CHAPTER 9

Gambling and Safe Play
y§LL MAN'S ECONOMIC STRIVINGS MOVE BETWEEN TfVO
-/jL apparently opposite poles: the urge to attain easy riches by gambling, and the need for security. Adventurers and stolid business men seem to represent two very distinct types, two philosophies, and two methods of fashioning the future, for while the former are gamblers by nature, or at best successful specu- lators, the latter not only avoid gambling and speculation but despise them as utterly immoral.
But between these two apparently irreconcilable attitudes, there is a close inner connection: gambling and certainty obey the same laws of probability — without these laws the most respectable Insurance Society and the most lurid casino alike would collapse. Gamblers are usually unaware of this fact; they expect to be paid their winnings, and rarely ask themselves where the money comes from.
Nor do people with insurance policies usually worry about being mere pawns in a very big and most complicated game, or about the fact that the security they are offered, no matter with what guarantees, is only safe within the bounds of probability, or, in this case, actuarial calculations. By paying their premiums, they "purchase" security from the "vendor" — the insurance company,^ which, in order to fulfil its obligations, has to run great risks itself. These risks are in many respects much more difficult to foresee than the risks of the gaming table. Insurance companies have to play with life and death, with catastrophies of all kinds, with accidents — in short with events whose frequency and gravity can only be estimated very approximately.
Insurance companies usually emphasise that they are merely trustees of wealth and thus clearly distinguished from gambling houses. This claim, too, is only true to some extent, for gambling houses must also keep large funds "in trust" if they are not to become insolvent during the first heavy setback. * Maurice Fauque: Les Assurances (Srd ed. Paris 1954), p. 14.
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The essential difference between these two institutions is that insurance companies contract to pay for certain accidents (or at fixed dates) while the gaming tables pay for sheer luck.
The gambler's mentality
It would seem that gamblers worry far less about the future than anyone else. However their outward fatalism rarely gives a true picture of their real mental attitude. The vast majority of gamblers, far from being fatalistic, believe that they can persuade fate to be particularly kind to them — which is the main reason why they gamble at all. Not a few feel like great conquerors who challenge fate in order to vanquish it. Fate must be forced to submit to them. A gambler who gets up from the table with neither a scoop nor with empty pockets, is no real gambler at all, but only a looker-on. Gambling means winning or losing to the limit, and not just staking for fun.
Many big-time gamblers have therefore tried to turn gambling into an art. Such men do not just stake on any number at roulette, but choose their numbers in accordance with a simple plan. Thus they will stake on red only after black has turned up a given number of times, or vice versa, on the assump- tion that, in the long run, both colours turn up an equal number of times.
Other players try to win by "doubling up". Having lost the first round, they double their original stake in the second round and so on. If the odds are even (as they are when staking on red or black in roulette) they will have staked 1+2 units in the second round in order to win back 2x2 units, and 1+2+4 units in the third round to win back 2X4 units, etc. — i.e. they will never win back more than double their original stake. Unfortunately, if they have a long run of bad luck, the stakes have to be so great, that all but the richest gamblers usually run out of cash. No wonder then that "doubling up" has ruined so many roulette players.
Almost all games of chance — except such very simple ones as tossing coins and throwing dice — allow of combinations that apparently increase the chances of winning. Many so-called
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"scientific" gambling systems are based on the belief that cer- tain lucky numbers which have turned up in the past, are likely to turn up again in the future. What happened once is bound to happen again, remains the motto of most gamblers — hence their preoccupation with cataloguing past results. In practice, how- ever, probablitity is not as straightforward as it appears in mathematical books, since in order to apply its laws successfully, a continuous record of a vast number of games at a particular table must be kept. Now, not even the most indefatigable gamblers spend 24 hours a day observing and recording games at one table, and they must therefore rely on the special sheets published by the big casinos, or organise an intelligence service of their own.
