probability in practice? This involves practical skills not very different from the practical skills required to estimate with accuracy the length of a golf shot, the number of carpenters you will need to build a house, or the time it will take you to walk to a friend's house; we will consider elsewhere some ways to improve your practical skills in estimating probabilities [references in book or elsewhere]. For now, let us simply categorize and consider in the next section various ways of estimating an ordinary garden variety of probability, which is called an "unconditional" probability. Consider the probability of drawing an even-numbered spade from a deck of poker cards (consider the queen as even and the jack and king as odd). Here are several general methods of estimation, the specifics of which constitute an operational definition of probability in this particular case: 1. Experience. The first possible source for an estimate of the probability of drawing an even-numbered spade is the purely empirical method of experience. If you have watched card games casually from time to time, you might simply guess at the proportion of times you have seen even-numbered spades appear - say, "about 1 in 15" or "about 1 in 9" (which is almost correct) or something like that. (If you watch long enough you might come to estimate something like 6 in 52.) General information and experience are also the source for estimating the probability that the sales of radios this December will be between 200 and 250, based on sales the last two Decembers; that your team will win the football game tomorrow; that war will break out next year; or that a United States astronaut will reach Mars before a Russian astronaut. You simply put together all your relevant prior experience and knowledge, and then make an educated guess. Observation of repeated events can help you estimate the probability that a machine will turn out a defective part or that a child can memorize four nonsense syllables correctly in one attempt. You watch repeated trials of similar events and record the results. Data on the mortality rates for people of various ages in a particular country in a given decade are the basis for estimating the probabilities of death, which are then used by the actuaries of an insurance company to set life insurance rates. This is systematized experience, - called a frequency series. No frequency series can speak for itself in a perfectly objective manner. Many judgments inevitably enter into compiling every frequency series - deciding which frequency series to use for an estimate, and in choosing which part of the frequency series to use, and so on. For example, should the insurance company use only its records from last year, which will be too few to provide as many data as would be liked, or should it also use death records from years further back, when conditions were slightly different, together with data from other sources? (Of course, no two deaths - indeed, no events of any kind - are exactly the same. But under many circumstances they are practically the same, and science is only interested in such "practical" considerations.) In view of the necessarily judgmental aspects of probability estimates, the reader may prefer to talk about "degrees of belief" instead of probabilities. That's fine, just a long as it is understood that we operate with degrees of belief in exactly the same way as we operate with probabilities; the two terms are working synonyms. There is no logical difference between the sort of probability that the life insurance company estimates on the basis of its "frequency series" of past death rates, and the manager's estimates of the sales of radios in December, based on sales in that month in the past two years. [4] The concept of a probability based on a frequency series can be rendered meaningless when all the observations are repetitions of a single magnitude - for example, the case of all successes and zero failures of space-shuttle launches prior to the Challenger shuttle tragedy in the 1980s; in those data alone there was no basis to estimate the probability of a shuttle failure. (Probabilists have made what some rather peculiar attempts over the centuries to estimate probabilities from the length of a zero-defect time series - such as the fact that the sun has never failed to rise (foggy days aside!) - based on the undeniable fact that the longer is such a series, the smaller the probability of a failure; see e.g., Whitworth, 1897/1965, pp. xix-xli. However, one surely has more information on which to act when one has a long series of observations of the same magnitude rather than a short series). 