# Read PDF Chance and Luck: The Laws of Luck, Coincidences, Wagers, Lotteries, and the Fallacies of Gambling

The prize and the chance everything I want. Kazushige, living in Japan, is people to change their life into better one, though addicted to horse racing.

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The Linda, who lives in Australia. By the time it They are a few examples of hundreds was published, America began to heal from the millions of people around the world shock after World War II. The brutality shown in who are addicted to gambling. This the story is far from the expression needed by phenomenon occurs as the society American society to make recovery. It is also inseparable from the role of a certain government that provides gambling facilities and Impact of American Lotteries on the makes this activity flourishing and Development of Lotteries around the World, growing even larger.

This around the world can play gambling view is based on facts that many forms freely. This betting and racing, and online place has become interesting for gambling. Gamblers have their own visitors around the world to enjoy the reasons for gambling. One reason is to entertainment facilities provided. The Vegas—which in attracted over other one is to enjoy the challenge or 30 million visitors per year to its competition.

Whatever the reasons, , hotel rooms and myriad casino gambling can be really addictive and entertainment facilities—had BigDeal, There are some become an ideal tourism destination examples that gambling occurs around resort, centered around casinos. South Africa has problems, but then it was closed Sun City as its luxury casino and resort, after its inability to meet the and China has Macao which territory's required funds and being protested economy is heavily dependent on by the society.

In Southeast b. It used coupon Bayfront and Sentosa which is called which was printed for about 4 Integrated Resorts Loong, Following the widespread increasingly c. Not a few of Indonesian of gambling, but many also found people are flocking to the resorts to otherwise. The result actually almost similar to the lottery is that Indonesian government revenue or toto in Spain in in which to is affected because of the great guess the outcome of football amount of money flowing out of the matches over 14 professional clubs country as a result of the proliferation in the top division of that country.

These and filled in 10 million chances to are policies which have been issued by get the main prize. Those policies are Singapore Lottery. This illegal gambling proofs that even though Indonesia is a emerged as a result of addiction and religious country, both government and assumption that lotteries served as the society are trying to preserve the solution of financial problems. Indonesian People Views toward Conclusion Lottery Gambling in Their Country It seems a common thing nowadays As gambling has been occurred in American that gambling can be found anywhere society since 17th century, this practice has and anytime in Indonesia.

Although this become a tradition which can be found in activity is illegal and is incompatible American social life. As tradition, gambling with values and norms of Indonesian practice has developed from time to time. People society, people still do it as a form of played gambling in the past conventionally.

It is lifestyle. The government attempted to developed, then, that people build a machine to make positive results from issuing the help them play gambling, as well as to make mentioned policies. However, the gambling more interesting. Nowadays, people negative impacts of these policies even gamble creatively by using internet. They occurred around the country.

This preservation leads Middle class society and poor United States of America financially from society were carried away by the lure tourism sector which is influenced by the to make them rich instantly. Those development of Las Vegas as casino resort. With the fast growing and widespread of ahead of the coupon drawing. Not only these kind of resorts, Indonesian government that, education and health issues are struggled to suppress funds people spent for also affected which might lead to gambling or just travelling to those resorts by worse situation for young generations.

These policies also also socially. Crime was increasing and have positive and negative impacts for became a major issue, especially fraud. Indonesian people. However, as this lottery With many people protested this form demanded a lot of money to buy coupons, people of lottery practice, the government began addicted. Most of their monthly or daily eventually erased it in However, income was used to buy coupons causing another problem arouse for illegal inability to fulfill their proper daily needs. Accessed on: 2 Oct Lottery gambling is a game, which for some Fibiona, Indra.

Accessed on: 1 Oct After all, lottery gambling makes people Accessed on: 5 July Accessed on: logy Division. Available at: 5 July Accessed on: 12 London. June Accessed on: 3 Integrated Resorts. Available Guide. Centre du Jeu Excessif. Accessed on: 4 Oct. Dunstan, Roger.

Gambling in California. McDowell, Tremaine. American Studies. Sacramento: California Research Bureau. Minneapolis: The University of Available at: Minnesota.

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Access- dari Amerika American Bouquet. Eadington, William R. Forthcoming in Contem- Pengawal.

Tarik Global. Pearce and Richard W. Accessed on: 1 Oct Hance and Luck: A. Accessed Steinmetz, Andrew Esq. The Gaming on: 9 June Issues, and Society. CLIO, Inc. Related Papers. A profile of problem gambling clients in Northern NSW. On the other hand, luck, fortune, and, most importantly, coincidence have a distinctly different temporal structure as opposed to the first group of concepts. They are a posteriori in nature, and often not quantifiable. Confusion of these two basic sets of categories introduces a great potential for bias of inference.

Both Bayesian and frequentist theories of probability are affected, if they neglect to make this distinction. We shall briefly look at what the two theories have to offer, and we shall take important ideas from both theories for subsequent chapters. Bayesian inference will return in the fourth chapter, whereas condition of homogeneity for the frequentist theory of probability is an important element in the theory of explanation in the second chapter.

The Appearance of Chance. The accidental is an ontological concept, defining a condition of a world, and of a lawful world. Only in this lawful infinity can freedom be found or the secular become sacred. The vocabularies of many European languages have the ability to express the accidental occurrence of certain events, having either a positive or negative influence on those who are subject to it. In English these are chance , coincidence , hazard , risk , probability , randomness , luck , and fortune , with activities corresponding to them, such as betting , gambling , and lotteries.