But even when they have all the information they need, gamblers are not usually strong-minded enough to follow a monotonous system, particularly when they are losing. Thus they may decide not to stake when the system decrees that they should, or go to another table to try their luck there. Since they have no records about what has just been happening at the new table, they are forced to gamble like the other punters whom they despise as amateurs.
Gamblers par excellence
The question therefore remains whether the systematic gambler has greater chances of success than his happy-go-lucky counter- part. What we know of famous gamblers and great coups at the gaming tables leads us to suspect that the question must be answered in the negative. The biggest of coups, in which gamblers won vast fortunes within a few hours and "broke the bank", were made by gamblers in the true sense of the word, and not by followers of systems. Theirs was the victory of dis- continuity over continuity, of chance over calculation, of the im- probable over the probable.
To give but a few of the most famous cases: in the middle of the 19th century, casinos throughout Europe dreaded the name of the Prince of Canino — the pseudonym of Charles-Lucien Bonaparte, a cousin of the then President of the French Republic
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who later became Napoleon III, His great connections alone procured the Prince the necessary credits to play with unheard- of stakes. He would never stake less than the permitted maxi- mum— which was much higher than it is today — and always accompanied his main bet with a side bet on one of the two colours. In this way he could make his money last longer, but not cover the losses from his main bets, had these been persis- tent. He was therefore a pure rather than a systematic gambler, and this despite his being well-versed in the mathematics of his day.
But though Charles Lucien Bonaparte often lost consistently for a few hours on end, his tremendous reserves always enabled him to recoup his losses and to leave the table as the winner. His greatest trump was, in fact, his ability to get up when he had won enough, and not to strain his luck, as most gamblers in his place would have done. He would also allow some time to elapse after his last victory, and then begin anew in another casino. He might therefore be called the discontinuous gambler par excellence.
His most sensational coup was made in Bad Homburg where he won 560,000 gold francs — the equivalent of ^185,000 today — within one week and broke the bank. The directors of the casino, the famous brothers Blanc who subsequently became the directors of the Monte Carlo casino, were forced to borrow money from Rothschild's Bank and to reduce their maximum stakes.^ However, this emergency measure proved of small avail, for in 1860, eight years after the Bonaparte episode, a Spaniard by the name of Thomas Garcia managed to relieve the Homburg casino of 800,000 gold francs ( ^270,000) by working with a number of accomplices all of whom staked on the same number, thus exceeding the permitted maximum stake. Once again the casino was in serious trouble, but Garcia was too much of a gambler to be satisfied. Instead, he returned time and again to the tables, until he had lost every penny of his winnings,
Monte Carlo, too, which became the world's leading casino in the 1860's and remained so until 1914, when Latin American and Asian casinos raised their maximum stakes, was the scene of
^ E, Caesar Conte Corti: Der Zavberer von Homburg und Monte Carlo (Berlin 1955), pp, 66-71.
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many a desperate struggle to break the bank. Those legendary Russian Grand Dukes and Princesses — ^like the Princess Suvorov who during one afternoon lost 300,000 francs, won 700,000 francs, and finally got up with a net loss of 300,000 francs — were not the bank's real enemies. With or without a system, they would usually stay at the table until their stake-money had run out, to return again after an interval. Much more dangerous, were such reckless gamblers as Vincenzo Bugeja, a Maltese merchant, who entered into the battle with a million gold francs and forced the management to send to Paris for additional funds — despite the fact that the casino had a cash reserve of over two million francs.
"The man who broke the bank of Monte Carlo", however, was, in fact, a myth put out by the casino to lure new customers to its tables. In actual fact, casinos became more and more care- ful, restricting maximum stakes and always keeping large enough reserves to match up to even the richest and most obstinate gamblers. "Breaking the bank" came to mean no more than that a given table had to interrupt the game until suflficient cash was fetched from the office. Continuity, therefore, always scored over discontinuity.
Gaming systems
Nevertheless the organisation of gambling houses involves a most complex and difficult application of probability laws, for while these laws are the same for all games of chance, their operation differs from case to case.