2. Simulated Experience. A second possible source of probability estimates is empirical scientific investigation with repeated trials of the phenomenon. This is an empirical method even when the empirical trials are simulations. In the case of the even-numbered spades, the empirical scientific procedure is to shuffle the cards, deal one card, record whether or not the card is an even-number spade, replace the card, and repeat the steps a good many times. The proportions of times you observe an even-numbered spade come up is a probability estimate based on a frequency series. You might reasonably ask why we do not just count the number of even-numbered spades in the deck of fifty-two cards. No reason at all. But that procedure would not work if you wanted to estimate the probability of a baseball batter getting a hit or a cigarette lighter producing flame. Some varieties of poker are so complex that experiment is the only feasible way to estimate the probabilities a player needs to know. The resampling approach to statistics produces estimates of most probabilities with this sort of experimental "Monte Carlo" method. More about this later. 3. Sample space analysis and first principles. A third source of probability estimates is counting the possibilities - the quintessential theoretical method. For example, by examination of an ordinary die one can determine that there are six different numbers that can come up. One can then determine that the probability of getting (say) either a "1" or a "2," on a single throw, is 2/6 = 1/3, because two among the six possibilities are "1" or "2." One can similarly determine that there are two possibilities of getting a "1" plus a "6" out of thirty-six possibilities when rolling two dice, yielding a probability estimate of 2/36 = 1/18. Estimating probabilities by counting the possibilities has two requirements: 1) that the possibilities all be known (and therefore limited), and few enough to be studied easily; and 2) that the probability of each particular possibility be known, for example, that the probabilities of all sides of the dice coming up are equal, that is, equal to 1/6. 4. Mathematical shortcuts to sample-space analysis. A fourth source of probability estimates is mathematical calculations. If one knows by other means that the probability of a spade is 1/4 and the probability of an even-numbered card is 6/13, one can then calculate that the probability of turning up an even-numbered spade is 6/52 (that is, 1/4 x 6/13). If one knows that the probability of a spade is 1/4 and the probability of a heart is 1/4, one can then calculate that the probability of getting a heart or a spade is 1/2 (that is 1/4 + 1/4). The point here is not the particular calculation procedures, but rather that one can often calculate the desired probability on the basis of already-known probabilities. It is possible to estimate probabilities with mathematical calculation only if one knows by other means the probabilities of some related events. For example, there is no possible way of mathematically calculating that a child will memorize four nonsense syllables correctly in one attempt; empirical knowledge is necessary. 5. Kitchen-sink methods. In addition to the above four categories of estimation procedures, the statistical imagination may produce estimates in still other ways such as a) the salesman's seat-of-the-pants estimate of what the competition's price will be next quarter, based on who-knows-what gossip, long- time acquaintance with the competitors, and so on, and b) the probability risk assessments (PRAs) that are made for the chances of failures of nuclear power plants based, not on long experience or even on laboratory experiment, but rather on theorizing of various kinds - using pieces of prior experience wherever possible, of course. Any of these methods may be a combination of theoretical and empirical methods. Consider the estimation of the probability of failure for the tragic flight of the Challenger shuttle, as described by the famous physicist Nobelist Richard Feynman. This is a very real case that includes just about every sort of complication that enters into estimating probabilities. ...Mr. Ullian told us that 5 out of 127 rockets that he looked at had failed - a rate of about 4 percent. He took that 4 percent and divided it by 4, because he assumed a manned flight would be safer than an unmanned one. He came out with about a 1 percent chance of failure, and that was enough to warrant the destruct charges. But NASA [the space agency in charge] told Mr. Ullian that the probability of failure was more like 1 of 105. I tried to make sense out of that number. "Did you say 1 in 105?" "That's right; 1 in 100,000." "That means you could fly the shuttle every day for an average of 300 years between accidents - every day, one flight, for 300 years - which is obviously crazy!" "Yes, I know," said Mr. Ullian. "I moved my number up to 1 in 1000 to answer all of NASA's claims - that they were much more careful with manned flights, that the typical rocket isn't a valid comparison, et cetera...". But then a new problem came up: the Jupiter probe, Galileo, was going to use a power supply that runs on heat generated by radioactivity. If the shuttle carrying Galileo failed, radioactivity could be spread over a large area. So the argument continued: NASA kept saying 1 in 100,000 and Mr. Ullian kept saying 1 in 1000, at best. Mr. Ullian also told us about the problems he had in trying to talk to the man in charge, Mr. Kingsbury: he could get appointments with underlings, but he never could get