We shall look into the etymology of some of these concepts for a hint of a proper application. In English the word luck expresses a fortunate and, at least partially, unanticipated occurrence. Proverbs in several languages celebrate the inner connection between the conditions of fortune and happiness. Is being happy, i. In other words, can we claim moral appraisal for being lucky?

We shall answer this question in the second chapter when we deal with the issue of moral luck. There is another aspect to luck. In Dutch, which has evolved from Middle High German, the verb lukken or gelukken means "to succeed with a certain help of luck. In Dutch, one says that the revolution lukte lucked out , rather than that the rebels did. Apparently, luck acts as an impersonal force. It refers to certain essential features of chance, as for instance, being out of the subject's control.

Moreover, luck is always attributed a posteriori. Both chance and coincidence both have their etymological roots in the Latin verb cadere , to fall, and its conjunction, cadens , falling. It reflects the idea that chance is something that falls down from the heavens onto the people. Chance has often been considered to be authored by God s in heaven and thus came quite inappropriately, as we shall argue to be connected with fate and necessity.

Hazard's etymological derivation lies in the Arabic word for a die, az-zahr. Gambling is an old practice. It goes back at least as far as ancient Egypt where people used four-sided astragali made from animal heel-bones. Gambling, from ancient French gamen to play , has always and everywhere been pervasive among all social classes. Interestingly, the word wedding finds its origins in gambling. In Dutch wedden still means to bet. According to the historian Johan Huizinga , in Medieval Europe a wedding used to be a contract between two persons that were gambling. The contract specified that the one player promised to give away his daughter if he lost the gamble.

In an attempt to produce some income in the early sixteenth century Florence organized a lottery , which it called La Lotto. Although not the first lottery, it became a model for all subsequent lotteries and its name became a noun. In a lottery the prizes are specified in advance and the random activity lies in the distribution of these prizes. When we shall discuss the issue of distributive justice, we shall use the model of a lottery to present the problem of whether fairness exists in the presence of chance.

The concept of risk was first employed in the fifteenth century merchant vocabulary in Central and Western Europe, where it stood for financial speculation. The concept developed in the Italian city states, probably from the verb rischiare , meaning to endanger or to wager. The verb itself originates, via risicare , from the Greek r i z a , which means cliff. Risicare came to mean to go around the cliff , much in the same sense as the noun clipper is intended, and consequently they acquired their contemporary meaning, rischiare , to speculate.

Until the nineteenth century the use of risk was exclusively in the domain of economics. During the same period there was an important shift of meaning. Whereas previously the idea of risk had been connected with the hope and expectation of the one who financed the operation in the individual who executed it, in the course of the seventeenth and eighteenth century the newly gained understanding of probabilities and mathematical expectations transformed the concept of risk into long run average wins and losses away from individuality.

Aristotle's definition of a human being as a rational animal describes a particular human tension constitutive of its being. The tension stands for the complex interaction between spirit and matter. Reason, by its very nature, is colonial, imperial. It aims to take hold of whatever is presented to it. Also, reason is reflexive, bends back on itself, and in this move it can recognize its boundaries; these boundaries are not to make life more interesting, but they constitute a fundamental separation of the mind with the beyond, i.

One such transcendent aspect is luck. Our laws of chance go only so far. The cognitive separation of the present from the future, and that of knowledge from ignorance constitute two domains of luck, which present themselves as transcending aspects to reason, which in reflection recognizes them as its boundary. That luck transcends the imperial grasp of reason, i. We can find pleasure in the unexpected, at least if it is at some spatial or temporal distance. Unless a person is suicidal, she would not consider it pleasurable to be caught by surprise in a hurricane at sea.

If she identifies herself with a character in a movie that is caught in a similar hurricane, she might find it suspenseful. This is Aristotle's catharsis, the purification of the emotions, at its best. Or, if she re-experiences the event while telling it to friends, after all has been said and done, there could very well be a pleasurable aspect to it.

Otherwise, there is something stressful about luck and the unexpected. Both confront us with our rational limit. Rescher misconstrues the psychological dimension of luck as its ontological essence. We need and apparently do actually have a balance a world that is predictable enough to make the conduct of life manageable and by and large convenient, but unpredictable enough to make room for an element of suspenseful interest.

The emotional aspect of luck should be distinguished from the nature of luck. It is rather a platitude to assert that "we would find it horrible to live in a luckless world," as Rescher does. He confuses the psychological realm with the epistemic realm. It might sound appealing to assert that we would not want to eliminate luck from our lives. However, in a luckless world there would be no we. Luck is the cutting edge of the time line on which human beings experience themselves; luck is the expression of human finitude, both in time and in rational capabilities.

It is a mere phantom to think that we need unpredictability or otherwise we would bore ourselves to death. Unpredictability does not exist to suit human needs.

## Table of Contents

It exists because otherwise there would be no finitude and certainly no humankind. When De Mere asked Pascal how the stakes should be divided between two gamblers if a game of chance is interrupted, he unknowingly initiated new theoretical grounds. First of all, is it possible to subject that what is a matter of mere chance to solid calculations?

And, how much does one risk to lose or stand a chance to win in the face of uncertainty. In answering to those questions, Pascal developed the notion of expectation as a type of fair exchange or contract: a certain amount for which it is fair to forego the gamble. During the same period, seventeenth century Europe faced a spiritual crisis. The ideal of certain knowledge was undermined by reformation and skepticism and religious belief was under attack.