By and large, gambling falls into one of three categories. The simplest is common gaming in which all the stakes go into a common bank, and where the total amount — usually after deduction of a commission — is distributed among the partici- pants according to a fixed scale. This is the principle of the foot- ball pool in which the total investment and the odds vary from week to week, but in which the directors run no risks, provided only that their overheads are covered.
The second type is the lottery, in which the total number of tickets, and hence the total stake are fixed in advance. Here, too.
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the total stake is redistributed to the participants, after the organisers have deducted their own share to cover costs, etc. In most countries, lotteries are run by the state, whose institutions benefit from the additional revenue obtained. Thus only the gamblers and not the organisers are thrown at fate's mercy, and if they fully appreciated the odds against them, they would be bound to give up buying lottery tickets altogether. True, the government does not conceal any figures, and gamblers might argue that, if only they lived long enough and never missed a single draw, they would be bound to win in the end.
The third way, the gaming table, seems to involve greater risks for the organisers and to be fairer to gamblers, for here either party is liable to lose. In fact, the organisers run even greater risks, for while the gamblers can restrict their stakes, the organisers cannot restrict the number of players. While the gambler has to contend with only one unknown: the winning number, the bank must contend with two unknowns: the winning number and the unpredictable total stake on this number.
The fact that popular casinos nevertheless make good profits is, as we have seen, due to probability laws. To the individual gambler, the chances of acting according to these laws are small and rather unattractive. Thus, during 1000 million games of roulette, the chances of black (or red) turning up on SO succes- sive occasions are only 1 : 1000,000,000 and even if the gambler were to play 1000 games a day — which would take him sixteen hours at the rate of one game per minute — ^he might have to wait 2700 years before such an event occurred. Casino records do, in fact, show that the maximum run on a given colour is 24,^ though probability considerations show that two runs of 25 may be expected every week, if 1000 games a day are played.
Time and probability
Such contradictions between theory and practice are clearly perturbing to mathematicians. Hence many of them have always 1 Emile Borel: Les prohabilitis et la vie (Paris 1950), p. 21.
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dismissed probability theory as too vague to be anything but a perversion of true and precise mathematics, and this despite the fact that some of the greatest mathematicians of all time — Bemouilli, Laplace, and Gauss — ^played a leading part in its formulation. Some mathematicians have even questioned the validity of the fundamental theorem of probability, i.e. Bernou- illi's Theorem (which states that the probability of an event is a function of its occurrence in the past), calling it a mere tautology which throws no fresh light on anything. Others again, have called probability theory "unrealistic", and its applications too full of practical errors to be worth anyone's while.
Yet others — and this is perhaps the most telling reproach — maintain that, after three centuries, probability theory is just where it was when it began. This, for instance, is the contention of the Rumanian mathematician Pius Servien, according to whom probability theory is bad science and bad mathematics, and relevant exclusively to the throw of dice, without, even in that field, being able to predict anything precise about any one throw. ^ Probability theory tells us nothing about the nature of chance, or about any possible laws governing it. In other words, we are no further than Voltaire was when he said that what we call chance may well be the unknown cause of a known effect. 2
What astounds non-mathematicians most, is the cavalier manner with which probability theoreticians treat the concept of time, and the way in which they effect a complete separation of time from number. For probability it makes no difference whether a long series of events occurs during one hour or during one century. Thus if we play roulette once a year or once a minute, our chances of success are the same, since only the frequency of a given number turning up, and not the interval between two successive throws, affects our luck. For this very reason, probability calculations, unlike predictions based on periodic phenomena, seem completely irrelevant to human action and thought, which must be time-bound or else stray into fantasy. Instead of telling us that it will be our turn tomorrow, the theory merely promises us that our turn will come some day.
1 Pius Servien: Science et hasard (Paris 1952) pp. 59 and 185.
2 Voltaire: Article on Atheism in the Dictionnaire pbilosopbiqne ( 1764).
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State draws
All these objections notwithstanding, probability theory has maintained and proved itself in the most diverse fields. From the gaming table it has advanced into public and private finance, both as an incentive and also as a great leveller.
