Home > 6. Chap. 2

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Resampling: The New Statistics

��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,

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.

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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:

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Resampling: The New Statistics

2.

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Chapter 2—Basic Concepts in Probability and Statistics, Part 1

The ��Meaning�� of ��Probability��

A probability estimate of .2 indicates that you think there is twice as great a chance of the event happening as if you had estimated a probability of .1. This is the rock-bottom interpre- tation of the term ��probability,�� and the heart of the concept. 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 ex- ample, if someone asks you your name, 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 pre- diction will 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 prob- ability 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 use- ful when you have a sample from a larger set of data—a ��uni- verse��—and you want to know the probability of various de- grees 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 1997. 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. I��ve said that a probability statement is about the future. Well, usually. Occasionally you might state a probability about your future knowledge of past events—that is, ��I think I��ll find out that...��—or even about the unknown past. (Historians use probabilities to measure their uncertainty about whether events occurred in the past, and the courts do, too, though the courts hesitate to say so explicitly.) Sometimes one knows a probability, such as in the case of a gambler playing black on an honest roulette wheel, or an in- surance company issuing a policy on an event with which it has had a lot of experience, such as a life insurance policy. But

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Resampling: The New Statistics

often one does not

Digression about Operational Definitions

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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Chapter 2—Basic Concepts in Probability and Statistics, Part 1

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

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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Resampling: The New Statistics

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

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Chapter 2—Basic Concepts in Probability and Statistics, Part 1

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

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Resampling: The New Statistics

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

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Chapter 2—Basic Concepts in Probability and Statistics, Part 1

fore limited), and few enough to be studied easily; and 2) that the probability of each particular possibility be known, for ex- ample, 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

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Resampling: The New Statistics

took that 4 percent and divided it by 4, because he as- sumed a manned flight would be safer than an un- manned 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

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Chapter 2—Basic Concepts in Probability and Statistics, Part 1

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

*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

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Resampling: The New Statistics

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

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