Pascal made a strong attempt to resist this charge with the new means that he had created for the problem of dividing a stake. Pascal introduced a wager betting on the existence of God. Pascal showed that it was in the gambler's selfish advantage to act as if he believed in God, because the expected gain by believing in God was much higher than by not believing.

The development of the concept of probability has shown two essential features that were already present in its seventeenth century origins: the connection of probability and the magnitude of harm or profit the issue of risk and interrelatedness of the problems and solutions of probability. Pascal was confronted with two distinct issues that he tried to solve in a similar way using revolutionary new concepts.

These origins of probability as a scientific concept foreshadowed the way in which probability would conquer, or rather, colonize the world in the course of the following centuries. Pascal showed that the fruitful application of one concept in one realm, can be used to solve another problem in a completely different realm.

Probability theory hopped from one field to another, sometimes bluntly applying the results it had found in one to problems of the next. Probability acted like a parasite, settling there where it could gain most. From Pascal's time on the dice were, literary and figuratively, rolled. Probability came to be applied in a growing number of areas: it assisted gamblers, it modeled uncertain evidence in a courtroom and it described the life-time of people, which enabled insurance companies to be more competitive.

In the beginning the theory was thought to describe the reasonable intuitions of an impartial judge or a canny merchant. In the eighteenth century we can see a shift in the direction of prescription rather than description. People as for instance Laplace showed that probability theory often superseded intuitions, and by the beginning of the nineteenth century it had become a tool rather than a model for enlightenment. In the course of the nineteenth century, two branches started to grow out of a common root: the calculus of probabilities.

This movement resulted in a complete separation in the second quarter of the twentieth century, when Kolmogorov wrote his Foundation of the Theory of Probability and when around the same time Fisher established the first sound theory of statistics. In the same period, Von Neumann and Morgenstern applied the newly gained insights in the development of Game Theory, which explicitly dealt with making decisions under uncertainty, thereby completing the circle back to Pascal's problem of the wager. This keen observation by Pasteur will be the central theme of this section.

Chance only exists by virtue of anticipation. Coincidence will be distinguished from chance and it will be shown that chance not coincidence should be the concern of the statistical sciences. Another candidate for expressing the randomness of life is concept of coincidence. On first sight it seems to be an appropriate term for describing the essential features of chance.

Well, understand me well, it was indeed unlikely to happen prior to the facts, but after the facts the concept of probability loses its validity. Coincidence functions in a distinctively different temporal-semantic framework than chance. Chance is only meaningful without information about the actual occurrence or non-occurrence of the event, typically because the event lies in the future.

Coincidence, on the other hand, pertains to either present or past. Coincidence is derived from the Latin stem: con incidere , to fall together. The contemporary use of coincidence has preserved that meaning. A circumstance is called a coincidence when two, unrelated rather than unlikely, events fall together, i. I returned to my hometown, and met my favorite Latin teacher in a shop.

That was a coincidence. He and I happened to be in the same store at the same time; to be sure, it would have been a similar coincidence if I had met an old school friend there. It would have aroused in me the same kind of surprise and would have stimulated me to say the same expression: "What a coincidence!

The semantic structure of coincidence is different from that of chance. Coincidence is essentially a posteriori , while chance is an a priori property. For sure, it is possible to anticipate a coincidence in saying that it would be a coincidence to meet an old friend in a bar in New York. And, similarly, we can talk about chance after the fact, in saying that it was unlikely that that specific high school teacher would be in that store at time.

In doing so, we can easily create misconceptions that are so prevalent. Often people think of a coincidence as something unlikely, but we claim that this is not so. Is it unlikely that I would meet an old friend in a bar in Soho if every afternoon I have a coffee and often in the evening I have a beer there? A little probability theory suffices to show that over the length of my lifetime I am, in fact, very likely to meet an old friend there.

Nonetheless, it is completely justified call this a coincidence, because if I will meet John there three years from now, then my presence and his presence will coincide, will happen to fall together.

The character of something odd pertains to the nature of a coincidence, but that is not equivalent with being unlikely. It is the rarity or uncommonness of the combination of two events that yields a surprise, and that surprise makes us exclaim: "What a coincidence! Coincidence and explanation. Surprise requires the upsetting of anticipation, and thus assumes both form and content.

It is the mark of an experience for which data are presumed. It does not challenge them in principle, nor demand an explanation of them in principle. The determinist model of causality has often been taken as a model for explanation. In Chapter 2 we shall unnerve such an attempt and put forward a more modest, probabilistic model. Here we shall put forward a formal formulation of this probabilistic model of explanation in order to show that coincidence can be understood in terms of explanation, and vice versa.

The complementarity of explanation and coincidence foreshadows the main idea of next section, namely, that coincidence cannot be basis of statistical inference. Instead, genuine randomness is required. Before we get to a formal definition of coincidence, we should address the question how the idea of rarity pertains to a coincidence? The argument that a too incoherent coincidence or surprise is "the antithesis of the objective," relies on a deterministic view of the world.

This idea itself is, however, incoherent. Instead, the view of the world in which we live is full of uncertainties, chaos and choice. If one is committed to a frequentist definition of probability, as we are, then does rarity not automatically imply a low probability? In the next section we shall argue that what instigates a surprise has a problematic relationship with the concept of probability.