Gambling which seems so disreputable in the casinos, takes on a respectable guise in high finance, so much so that only the most uncompromising of moralists object when the state pays off its loans by public draws, thus favouring some of its creditors at the expense of others. Even non-speculative investors bear this injustice gladly, for it combines security with a small flutter.
Governments may also cash in on the gambling urge in other ways. Thus Premium Bonds, which bear no interest at all, are safe investments with enticing one-way risks. The investor can lose nothing (except interest) and stands to gain a (tax free) bonus, which may be vastly greater than his original investment. In this way, the government has brilliantly balanced the gambler's with the stolid business man's instincts — two instincts that are apparently poles apart. Many respectable citizens, who hate gambling like the plague and have never set foot on a race- course, seem to have no scruples of conscience about speculating in this way.
Oddly enough. Premium Bonds are most popular of all in the Soviet Union where public gambling is most strictly forbidden, but where those who contribute to their country's welfare by subscribing to these bonds, may become wealthy overnight — at least by Soviet standards.
In April 1957, however, economic difficulties forced the Soviet Government to suspend repayment of the Bonds, together with draws, imtil 1983. By that distant date, which only a few of the old creditors will live to see, the government hopes to have repaid all previous loans. As far as her lotteries are con- cerned, therefore, Russia plans far longer ahead than in the economic sphere. In the Soviet Union, the smile of fortune is apparently eternal, and gratification can therefore be postponed ad lib.
The speculative element of lottery loans is quite distinct from that of Stock Exchange transactions which are generally bound
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up far more closely with complex economic processes. All attempts so far to apply probability theory to market fluctua- tions have ended in abysmal failure, precisely because the stock market is governed by such dynamic factors as economic developments, political and psychological trends, and not by the simple situations characteristic of lotteries, roulette, and card games played for money ( baccarrat, vingt-et-un, etc. ) .
Mathematical forecasts are most reliable where the facts are clearly circumscribed, but fail wherever subjective factors and fluctuations play a large part, and where moods and whims may be more important than sober logic — in short wherever psycho- logical factors are decisive. In America, two European scholars, a Hungarian mathematician and a Viennese economist have tried to establish a theory of games from purely logical con- siderations,^ but though their profound book penetrated into the mentality and strategy of gamblers like none before or since, it contributed little to the art of predicting the actual results of speculative acts.
Shipwreck and conflagration
The greatest and most important field of mathematical fore- casting is insurance. In gambling, probability theory may prove of practical help here and there, but not even the biggest casinos bother to employ a full-time mathematician. They know that probability will work in their favour even without such help. In Insurance Companies, on the other hand, mathema- ticians play an essential role, so much so that there is a special branch of actuarial statistics which deals with mortality tables, premium rates, etc. Probability is the heart of insurance, and though Insurance Companies must gamble on the future, they do their utmost to reduce their own risks and thus to protect their policy holders. So successful are they, that insurance com- panies rarely if ever go bankrupt.
Though modern insurance is unthinkable without probability estimates, insurance was born independently of, and long before, probability theory. If we ignore the very ancient practice
^ O. Morgenstern and J. Von Neumann: Theory of games and economic hehaviotir^Pvmceton 1944).
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of covering funeral costs by public subscription, insurance proper may be said to date back to the late Middle Ages, when the first insurance contracts were signed.
These first covers were extended to ships and shipping, which were then exposed to the double danger of natural catastrophes and piracy. While such large maritime associations as the Ger- man Hansa and the English Company of Merchant Adventurers could readily bear the occasional loss of a few ships, smaller companies were easily ruined by such disasters, and though no prophet was needed to point this out, it took a long time before appropriate action was taken. Mediterranean shippers and merchant-sailors were far too individualistic and too envious of one another to make common cause. True, they would occasion- ally sail out on common trading expeditions and found new colonies, but to make themselves responsible for one another's losses went against their sense of business propriety.
