[LADY LUCK, like a tossed coin, has two aspects; one is based on personal, applied superstition, the other derives from a scientific study of probabilities. In this article Dean Clark, of Queen’s University, Kingston, Ontario, discusses both phases, but, like a good scientist, devotes most of his time to the theory of probabilities. As the result of an extraordinarily patient and careful series of experiments he demonstrates how closely empirical findings approach the formulae of pure mathematics. What are the odds against a bridge hand of thirteen spades? — THE EDITORS]

## I

To believe that good luck follows some people and that bad luck follows others is pure superstition, which is merely an irrational approach to a situation, as opposed to a scientific approach. A gambler usually believes that if he is unlucky for a time his luck is maturing, and presently it must change. He believes that if several bets have gone against him he must win presently. Most people hold a similar view perhaps unconsciously, and, if asked to bet on the turn of a coin after, say, five heads have come up, would be willing to give odds against a sixth head. They believe in the maturity of chance. When the game is going badly, a bridge player may be seen to get up from his chair, walk around it three t imes, and sit down again, all to change his luck. A poker player may for the same purpose ask for a new ‘deck.’ This may not be entirely irrational.

Roulette is played by systems by many of its devotees. Some of these are highly ingenious, but in the long run produce no better results than random play. In the long run, the ‘house’ wins. Some people play only when the stars are propitious.

Then there are people who have great faith in their own luck. They may even convince others that they are in some manner possessed of special gifts. They are usually optimists. If a group of people decide to play a slot machine the lucky man may be selected to operate the machine. If he fails to win they say, ‘Oh well! He can’t always win,’ and the whole affair is forgotten. If he wins, then their belief in his luck is substantiated.

Superstitions are associated with sports of many kinds. Baseball players and followers of the race track are proverbially superstitious and resort to absurd practices to bring good luck. If superstitions were confined to such people, the subject would have less interest. But who is free altogether from superstition in some form? Queer beliefs have come down to us from antiquity and have been kept alive by writers of all times. These beliefs are nearly ineradicable — at least they will be with us for many years to come. A black cat crossing one’s path at the beginning of a journey is supposed to be a very bad sign. The writer had the amusing experience of having three black cats cross his road during the first few hours of a motor trip. Nothing untoward happened; tires remained intact, and gas and oil did all that was expected of them. To walk under a ladder is supposed to be very unlucky — particularly if someone is at work on the ladder with heavy tools or with paint. Astrology and Numerology have their devotees in large numbers. The number 13 is so ominous that some hotels omit room 13, and indeed the thirteenth floor may be missing. Sailors have a strong feeling against sailing on Friday. And Friday the thirteenth is supposed to be very bad. For most of us it is exactly like any other day.

Many people are very unhappy if they find that there are thirteen people in a party. Sometimes, indeed, either someone leaves or a fourteenth is brought in to allay fears. It is supposed that if thirteen are present one of them must die in the near future. How many people died last year who had sat down at table with two, three, or any number except thirteen? The breaking of a mirror is a very bad omen. It is said that Napoleon once broke a mirror which was hanging above a portrait of his wife Josephine. He was so distressed that he sent a courier to find out if all was well. There was no occasion for alarm.

Farmers have great faith in the good and bad effects of the moon on planting, rainfall, and so forth. Storms or clearing weather are associated with the changes of the moon. There are really no changes; the moon passes continuously through the different phases which, according to the calendar, occur every seven days. So any weather change can be not farther than four days from the so-called change of the moon. A statistical study of the weather records shows that there is no relationship whatever between the phases of the moon and weather. Railroad men have a firm belief that if there is an accident on the road there will be two more in the immediate future. If there are not, the first is forgotten. If one or two more take place, then they say, ‘I told you so.’ ‘Misfortunes never come singly’ is another familiar superstition. They do come singly, as we all know. If more than one occurs, then we remember the adage.

Disraeli is said to have made a practice of writing the names of people who had injured him on slips of paper, which were then put away in a drawer. When the men whose names were thus filed disappeared from the scene, the slips were removed. There is a slight odor of superstition here, and we are reminded of the practice of some savages of st icking pins in the effigy of an enemy to injure him.

Almost all of these superstitions, and particularly those that have to do with affecting one’s luck, are egotistical. The holder thinks himself of very great importance in the vast mechanism of Nature. Each of us is the centre of a universe, and therefore each is of especial importance (to himself). It is inconceivable that the universe, a vast machine of which the earth is a small part, — but still a part, — could have its operations interrupted or diverted so as to make a single individual suffer or to benefit him. How many of us at one time or another have not shared the feeling of Margaret Halsey in her rather caustic little book, With Malice Toward Some? She was passing through a Norwegian fiord in a small boat which had to sail under some overhanging rocks. She imagines the rocks saying to each other: ‘Here comes Peg Halsey after all these years.’ Most of us have an instinctive feeling that the rocks may fall just as we pass under them in spite of the fact that they have been hanging there for thousands of years. Anyone else would be perfectly safe.