Here we shall show that a coincidence does not necessarily be connected with a low probability. Any attempt to specify a proper level of unlikeliness for a coincidence is doomed to fail, as the attempt to specify a proper level of likeliness for what can count as a fact has failed. If one draws a card out of a deck consisting of the four cards as shown in Figure 1, then it would be a coincidence if the card is a face card of Hearts a face card of Spades would have been more in line with the expectation , whereas it would not be a coincidence if the card was a number card of Hearts because Spades did not have any number.

However, both events have the same probability, one fourth, of happening. Apparently, it is impossible to specify an absolute level of unlikeliness for something to be a coincidence. How drawing a face card of Hearts is a coincidence. We take a hint from Owens' approach, in which a coincidence is coined in terms of independence between the constituents of the coincidence-event. An event is a coincidence, if and only if, it can be naturally divided into parts which are such that the temporarily prior conditions necessary and sufficient for the occurrence of one part are independent of those necessary and sufficient for the occurrence of the other.

This definition seems to grasp at least one aspect of a coincidence. The fact that my birthday and my uncle's birthday happen to fall on the same day is a coincidence. The constituent conditions of the two components of this event are clearly independent of one another. In our definition of coincidence we shall go one step further. It is a matter of a coincidence that Brenda will come home safely if she drives drunk because she comes home safely despite her drunk driving. Similarly, if an unqualified doctor performs an operation the patient's recovering will be a coincidence because the patient recovers despite the fact that an unqualified doctor operated on her.

Therefore, the definition of a coincidence should include besides the notion of independence also the idea that the components of the event could occur despite one another. We define a coincidence as an event that can be analyzed into two or more components, such that the occurrence of one component reduces the probability of the others, i. If E 1 and E 2 are non-vanishing events, then condition i implies condition ii , and vice versa.

In the above examples these conditions are fulfilled. Thus it can be called a coincidence that one comes home safely while driving drunk. The probabilistic definition grasps the salient features of coincidence. The definition of a coincidence can easily be extended to the multi-component case.

With the newly gained insight in the nature of coincidence it is possible to make a connection to the concept of explanation. In the next chapter we shall study the theory of explanation in greater detail. Here we shall provide the basic idea of it. An event E 1 explains another event E 2 , if it is more likely for E 2 to happen in the presence of E 1 than by itself alone, i. So we say that driving drunk explains having an accident, and driving drunk does not explain arriving home safely.

The latter is said to be a coincidence. Therefore, two events are a coincidence, if and only if, the events do not explain each other. This follows immediately from the definitions of coincidence and explanation. The idea of complementarity of explanation and coincidence will be the central theme in the following section on statistical inference. Inference from coincidence. In science there are many instances in which coincidence has played a constitutive role in discovering scientific explanations that furthered the progress of science.

Antoine-Henri Becquerel , a member of a famous French family of physicists, happen to discover certain rays or radioactivity as we, with Marie Curie, would be calling it when he was working with his photographic plates. He published several articles on what was called "Becquerel rays" in and , but he left the field because his rays did not seem as interesting as previously discovered X-rays. He himself did not see the importance of upon what he had stumbled.

It had just been a coincidence. A similar coincidence occurred to Alexander Fleming , when he discovered antibiotics while experimenting with yeast molds. The history of statistics began in the nineteenth century in many other branches of science. The Belgian sociologist Adolphe Quetelet applied probabilistic reasoning in social phenomena. This discussion was subsumed in Darwin's evolution theory, from which the statistically oriented Biometric School and the more probabilistic Mendelians sprang. Even in physics statistics was introduced to describe the behavior of masses of atoms. In these last decades of the nineteenth century statistics made its first clumsy, baby steps in an attempt to get rid of coincidence in science and expand the region of explanation.

When Fisher in the twenties and thirties of this century developed a consistent statistical theory in the field of agriculture, statistics became a separate science. In many experimental sciences statistics has become an irreplaceable tool. Statistics has in many cases even changed the nature of the methodology in those fields. Statistics became a normative criterion of the maturity of several experimental sciences. Psychology is an example of a science that has been changed radically over the course of the last sixty years under the influence of statistics.

It has been the promise of statistics to reveal genuine explanation in the region where there used to be mere coincidences. We concentrate on the issue of statistical inference to illustrate in what way coincidence enters the field of science in modern times and to argue that it is essential to make a distinction between coincidence such as the scientist might perceive a particular result of an experiment and chance in the way that the statistician should perceive the same result.

For the scientist a regularity seems meaningful, but for a statistician regularity should be compared carefully with a chance result it may be a mere coincidence. In statistical inference conclusions are drawn on the basis of data that are available to the scientist. The data that are presented to the scientist allow him to find regularities and check whether these regularities are significant or could be contributed "to chance. In hypothesis testing the procedure is a little more complex. The statistician has to define a certain operational feature of the characteristic in which she is interested.

This is called the test-statistic, whose value can empirically be determined on the basis of the data for instance, in order to see whether a coin is somehow manipulated, she could count the number of head vs. Then the statistician determines whether this number indicates enough evidence to reject the belief into the presence of the characteristic under scrutiny. Statistical inference by means of hypothesis testing decides whether the "conservative" hypothesis can be rejected on the basis of the pre-specified level of significance and the evidence as contained in the data.

In order to avoid the confusion of true probabilities and what can be coined as "surprising coincidence," it is essential to specify levels of significance and the hypotheses prior to the collection of data. Negligence of this procedure may make the scientist susceptible to a hindsight bias. It is not uncommon in the practice of science that hypotheses are specified only after the experiments have been done and after the experimenter has gathered a sense of the responses.