As insurance against maritime losses involved major financial risks, it is not surprising that the only people who were initially prepared and able to take it up were wealthy Italian bankers — the great Florentine houses of the Bardi, the Peruzzi and the Frescobaldi. The first policies were restricted to one sea- voyage or to a limited period of time, but as insurance expanded, as losses could more easily be recouped from current profits, and as experience increasingly showed what risks were the most serious, policies became more and more elastic. Meanwhile, all maritime insurance had been placed under public supervision, for the protection of all the parties concerned. In the great ports of Genoa, Barcelona, Bruges, Antwerp, and London special assurance offices saw to it that all business was conducted pro- perly, so that fraudulent claims on the one hand, and delay in payment on the other, became increasingly rarer. By the 16th century, maritime insurance was an established branch of big business, with rigid laws and customs. The oceans were as stormy as they ever had been and piracy was still rife, but the financial risks were fast diminishing.
The second great branch of insurance, fire insurance, only came into its own during the 17th century, particularly in England. The Great Fire of London in September 1666, in which 13,000 houses were destroyed, was the signal that some- thing had to be done about a possible repetition of such a
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disaster. True, no insurance company could have made good that damage — as no modern insurance company could rebuild a bombed-out city from its own funds — but the Fire of London drove home the point that it was better to have some restitution than none at all.
Since the Great Fire remained a unique event, insurance companies did well out of this new demand, and fire insurance became another rewarding field for private enterprise.
Wagers with death
The third branch of insurance, life insurance, was bom later and had a much greater struggle to see the light of day. Here was an entirely novel idea, for while shipwreck and fire, however frequent, were exceptional events, death was anything but that. Life Insurance was therefore looked upon by many as direct interference with the will of God.
Still, the idea caught on, particularly when the probability experts pointed out that death, though inevitable, could strike some earlier than others, and was therefore an excellent object of insurance, and provision against it a commendable way of ensuring the welfare of bereaved families. Those who lived longer, and paid in more money therefore helped their fellow-men — a proposition to which not even the staunchest puritan could object. True, life insurance is a gamble with death, but the odds can easily be worked out from mortality tables, so that no one loses in the end.
Actually, the first beginnings of life insurance go back to the 17th century, long before mortality tables were known. It was then that a number of Italian cities asked their citizens to subscribe to life annuities, the returns from which varied with the subscriber's age. Young people who had a long life- expectancy were repaid at the rate of 5 % per annum, while old people were paid up to 12|% per annum. In 1663, an Italian banker, Lorenzo Tonti introduced this system to France, where so-called tontines, i.e. loans by public subscriptions, the last survivor obtaining all the remaining funds, exist to this day.
Tontines were, however, a rather crude way of assessing life
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risks, and at bottom, merely a variation of the common mediaeval custom of providing for the future. Thus well-to-do and con- siderate fathers would deposit money in their children's name, often at birth, so assuring their descendants of a life-long annuity. A young gentleman with so provident a father was made for life, while his sister was sure to be a good match for anyone. True, if the original capital fell by the wayside for one reason or another, so did the annuity, but a father who had made such provision was judged by one and all to have done every- thing in his power to safeguard his descendants. His was the peak of financial foresight.
Only in the second half of the 17th century, could life insurance proper make its debut, for it was then that research into English church registers led to the compilation of the first mortality tables from which the average life expectancy of a man in his thirties, forties or fifties could be calculated. While no indivi- dual prognoses could be based on these tables, predictions for large groups proved uncannily correct. Shrewd financiers realised at once that this newly acquired knowledge could be put to profitable use, since all people, irrespective of age, could now be insured against death. Naturally old and ailing people had to pay higher premiums, but even if they died the day after a policy was issued to them, their dependents would be paid the full amount.
