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# 6. Chap. 2

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��Uncertainty, in the presence of vivid hopes and fears, is pain- ful, 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 Nature and Meaning of the Concept of Probability The Various Ways of Estimating Probabilities
Introduction
Basic Concepts in Probability and Statistics, Part 1
The central concept for dealing with uncertainty is probabil- ity. Hence we must inquire into the ��meaning�� of the term probability. (The term ��meaning�� is in quotes because it can be a confusing word.) You have been using the notion of probability all your life when drawing conclusions about what you expect to happen, and in reaching decisions in your public and personal lives. 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 an- swers to these questions rest on the probabilities you estimate. And you act on the basis of probabilities: You place your blan- ket 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 administra- tor judges that there is a low probability that all the other am- bulances will ever be in use at once. NASA decides whether or not to send off the space shuttle this morning as scheduled.
CHAPTER
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Chapter 2—Basic Concepts in Probability and Statistics, Part 1
This chapter discusses what is meant by such key terms as ��probability,�� ��conditional�� and ��unconditional�� probability, ��independence,�� ��sample,�� and ��universe.�� It discusses the nature and the usefulness of the concept of probability as used in this book, and it touches on the source of basic estimates of probability that are the raw material of statistical inferences. The chapter also distinguishes between probability theory and inferential statistics. (Descriptive statistics, the other main branch of statistics, was discussed briefly in the previous chap- ter.)
The nature and meaning of the concept of probability
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. It is confusing and unnecessary to inquire what probability ��really�� is. (Indeed, the terms ��really�� and ��is,�� alone or in combination, are major sources of confusion in statistics and in other logical and scientific discussions, and it is often wise to avoid their use.) Various concepts of probability—which correspond to various common definitions of the term—are useful in particular contexts (see Ayer, 1965; Schlaifer, 1961, Chapter 1; for perhaps the best summary of the probability concept, see Barnett, 1973, Chapter 3). This book contains many examples of the use of probability. Work with them will gradu- ally develop a sound understanding of the concept. There are two major concepts and points of view about prob- ability—frequency and belief. Each is useful in some situations but not in others. Though they may seem incompatible in prin- ciple, there almost never is confusion about which is appro- priate in a given situation. 1. Frequency. The probability of an event can be 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 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 shortcom- ing: Sometimes you must bet upon events that have never or only infrequently taken place before, and so you cannot rea- sonably reckon the proportion of times they occurred one way or the other in the past.)

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2. Belief. 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. (Notice a short- coming of this concept: 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.) The connection between gambling and immorality or vice troubles some people about gambling examples. On the other hand, the racy aspects can give the subject a special tang. There are several reasons why statistics use so many gambling ex- amples—and especially tossing coins, throwing dice, and play- ing cards: 1. Historical. The theory of probability began with gambling examples of dice analyzed by Cardano, Galileo, and then by Pascal and Fermat. 2. Generality. These examples are not related to any particular walk of life, and therefore they can be generalized to applica- tions in any walk of life. Students in any field—business, medi- cine, science—can feel equally at home with gambling ex- amples. 3. Sharpness. These examples are particularly stark, and unen- cumbered by the baggage of particular walks of life or special uses. 4. Universality. Every other text uses these same examples, and therefore the use of them connects up this book with the main body of writing about probability and statistics. Often we��ll begin with a gambling example and then consider an example in one of the professional fields—such as business and other decision-making activities, biostatistics and medi- cine, social science and natural science—and everyday living. People in one field often can benefit from examples in others; for example, medical students should understand the need for business decision-making in terms of medical practice, as well as the biostatistical examples. And social scientists should un- derstand the decision-making aspects of statistics if they have any interest in the use of their work in public policy.