Historically, this is the way in which science progressed to its current height. From Francis Bacon's tables full of data of his experiments, to Isaac Newton, who saw the apple fall from the theory, science has been involved in a form of reasoning that fitted its conclusions to what it saw happening. This form of reasoning has generally been called inductive reasoning. Rao observes that for a long time "inductive reasoning remained more as an art with a degree of success depending on an individual's skill, experience and intuition.

Statistics has generally been considered the continuation and perfection of inductive reasoning statistics is sometimes called inductive logic. There is, however, also an important distinction between the old and the new form of induction. Coincidence could be a constitutive element of the old form of induction as the examples above showed. However, coincidence in the new model principally biases the stochastic models, and as a result warps the scientific conclusions. In the case that a statistician has seen the data, subsequent inference has the immanent danger of bias.

For instance, the to be tested hypothesis and the test statistic should be chosen under the veil of ignorance, i. If this is not the case, then although no warning shows up in the quantitative analysis the numbers lose their probabilistic significance. One could try to incorporate explicitly the bias in one's analysis by formulating the hypothesis as a joint hypothesis stating instances in which one's surprise would instigate further statistical testing.

What counts as a surprise depends on human psychology, and would probably be impossible to explicate. For sure the joint hypothesis will bring down the significance maybe to a level of insignificance. We shall show that basing either the formulation of the hypothesis or the choice of the test statistic on the experimental data will, in unpredictable ways, decrease the significance of one's test without becoming quantitatively visible in one's calculations. In the following simple example we shall test whether a certain coin is fair.

We state the conservative hypothesis H 0 as "The coin is fair. Table 1. Coin-flip data I. An experimenter looks at this sequence and decides to take the amount of tails that finish a sequence of three as the test statistic. It may seem silly, and it is definitely not the most powerful test, but it is a perfectly fine test statistic.

The experimenter observes that 5 tails occurred at the end of the 5 sequences. However, in reality, it was the scientist who was biased. In the twenties and thirties the British statistician Sir Ronald Aylmer Fisher wrote about the importance of a proper design of statistical experiments. Even thirty years before that the American philosopher Charles Sanders Peirce made these observations;. If the major premiss, that the proportion of r of the M 's [e.

But if we draw the instances of the M 's first, and after the examination of them decide what we will select for the predicate of our major premiss, the inference will generally be completely fallacious. Proper probabilistic inference can only result from an intentional, a priori mental process. It is essential that, in Peirce's terms, the predicate of the major premise is determined before information is gathered.

Still, people might be confused and might be wondering whether it was a coincidence that the last of all sequences of three was a tail. Indeed it is a coincidence ; we do not deny that. It is surprising to see that five tails constitute the end of each sequence.

## Chance and Luck: The Laws of Luck and Coincidence by Richard A. Proctor

It might motivate to do further experimenting specifically aimed at determining whether there is significant evidence for the fact that the third of a sequence of coin tosses with this coin is a tail. From the moment that we specify our hypothesis and test-statistic, the semantic structure of the future chain of events changes; the sequence of coin tosses loses its semantic openness that allowed it to yield surprises.

Surprise is an essentially a posteriori concept that goes hand in hand with coincidence. The expression goes that life is full of surprises , and that is true in a very literal sense of the term. Retrospectively we can always describe a certain sequence of events as being exceptional in some sense.

Certainly, several of these descriptions are considered rare from a prior probabilistic point of view, but that a rare description of the actual sequence of events is possible is almost certain. The conceptual confusion between chance and coincidence has played a disruptive role in the acceptance policy of publications in scientific journals. This policy led to some rampant examples of fraud with test-statistics and hypotheses, not always out of evil will, and, in fact, often due to striking misunderstandings of statistics.

However, even if no hindsight bias were present in the construction of the null hypothesis or the choice of the test statistic, this policy would still introduce another bias. Think of a test of a certain null hypothesis that is actually true although the statistician does not know this fact because it is epistemically hidden from her.

Assume that a generation of scientists attempts to disprove the null hypothesis and that they all use the same test-statistic, which rejects the null hypothesis with probability 0. Thus, each experiment can be considered a flip of a coin, which with probability 0. Although most journals have released this bias-inducing policy, it is still not uncommon that articles that do not reject the null hypothesis fail to be published. These articles are dropped somewhere in the process of experimenting, writing, and reviewing. Probability is often misunderstood. The origin of the difficulty is the asymmetry of probability with respect to time.

A probability is a property of an event that changes when it is observed. A probability depends on the state of mind of the observer, or, more carefully, the conclusion we can draw from the occurrence of a certain random event depends on the antecedent state of mind of the observer. Someone is carelessly tossing a coin, and suddenly she notices that head showed up ten times in a row. She is surprised.

Is her surprise justified? Yes, because surprise is a mark of coincidence. However, surprise is untouched by chance because chance possesses the aspect of anticipation, which is exactly that what a surprise lacks. Any sequence was equally likely to occur and a sudden sequence of heads is to be expected if one keep tossing a coin.

She would commit a hindsight fallacy, if she without prior formulation of a hypothesis believes that the occurrence of ten heads could lead to the same significant conclusion under a prior hypothesis. Is it not a common practice in science that the scientist attempts retrospectively to find regularities in the data? Our point is relatively simple and can be expressed in straightforward mathematical language. If a scientist formulates his hypothesis on basis of the data and then tries to test whether this pattern could be attributed to chance, she does not calculate the probability that this specific pattern of data occurred, P pattern , which might be small.