It was a lucky coincidence that at the very time when the first mortality tables were being computed, England learnt of probability theory from Holland and France.^ Fifty years later when she first became acquainted with Bernouilli's Ars con- jecturandi (The Art of Conjecture) — the result of twenty years of work — she had all the elements needed for putting Life Insurance on firm ground.
In England, Life Insurance soon developed into a flourishing business, but on the continent it ran into unexpected difficulties. The authorities, while recognising that life insurance policies in favour of one's dependents were praiseworthy indeed, felt that the benefits might encourage unscrupulous persons to despatch their benefactors to an untimely grave. One of the results of fire insurance had been an increase in arson, and no government
"^ De Witt: De vardye van de lif-renten na proportie van de los-renten (Den Haag 1671)
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wanted to see such avarice being directed at life itself. These fears, and the religious objections we mentioned earlier, caused the governments of France, Prussia, and Genoa to prohibit all forms of life insurance^ for a whole century. France gave way after the Napoleonic wars, and Germany followed in 1827.
From then on vast amounts of money poured into insurance offices. Life insurance companies built up tremendous reserves which they invested in all sorts of sound enterprises, gradually becoming the financial empires they are today. In the United States, where more than half the adult population is insured, the total value of life insurance policies was 450,000 million dollars in 1956, and insurance companies had assets to the tune of 100,000 million dollars. In Switzerland 170 people in every 100 households carry life insurance, and in other West European countries, too, life insurance figures run into millions of pounds. If some disaster were ever to kill off a large proportion of the population, no insurance company could possibly meet its obligation. But the probability of such a disaster is small enough for insurance companies to be able to say with good consciences that they offer mankind as much security as is within the bounds of reason.
From the cradle to the grave
Nevertheless, some countries have thought it best to make doubly sure. Since wars and other catastrophies invariably require the government to intervene, and since, moreover, the West, at least, has come to realise its responsibility to those who are too poor to provide for their own future, social insurance against accidents, disease, and unemployment have become matters of state in most European countries.
It was in England that the most comprehejisive plan ever to offer some measure of security from the cradle to the grave was first put into practice. While the Second World War was still in full swing and while bombs were pouring down on London, an Oxford don, Sir William Beveridge, worked out a plan for insuring the entire population against occupational and economic ^ H. E. Raynes: A History of British Insurance (London 1950), p. 11 9.
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hazards. The price was to be a relatively small weekly contribu- tion to a National Insurance Fund, which would not only pay compensation for accidents, illness, and unemployment, but also contribute to such happy, if expensive, events as birth and marriage. In addition, the Fund would also pay family allow- ances and provide a pension for all men over 65 and all women over 60 years of age. According to Beveridge's somewhat optimistic estimate, the total cost of this scheme would have been no more than 697 million pounds in 1945, and 858 million pounds in 1965, when the plan was expected to be in full operation.^
The Beveridge plan aroused tremendous excitement not only in England but also in all the allied countries. For some months, the name of Beveridge was better known than that of even the most famous generals. But then the government had second thoughts and refused to implement the plan, and it was not until after the war that some of its main suggestions were carried into effect.
England is also a pioneer in another field of insurance: the all- risks policy. Thus Lloyd's, for instance, will underwrite almost any kind of risk imaginable. Film stars can insure their eyes or their lovely legs, acrobats can insure their biceps, peace-lovers can insure against a world war, and neurotics against artificial satellites dropping on their back gardens. In short, as long as men are willing to pay the appropriate premium, Lloyd's is willing to give them almost any kind of cover. But what does the word "appropriate" really mean in such cases? There is, in fact, no objective standard for judging many risks, since probability theory can only be applied to large-scale phenomena susceptible to statistical analysis. The weirder the risk, therefore, the more an insurance policy becomes a straight bet, much as mediaeval maritime policies were, in which the insurers said to themselves that if the captain can brave a voyage, the risks cannot be all that great.