through to Kingsbury and find out how NASA got its figure of 1 in 100,000 (Feynman, 1989, pp. 179- 180) Feynman tried to ascertain more about the more about the origins of the figure of 1 in 100,000 that entered into NASA's calculations. He performed an experiment with the engineers: ..."Here's a piece of paper each. Please write on your paper the answer to this question: what do you think is the probability that a flight would be uncompleted due to a failure in this engine?" They write down their answers and hand in their papers. One guy wrote "99-44/100% pure" (copying the Ivory soap slogan), meaning about 1 in 200. Another guy wrote something very technical and highly quantitative in the standard statistical way, carefully defining every- thing, that I had to translate - which also meant about 1 in 200. The third guy wrote, simply, "1 in 300." Mr. Lovingood's paper, however, said, Cannot quantify. Reliability is judged from: * past experience * quality control in manufacturing * engineering judgment "Well," I said, "I've got four answers, and one of them weaseled." I turned to Mr. Lovingood: "I think you weaseled." "I don't think I weaseled." "You didn't tell me what your confidence was, sir; you told me how you determined it. What I want to know is: after you determined it, what was it?" He says, "100 percent" - the engineers' jaws drop, my jaw drops; I look at him, everybody looks at him - "uh, uh, minus epsilon!" So I say, "Well, yes; that's fine. Now, the only problem is, WHAT IS EPSILON?" He says, "10-5." It was the same number that Mr. Ullian had told us about: 1 in 100,000. I showed Mr. Lovingood the other answers and said, "You'll be interested to know that there is a difference between engineers and management here - a factor of more than 300." He says, "Sir, I'll be glad to send you the document that contains this estimate, so you can understand it."* *Later, Mr. Lovingood sent me that report. It said things like "The probability of mission success is necessarily very close to 1.0" - does that mean it is close to 1.0, or it ought to be close to 1.0? - and "Historically, this high degree of mission success has given rise to a difference in philosophy between unmanned and manned space flight programs; i.e., numerical probability versus engineering judgment." As far as I can tell, "engineering judgment" means they're just going to make up numbers! The probability of an engine-blade failure was given as a universal constant, as if all the blades were exactly the same, under the same conditions. The whole paper was quantifying everything. Just about every nut and bolt was in there: "The chance that a HPHTP pipe will burst is 10-7." You can't estimate things like that; a probability of 1 in 10,000,000 is almost impossible to estimate. It was clear that the numbers for each part of the engine were chosen so that when you add everything together you get 1 in 100,000.(Feynman, 1989, pp. 182-183). We see in the Challenger shuttle case very mixed kinds of inputs to actual estimates of probabilities. They include frequency series of past flights of other rockets, judgments about the relevance of experience with that different sort of rocket, adjustments for special temperature conditions (cold), and much much more. There also were complex computational processes in arriving at the probabilities that were made the basis for the launch decision. And most impressive of all, of course, are the extraordinary differences in estimates made by various persons (or perhaps we should talk of various statuses and roles) which make a mockery of the notion of objective estimation in this case. Working at a practical level with different sorts of estimation methods in different sorts of situations is not new; practical statisticians do so all the time. The novelty here lies in making no apologies for doing so, and for raising the practice to the philosophical level of a theoretically-justified procedure - the theory being that of the operational definition. The concept of probability varies from one field of endeavor to another; it is different in the law, in science, and in business. The concept is most straightforward in decision-making situations such as business and gambling; there it is crystal- clear that one's interest is entirely in making accurate predictions so as to advance the interests of oneself and one's group. The concept is most difficult in social science, where there is considerable doubt about the aims and values of an investigation. Most philosophical discussion focuses on the roles of probability and statistics in physical science - which is just one of the many types of situations where these concepts are used, and certainly one of those where the waters of thought have been most muddy. THE DUALITY OF PROBABILITY AND PHYSICAL CONCEPTS An important argument in favor of approaching the concept of probability with the concept of the operational definition is that an estimate of a probability often (though not always) is the opposite side of the coin from an estimate of a physical quantity such as time or space. For example, uncertainty about the probability