She calculates the probability that this pattern would occur given the data, P pattern data , which could be as high as one. We shall formulate our objection in a more subtle way: the method that makes use of a retrospective study of the data cannot reach the same significance level as a prior formulation of the hypothesis.

Kant recognized that the human mind is teleologically organized. Human beings tend to interpret the world around them as purposeful. The scientific mind performs a similar action when observing data: it tries to find regularities. This is an important statement as it makes clear that when we study the sequence of coin tosses we do not only look for a sequence of heads, but for any kind of regularity.

Our retrospective hypotheses should therefore be formulated as follows:. And the alternative hypothesis thus covers a wider region, which has as necessary consequence that the significance of the test cannot be as sharp as before. It is a matter of psychology to determine what our minds count as a surprise and would induce us to perform a statistical test. We perform, in fact, a test with a joint hypothesis, for instance:. H 1 : more heads than tails, or, more tails than heads, or, more switches from heads to tails and back, or, more groups of two, or Hardly any scientist, however, will find it interesting to test the fairness of a coin and therefore the relevance of my previous remarks may seem questionable.

We would like to clear ourselves from this accusation by showing that similar methods are still widespread in applied statistics. Look at the following excerpt of an introduction to applied statistics. When the author wants to indicate the importance of numerical and graphical data-description prior to statistical inference, he gives two examples. After the first example he continues,. Similarly, in developing an economic forecast of new housing start for the next year, it is necessary to use sample data from various economic indicators in order to make such a prediction inference.

In both of these examples involving an inference, description of the sample data is an important step leading toward the inference that we make. Thus no matter what our objective, statistical inference or data description, we must first describe the set of measurements at our disposal. As we have seen in the case of the coin, any prior knowledge about our actual data-set might affect our choice of test-statistic or it could lead to the inclusion of a certain variable in forecasting.

To what extent this happens is unclear. When the scientist observes the data in some form possible bias slips into her head. Take the following example of two data sets of length of hospital stays at two different hospitals. When we view the data in a graph it seems that hospital A generally keeps its patients longer for observation than hospital B does.

To see if we can make a significant inference, we perform a one sided test of the kind where we compare the two mean hospital stays:. Clearly, the choice for a one-sided test was inspired by the data themselves or a graphical description thereof which meant that we used the data more than one time for the same inference. The significance level of this inference, commonly called the p-value , will therefore misrepresent the true significance of the data.

The inference becomes a black box where data come in and meaningless numbers come out. The general drawback of seeing a description of the data before making inferences is that it affects significance levels in an unknown fashion. This discussion of statistical hypothesis testing pin-points a more general point.

Chance and coincidence are different entities. Something is not a coincidence because it is chancy. Chance has a sense of anticipation, whereas coincidence is a retrospective notion. This observation is not a philosopher's phantom; it is part of our everyday language. We say that it was a coincidence that Becquerel discovered radioactivity, or that I met a friend in a cafe in New York.

From our use of language it is clear that coincidence is retrospectively attributed to a certain event that is considered to be rare in some sense. Even when we say: "It would be a coincidence if I meet my friend in a cafe in New York," clearly the identification of can only occur after the event has actually happened. Chance, on the other hand, is an inherent aspect of a situation prior to any development. Coincidence: a problem for Bayesians and frequentists.

In this section we shall show that coincidence is a problem for both Bayesians and frequentists by means of a simple example. Whereas frequentists may go wrong in hypothesis testing, Bayesians may face an even greater danger. Bayesian theory argues that specifying a subjective numerical probability of an event is always possible, permissible and even inherently part of the furniture and functionality of the human mind. In what follows we shall argue that certain probabilities are fundamentally meaningless because the Bayesian failed to distinguish between a probability and a coincidence.

The argument here is aimed at the universal ambitions of the Bayesian theory an attitude that is coined Bayesianism. Bayesians have no conceptual problem, for instance, to construct the probability that Gauguin becomes a successful painter. Williams argues that any such attempt is absurd. Should Gauguin consult professors of art? Any subjective approach to chance may result in probabilistic bias if no distinction has been made to prior probabilities and posterior results. We shall quickly recapitulate the problem of hindsight in the case of hypothesis testing, by means of an example.

Bias is introduced in a statistical conclusions if one fails to construct a prior mental model of the results.

We shall continue to show that the situation is analogous for Bayesian estimation and that the universal aspirations of the Bayesian theory put the validity of its results in peril. We shall give two examples of the dangers of Bayesian theory, which rest essentially the confusion of coincidence with chance. Suppose Andy and Brenda are told that a coin is going to be flipped eight times. Andy decides to check whether the coin is biased toward changing sides. Brenda does not give it a second thought and just observes the experiment. They both receive the data as recorded in Table 2.

Table 2. Coin-flip data II. Andy starts testing his hypothesis. He wanted to check whether there was a bias to changing sides, thus, accepting the null-hypothesis:. According to conventional statistical theory, he should then evaluate the probability under the null-hypothesis that the observed sequence would happen. The so-called test-statistic is:.

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Note that T has the value seven, as the coin changed sides seven times. Andy calculates the probability that such an event could have occurred by chance , i. Brenda, on the other hand, did not make any hypothesis about the outcome. She can be surprised about the outcome, but she cannot make any probabilistic inference based on her surprise. She could formulate now the hypothesis that the coin is biased towards changing sides, but she cannot, on the basis of these same data, draw the same conclusion as Andy did.