Subjective and objective chances
We have just touched upon a crucial question which concerns
^ Sir William Beveridge: Report on Social Insurance and Allied Services, 17 Nov. 1942 — The Pillars of Security (London 1943), p. 73.
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not only insurance but science also, and particularly theoretical physics in which statistical phenomena play an ever-increasing role: where is the dividing line between subjective and objective probability estimates, where does mere speculation end and calculable probability begin?
Philosophers who have delved into this question and who have given all sorts of answers to it, fall into two groups: those who contend that the future is based on too many chance factors to be assessed in anything but degrees of pro- bability, and those who maintain that, since all nature (and all human actions ) follow fixed laws, the future is in principle, fully predictable, and that so-called probability or chance is only the measure of our own subjective ignorance.^ This attitude which was characteristic of 18th century rationalism and of 19th century natural philosophy, continues to hold sway to this day, particularly through the influence of the great English economist John Maynard Keynes. According to Keynes "objective chance" is always derived from "subjective chance", for whenever we attribute a coincidence to objective chance we simply state our ignorance of the law of connection between the two.^
However, when we climb down from the pedestal of philo- sophy, we find that the distinction between subjective and objective chance is not as great as philosophers would have us believe. Thus no one in his right senses will bother to say that the objective chance of next summer being warmer than last winter, is merely a reflection of the degree of our subjective ignorance. Predictions of this kind are admittedly based on past (subjective) observations, but that does not in the least affect their practical (objective) validity.
It is on such observations that most of what we call prob- ability is based. Probability statements are merely projections of the past into the future, on the assumption that the causes — no matter whether they are known precisely or not — will remain the same and will continue to have the same effects. Nor are quantitive predictions, characterised by degrees of probability, any more subjective than purely qualitative predictions. Thus, the statement that the probable deaths from heart and vascular
^ Henri Poincar^: Calcul des prohahilitSs (2nd ed. Paris 1912), p. 2. ^ John Maynard Keynes: A Treatise of Probability (London 1921), p. 288.
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diseases will account for 25 % of all deaths in the ensuing year, is no less objective than the statement that these diseases will be frequent causes of deaths. The only difference is that the quanti- tative statement makes the additional assumption that deaths from other causes will also remain stable.
Probability only becomes truly subjective in the absence of sufficient empirical evidence. Thus whenever scientists, econ- omists, or politicians have made generalisations from inadequate data, they have been subjective — and almost invariably wrong. The fault lies not with probability theory, but with those who forget that probability theory applies to aggregates and to aggregates alone.
Predicting the improbable
Unfortunately language itself lets us down in this respect, for when we say an event is probable, we assume that it is likely to happen, and never realise that the probability of its occurrence may be anything from 0 to 1 .
Mediaeval philosophy took over from Aristotle the distinc- tion of probability into four fixed categories, the highest of which was complete certainty, and the lowest, in which assertion and negation were equally balanced, was doubt [dubitatio, aporia). The second degree was suspicion {suspicio, hypokpsis), in which the assertion just outweighed the negation, the third was belief {opinio, doxa), in which the assertion seemed much more probable than the negation but was not completely proven. We, too, could do with a similar non-mathematical classification of degrees of probability.
Another thing that confuses the layman is the absence of any connection between aggregate predictions and their conse- quences for the individual. If health statistics show that 25 out of every 100 people die of heart or vascular diseases, or that the average life-expectancy in the United States is 77 years, this does not tell the individual anything about the possible causes of his own death or about his own life expectancy. Men being what they are, they are far more concerned with their own fate than with that of mankind as a whole, and are inclined to apply statistical predictions to themselves and to act accordingly.
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If the predictions fail to oblige, the whole of probability theory is dismissed as worthless, even though it has fulfilled its own promise: to forecast aggregate phenomena.
Wherever such aggregate predictions have been applied by governments or large institutions, they have shown their extreme usefulness, and have helped to turn mere chance into serious prognosis. Only when statistics are put to uses for which they were never intended, do they prove to be a useless — and sometimes a dangerous — ^toy.