that one will finish a task within 9 minutes is another way of labeling the uncertainty that the time required to finish the task will be less than 9 minutes. Hence, if an operational definition is appropriate for time in this case, it should be equally appropriate for probability. The same is true for the probability that the quantity of radios sold will be between 200 and 250 units. Hence the concept of probability, and its estimation in any particular case, should be no more puzzling than is the "dual" concept of time or distance or quantities of radios. That is, lack of certainty about the probability that an event will occur is not different in nature than lack of certainty about the amount of time or distance in the event. There is no essential difference between whether a part of length 2 inches long will be the next to emerge from the machine, or what the length of the next part will be, or the length of the part that just emerged (if it has not yet been measured.) The information available for the measurement of (say) the length of a car or the location of a star is exactly the same information that is available with respect to the concept of probability in those situations. That is, one may have ten disparate observations of an auto's length which then constitute a probability distribution, and the same for the altitude of a star in the heavens. All the more reason to see the parallel between Einstein's concept of time and length as being what you measure on a clock and on a meter stick, respectively - or better, that time and length are equivalent to the measurements that one makes on a clock or meter stick -- and the notion that probability should be defined by the measurements made on a clock or a meter stick. Seen this way, all the discussions of logical and empirical notions of probability may be seen as being made obsolete by the Einsteinian invention of the operational definition, just as were discussions of absolute space and time made obsolete by it. Or: Consider having four different measurements of the length of a model auto. Which number should we call the length? It is standard practice to compute the mean. But the mean could be seen as a weighted average of each observation by its probability. That is (.25 * 20 inches + .25 * 22 inches...) = mean model length instead of (20 + 22 + ...) / 4 = mean model length This again makes clear that the decimal weights we call "probabilities" have no extraordinary properties when discussing frequency series; they are just weights we put on some other values. The key difference between a probability and other related concepts such as length is that length can refer to the past, the present, or the future. But a probability refers only to the future (or to our future knowledge), and therefore cannot be measured in ways analogous to our measurement of length and weight and time, at least in those cases that are not aggregate measurements such as the distribution of heights, or error distributions such as the location of star. Savage's "Bayesian" view of probability as a magnitude that emerges implicitly from an assessment of the expected value (or the expected utility, in decisions with very large consequences) of a routine choice also fits in here, as I understand it. For example, one may consider a wager on a football game between Bolivia and Argentina at odds of 3-1; those odds then express implicitly an estimate of the probability of each of the two teams winning. Here, the terms of the wager - 3 to 1 odds - are not different than the terms of an even-money wager with New York being a ten-point favorite over Washington, where the point spread rather than a difference in odds expresses the assessed difference in skill. The point spread may seem to be a less probabilistic notion than odds, but they are of quite the same nature. In the course of preparing the point spread or the odds, however, the bet-maker almost surely has thought explicitly in terms of the probabilities of the teams winning the game. That is, a forecast underlies a statement of the odds, so we seem to be about back to the same place we were before. A probabilitiy is a statement about the future, which may be considered the same as a judgmnent about the future. There are many different methods that people find it useful to form such judgments, and the methods are quite different in different circumstances, and not easy to classify. A weather forecaster forms a probability of rain tomorrow on the basis of a wide variety of information, various mathematical models, records about local weather conditions in the past, records of his/her own performance, and much more. Frequency data and personal probabilities are inextricably mixed in here, as they are in sports forecasting and odds making, and in the sort of engineering forecast as the failure of the space shuttle Challenger's O-rings (see page 000). To my mind, all this makes a complete hash (in all senses of the word) of the dispute between frequentists and personalists and all other schools of thought about the "nature" of probability. It should be noted that the view outlined