The situation of Andy and Brenda is not uniquely a problem of hypothesis testing. The same issue features in the case of Bayesian inference. Let us assume that Andy and Brenda are now two Bayesians. Again, Andy is interested to see whether the coin is biased towards changing sides. He has no knowledge of the coin that is used for this experiment, so he considers the parameter of interest, p changing sides , as a uniform 0, 1 distribution. Having observed that 7 changes of sides take place, Andy updates his parameter, which becomes a beta 8,1 distribution.

For Brenda the situation is essentially different. She did not specify a prior distribution and thereby forfeited the opportunity to make a statistical inference. She cannot conclude as Andy did that the coin is biased towards changing sides. This conclusion may seem strange, but it becomes intuitive after grasping the following example, which is essentially the same as the coin-flip example. Seven numbers have been recorded in Table 3.

Each entry is either a one, heads, or a two, tails. Table 3. Coin-flip data III. A Bayesian is asked to make an estimate about the next number in the sequence. If we then retrospectively apply both g 1 and g 2 as priors, a paradoxical situation arises: the prior distributions on both p according to g1 and p according to g2 are uniform 0, 1 distributions. Having observed that seven numbers occur both according to g 1 and g 2 , the Bayesian statistician would believe that the posterior distributions on both p according to g1 and p according to g2 are beta 8,1 distributions.

Paradoxically, the predictions of the eight number are conflicting although each has the same high probability, i. The conclusion of these paradoxical calculations is that coincidence does not have a legitimate place in statistics, neither in Bayesian nor in Classical statistics. Moreover, probabilities do not belong to the furniture of the human mind. The mind can lease probabilities, but the rent to pay is prior attentiveness. Pasteur was more in the right than he might have known, when he said, almost one and a half century ago, that chance favors only an attentive mind.

Deliberative rationality: beating the coincidence. At the end of this section on coincidence we shall indicate how the concept of coincidence will function in the dynamics of the ethics of chance. Coincidence upsets the cognitive continuity of the world, because a coincidence indicates an absence of explanation. Cognitive expectations and explanations do not always coincide with the causality in a broad sense of the world.

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In the chasm between the two realms coincidence lures. However, coincidence does not necessarily indicate a lack of rationality. Quite on the contrary, it points us to an important aspect of rationality. Rationality itself, as we shall argue in this dissertation, should admit the possibility of the unexpected and the improbable, and should make its judgment to the largest possible extent resistant to the occurrence of both, that is, rationality should act in such a way that no matter what turns out it feels the least possible regret about its decisions.

The following two examples precisely makes this point. Media reported in the seventies and eighties cases of people who spent their entire incomes on building nuclear bomb shelters. Was it worth spending so many resources on preventing what never happened? The answer of this question is not as obvious as it may seem. It is not an unambiguous no. With hindsight people often feel justified to qualify those people as too cautious, bordering on the irrational. However, hindsight is a bad judge when one is confronted with uncertainties.

We would say that a person who refuses to give her wallet to the robber with a fake gun is lucky. However, if she is prepared to give him her wallet, then calling her overcautious or irrational afterwards is certainly not true. At a height of the Cold War in the eighties experts estimated an expected accidental nuclear war within three to fifteen years, given the false alarm rates and decision structure of that moment. That means an expected occurrence within one generation. Those preparing for this possibility were not unreasonable in their assumptions that the probability of an accidental nuclear war was considerable.

The coincidence that the nuclear threat diminished does not diminish the level of deliberate rationality of their decision. These examples foreshadow the general issue of deliberative rationality in later chapters, in which the concept of coincidence will play a key role. The tension between explanation and coincidence will result in the relevance of the latter for the issue of responsibility. In Chapter 2 we shall show the unjustified popularity of the contemporary notion of moral coincidence or moral luck with the tools that we have developed here.

In the final chapter of this dissertation the concept of coincidence or unknown risk will be important, and we shall prove its relevance for safety regulations and intervention. Theories of Probability. Probabilities are not readily available in the world around us. Expressing uncertainty, probability represents precisely what is epistemically unavailable to us.

Also the concepts chaos and free choice indicate a lack of predictability of the world. Probability is distinct from chaos and free will in that it presupposes some type of long run regularity. In this section we shall deal with questions such as how probabilities can be assessed and evaluated and to what extent long run regularities are relevant to this issue. Is every long run relative frequency a probability? What is the probability of a single event? There are several methods to assess a probability and they can be distinguished broadly in four different methods, i.

Each of these methods has its own criteria of assessment and evaluation. We shall advance an eclectic mixture of all four theories. It seems to us that the wide semantic range of chance should be reflected in an equally rich interpretative approach to probability. The quantum mechanical behavior of subatomic wave-particles is generally given a propensity interpretation, whereas a die, if not suspected of being biased, exhibits probabilistically logical behavior. The traffic in New York City has been modeled with a probabilistic, frequentist model.

Decisions by drivers have been replaced by impersonal, randomized events. A doctor interprets the posterior probability of having breast-cancer given a positive test-result of the mammogram as the level of epistemic certainty she has on the basis of the test alone. In the following sections we shall touch upon three separate issues of probability, its interpretations, methodologies and structure.