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The ��Meaning�� of ��Probability��

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often one does not know the probability of a future event. Therefore, our concept of probability must include situations where extensive data are not available. All of the many techniques used to estimate probabilities should be thought of as proxies for the actual probability. For example, if Mission Control at Space Central simulates what should and probably will happen in space if a valve is turned aboard a space craft just now being built, the test result on the ground is a proxy for the real probability of what will happen in space. In some cases, it is difficult to conceive of any data that can serve as a proxy. For example, the director of the CIA, Robert Gates, said in 1993 ��that in May 1989, the CIA reported that the problems in the Soviet Union were so serious and the situ- ation so volatile that Gorbachev had only a 50-50 chance of surviving the next three to four years unless he retreated from his reform policies�� (The Washington Post, January 17, 1993, p. A42). Can such a statement be based on solid enough data to be more than a crude guess? The conceptual probability in any specific situation is an inter- pretation 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 isola- tion, 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 estimat- ing a probability constitutes an operational definition of prob- ability.
An operation definition is the all-important intellectual pro- cedure that Einstein employed in his study of special relativ- ity 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

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consists of. For example, a crude attribute definition of a col- lege 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 organi- zation found in The World Almanac��s listing of ��Colleges and Universities.���� (Simon, 1969, p. 18.) P. W. Bridgman, the inventor of operational definitions, stated 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 phys- ics back and constituted the barrier that it took Albert 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 (includ- ing 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 Sci- ence, p. 476). A. J. Bachrach adds that ��the operational defini- tion of a dish ... is its recipe�� (Bachrach, 1962, p. 74). The language of empirical scientific research is made up of in- structions 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 ��opera- tional definition.�� An operational definition contains a speci- fication 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 operation- ally.
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 labora- tory 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

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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 hap- pen. Whether any proxy is a good or bad one depends on the wisdom of the person choosing the proxy and making the probability estimates. How does one estimate a probability in practice? This involves practical skills not very different from the practical skills re- quired 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 con- sider elsewhere some ways to improve your practical skills in estimating probabilities. For now, let us simply categorize and consider in the next section various ways of estimating an or- dinary garden variety of probability, which is called an ��un- conditional�� probability.
The various ways of estimating probabilities
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 casu- ally from time to time, you might simply guess at the propor- tion 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 Decem- ber will be between 200 and 250, based on sales the last two Decembers; that your team will win the football game tomor- row; that war will break out next year; or that a United States astronaut will reach Mars before a Russian astronaut. You sim-

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ply put together all your relevant prior experience and knowl- edge, and then make an educated guess. Observation of repeated events can help you estimate the prob- ability 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 par- ticular country in a given decade are the basis for estimating the probabilities of death, which are then used by the actuar- ies 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 objec- tive manner. Many judgments inevitably enter into compiling every frequency series—deciding which frequency series to use for an estimate, 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 much data as is preferable, 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 prac- tically the same, and science is only interested in such ��practi- cal�� considerations.) In view of the necessarily judgmental aspects of probability estimates, the reader may prefer to talk about ��degrees of be- lief�� instead of probabilities. That��s fine, just as 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 esti- mates of the sales of radios in December, based on sales in that month in the past two years. The concept of a probability based on a frequency series can be rendered meaningless when all the observations are repeti- tions of a single magnitude—for example, the case of all suc- cesses 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 fail- ure. (Probabilists have made some rather peculiar attempts

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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 undeni- able fact that the longer such a series is, the smaller the prob- ability 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 mag- nitude 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 num- ber 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 possibili- ties—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 de- termine 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 prob- ability estimate of 2/36 = 1/18. Estimating probabilities by counting the possibilities has two requirements: 1) that the possibilities all be known (and there-

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ing everything, 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 en- gineering judgment.�� As far as I can tell, ��engineering judgment�� means they��re just going to make up num- bers! 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

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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 every- thing together you get 1 in 100,000. (Feynman, 1989, pp. 182-183). We see in the Challenger shuttle case very mixed kinds of in- puts to actual estimates of probabilities. They include fre- quency 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 with different sorts of estimation methods in differ- ent 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 busi- ness. 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. In sum, one should not think of what a prob- ability ��is�� but rather how best to estimate it. In practice, nei- ther in actual decision-making situations nor in scientific work—nor in classes—do people experience difficulties esti- mating probabilities because of philosophical confusions. Only philosophers and mathematicians worry—and even they re- ally do not need to worry—about the ��meaning�� of probabil- ity. This topic is continued in the following chapter.
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