above has absolutely no negative implications for the formal mathematical theory of probability. In a book of puzzles about probability (Mosteller, 1965/1987, #42)), this problem appears: "If a stick is broken in two at random, what is the average length of the smaller piece?" This particular puzzle does not even mention probability explicitly, and no one would feel the need to write a scholarly treatise on the meaning of the word "length" here, any more than one would one do so if the question were about an astronomer's average observation of the angle of a star at a given time or place, or the average height of boards cut by a carpenter, or the average size of a basketball team. Nor would one write a treatise about the "meaning" of "time" if a similar puzzle involved the average time between two bird calls. Yet a rephrasing of the problem reveals its tie to the concept of probability, to wit: What is the probability that the smaller piece will be (say) more than half the length of the larger piece? Or, what is the probability distribution of the sizes of the shorter piece? The duality of the concepts of probability and physical entities also emerges in Whitworth's discussion (1897/1965) of fair betting odds: ...What sum ought you fairly to give or take now, while the event is undetermined, in exchange for the assurance that you shall receive a stated sum (say $1,000 if the favourable event occur? The chance of receiving $1,000 is worth something. It is not as good as the certainty of receiving $1,000, and therefore it is worth less than $1,000. But the prospect or expectation or chance, however slight, is a commodity which may be bought and sold. It must have its price somewhere between zero and $1,000. (p. xix.) ...And the ratio of the expectation to the full sum to be received is what is called the chance of the favourable event. For instance, if we say that the chance is 1/5, it is equivalent to saying that $200 is the fair price of the contingent $1,000. (p. xx.)... The fair price can sometimes be calculated mathematically from a priori considerations: sometimes it can be deduced from statistics, that is, from the recorded results of observation and experiment. Sometimes it can only be estimated generally, the estimate being founded on a limited knowledge or experience. If your expectation depends on the drawing of a ticket in a raffle, the fair price can be calculated from abstract considerations: if it depend upon your outliving another person, the fair price can be inferred from recorded statistics: if it depend upon a benefactor not revoking his will, the fair price depends upon the character of your benefactor, his habit of changing his mind, and other circumstances upon the knowledge of which you base your estimate. But if in any of these cases you determine that $300 is the sum which you ought fairly to accept for your prospect, this is equivalent to saying that your chance, whether calculated or estimated, is 3/10... (p. xx.) It is indubitable that along with frequency data, a wide variety of other information will affect the odds at which a reasonable person will bet. If the two concepts of probability stand on a similar footing here, why should they not be on a similar footing in all discussion of probability? Why should both kinds of information not be employed in an operational definition of probability? I can think of no reason that they should not be so treated. Scholars write about the "discovery" of the concept of probability in one century or another. But is it not likely that even in pre-history, when a fisherperson was asked how long the big fish was, s/he sometimes extended her/his arms and said, "About this long, but I'm not exactly sure", and when a scout was asked how many of the enemy there were, s/he answered, "I don't know for sure...probably about fifty". The uncertainty implicit in these statements is the functional equivalent of probability statements. There simply is no need to make such heavy work of the probability concept as the philosophers and mathematicians and historians have done. CONCLUSION In sum, one should not think of what a probability "is" but rather how best to estimate it. In practice, neither in actual decision-making situations nor in scientific work - nor in classes - do people experience difficulties estimating probabilities because of philosophical confusions. Only philosophers and mathematicians worry - and even they really do not need to worry - about the "meaning" of probability. **FOOTNOTES** [1]: I have long wondered why the concept of operational definition has not had a larger place in discussions by professional philosophers. I now speculate that as a powerful tool for resolving apparently-insolvable controversies that have raged for decades and even centuries, it threatens the livelihoods of some philosophers whose intellectual capital and reason for being would disappear with the disappearance of those controversies. See Deming (1989, Chapter 15) for spirited advocacy of operational definitions in statistics, and Simon 1969; 3rd ed. (with Burstein) 1985, for discussion of the concept. ENDNOTES