Besides an eclectic interpretation of probability, it is our claim that any probability, whatever interpretation suits best, possesses a frequency structure , i. One important methodological aspect of probability follows from our observations concerning coincidence.

Any Platonic image of probability, namely as the world as a list of probabilities out there , is misguided. This idea is defended by some personalists, logicists and frequentists, but is most frequently found among the former. Platonic ideas about probabilities can be found among logicists and personalists with respect to the issue of a uniform prior. In the following sections we shall examine the personal theory of probability and the frequency theory of probability.

Personal theory. Some things are thought to be more probable than others. It is more probable that the sun will rise tomorrow than that it will not. One holds this belief quite strongly and legitimately so. Probability captures, in some sense, the strength of belief, i. Personalist theory postulated that probability not only expresses the strength of belief, but that it is actually defined as such. Personalists associate probability with the subjective magnitude of seeming probable. The sun-rise example is particularly interesting, because it has been a focus of controversy between personalists and defenders of other theories.

We shall return to this example shortly. Personalists make a distinction between the construction and the evaluation of a probability. According to personalists, "the question when a probability statement is correctly made has two different meanings: 1 How should we make or construct well justified probability judgments? For example, when is a surgeon justified in saying that an operation will succeed with probability 0.

For example, how would we evaluate the surgeon's probability judgment if the operation succeeds, or if it fails? Methods of evaluation, such as calibration, are discussed only briefly. Personalists come in several flavors. After ground-breaking work by Definetti and Savage, different schools of personalist thought have developed. All personalists have in common that they define probability as a numerical measure of the strength of a belief in a certain event.

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Their picture of belief bears strong affinity with that of the British empiricists. John Locke argued that every belief is held with a certain "strength" in the human mind. The personalist theory of probability interpreted this strength as the individual's idea of the likelihood of the event. The current mainstream personal theory is the Bayesian theory, and we shall use the terms interchangeably. Bayesians believe that it is possible to make probability assessments even in the absence of frequency information.

However, a personal probability is not a mere opinion. It is an orderly opinion. The personal theory specifies consistency rules. One of the concerns of the theory is with the revision of a probability in the light of new evidence. Bayesian theory developed a calculus of beliefs that specifically deals with this issue. The original degree of belief is replaced by a new degree of belief when new evidence is obtained.

The personal theory of probability raises a number of issues. The great advantage of the theory, viz. The initial prior probability, i. Prior beliefs are subject to the individual's bias. Personalists defend a Peircean stance to truth, i. The possibility of wide variation of prior personal probability assessments has been recognized by the personalists as their Achilles' heel, and it has been become the aim of serious theory to show mathematically that in the light of new evidence different personalist probability inferences will converge to the same numerical value.

Several convergence theorems have been proven. In the following sections we shall focus on some important aspects of Bayesian theory. Bayesians have recognized that acting under uncertainty is essentially the same as making a bet. In constructing probabilities Bayesians have historically invoked several additional assumptions that are not directly related to any frequency ideas: the idea of risk neutrality and the principle of insufficient reason. We shall briefly discuss some Bayesian calculations.

Further mathematical details are placed in the appendix. We shall then return to the Sunrise example and show that the world cannot be considered as a list of probabilities. Some personalists make an explicit connection between probability and a specific behavioral attitude. Pragmatic personalists operationalized probability, the measure of believe, as a measure of willingness to bet.

The concept of a bet can include also non-monetary rewards and punishments. Any action under uncertainty can be interpreted as a gamble in a wider sense of the term and thus gambling is unavoidable. For instance, if Sarah goes out for a walk and does not bring her umbrella despite a negative weather forecast, she is, in fact, gambling: the stake is the nuisance of carrying her umbrella, whereas the uncertain pay-off is the event of getting wet by the rain.

In hypothesis testing, for instance, probabilities stand for the rate of accepting faulty inferences. According to these personalists, the identification of probability with the willingness to bet singles out an identifiable numerical value for a probability:. Thus, if I say that the probability of an event is one-third, I will be just willing to accept a bet in which I gain 20 cents if the event occurs and lose 10 cents if it does not, that is, a bet at odds of I shall be very happy to accept a bet on this event at more favorable odds but unwilling to accept a bet at less favorable odds than This definition of probability makes an important assumption: risk-neutrality of the agent.

It makes an explicit connection between probability and the willingness to be involved in a bet. That means that the gambling situation as such is assumed not to have any influence on the preference-structure and that the measure of preference of an event is defined by the expected benefit of the event. This definition indeed singles out a measure of probability. Intuitively, the procedure functions as follows.

A certain event yields twenty cents, but it is uncertain whether it is going to happen or not. Andrea tries to buy Brenda out of gambling on the event. With any amount less that a little over six cents Brenda feels more inclined to take the risk, whereas if Andrea offers her more than seven cents she prefers to take that rather than being involved in the gamble. Apparently at six and two thirds of a cent Brenda is indifferent between the gamble and the buy-out.

Assuming that Brenda is risk-neutral, the expected utility of betting and the expected value of taking the certain stake are equal. Assuming risk neutrality, the true, subjective probability can be defined as the fraction of the of the indifference utility and the utility of the event. However, not all personalists rely on this identification of rationality and risk-neutrality. In Savage's axioms, for instance, this idea is completely absent.

He defines probability as a subjective level of confidence under the conditions of transitivity, substitutability and monotonicity. One of the oldest controversies in the theory of probability, going back as far as the early nineteenth century, is whether probabilistic homogeneity can be the result of ignorance.