Birds and animals foretell weather, rheumatic joints give warning of approaching storms, only because they respond to present conditions which are the preliminaries of the storm. A storm is much more than rain and wind. It begins long before these obvious phenomena with falling barometer, rising temperature and humidity. These preliminaries begin almost imperceptibly and gradually pass into what we commonly call a storm. Different people begin to respond at different times according to their powers of observation and memory of past conditions. Earthquakes are supposed to be associated with ‘earthquake weather’ and comets with wars, but statistics show otherwise. If sensible people would establish a habit of recording in parallel columns (even mentally) signs or omens and the actual course of events supposed to be related to them, many superstitions would disappear, and with them many of the absurd fears so much a part of human lives. We ‘count the hits and forget the misses.’

## II

With human beings there seem to be two kinds of luck. We may give two explanations for a run of favorable or of unfavorable events. The intelligent, forward-looking person will see the trend of past events and by foresight avoid what would catch the unwary and may so shape his course as to seem to make future events conspire for his benefit. In reality he merely seizes his opportunities, and events are favorable to him. The unlucky person is unable to foresee the future and by blundering may do exactly the thing that wall bring disaster. There is a very real connection between ability and good fortune, between lack of ability and ill luck. Goethe makes Mephistopheles say: —

Wie sick Verdienst und Gluck verketten,
Das fallt den Toren niemals ein.

This is not the whole story, however. There are roughly two billion people in the world, and the population has been very large for a long time. We have here the large numbers where the application of statistical methods is so fruitful. We may all be regarded as taking part in a great probability experiment. Each will have his ups and downs of a minor character, and it may be agreed that there is a most probable kind of life although there are no measurements which may be made to express this life in precise terms. It is what we think of as the normal life and is the life which the large percentage of the population enjoy — or endure. There will be departures from this most probable life, and these departures, though small, will involve a large number of people. Greater departures will occur for smaller numbers of people, until we reach the point where very few people are concerned. These will rise to great heights or sink to very low depths. It is certain that some people will suffer great reverses in no way due to their lack of ability, and others will be elevated by circumstances with which they have but little to do.

This departure from the normal is our second kind of luck and may have no connection whatever with the victim’s mistakes or wisdom. From a study of a sample million people properly chosen to be representative, humanity as a whole may be catalogued with some accuracy, provided, of course, there is no cataclysm or other theory-shattering series of events.

If life insurance companies can construct mortality tables from a study of sample lives, we should be able to construct tables in many other fields. Statistics tell us the number of deaths per thousand men of a certain age and of insurable condition of health. We could also find, by statistical data accumulated for a sufficiently large sample, how many broken legs or arms will require treatment in the next month throughout the United States and Canada. We could find out how many will lose the sight of an eye or their hearing, or how many appendectomies will be performed. From data accumulated over a period of years it is possible to predict with fair accuracy how many students will fail in the final examinations of any year, how many will graduate, and how many will graduate with honors.

The luck of human beings involves many factors. Where only inanimate objects are involved, the whole matter is much simpler. At a recent popular lecture before a group of engineering students, each member of the audience was asked to take a coin from his pocket and report whether it showed a head or a tail. There were thirty-three heads and thirty-five tails. The result was about what was expected, — a little better, in fact, — but it gave a striking verification of an elementary principle in probability. A coin may lie in only two ways and, if it is perfect, neither face should show any predilection. So about one half of the coins in the experiment should show one face, and one half the other. We say that the probability of either face is 1 to 2.

If two coins are taken at random or are tossed into the air, in the usual manner, with each face of one either of the two faces of the other may appear. Thus there are four ways in which the two coins may fall, or, in the jargon of probability, there are four complexions. One of these is two heads, another is two tails, and then there are two in which a head and a tail come up together. The probability of two heads or two tails is therefore 1 to 4. In general, probability may be defined as the ratio of the favorable cases, or cases of some specified kind, to the total number of cases.

If three coins are tossed, with each of the four complexions of the two coins there may appear a head or a tail for the third; thus there are eight complexions. Only one of these is the case of three heads, so the probability of the three heads is 1 to 8. It is immaterial whether three coins are tossed simultaneously, or one coin three times, provided that the throws are entirely independent of each other — that is, the fall of each is absolutely uncontrolled.

If a coin is tossed very many times it would be expected that approximately one half of the trials will show heads and one half tails. The results would not be expected to be exactly alike, but with a large total number of trials the difference between heads and tails ordinarily will be small compared to the total.

The ordinary six-faced die, if fairly made and fairly cast, may fall so that any one of the six faces appears. The probability of any particular face, say a six, is therefore 1 to 6. If two dice are cast, with any face of the first die any one of the six faces of the other may appear, so that for the two there are 36 complexions. For three dice there are 216 complexions. So the probability of any particular doublet with two dice is 1 to 36, of any particular triplet with three dice is 1 to 216. With a large number of dice, then, the probability of all showing the same face is very small. Again, if a single die is cast a very large number of times, we should expect to find the six faces showing about an equal number of times.

In an ordinary pack of cards there are four suits of thirteen cards each. Consequently the number of different bridge hands which may be dealt is the number of combinations of fifty-two things taken thirteen at a time. By elementary algebra this is 635,013,559,600. The probability of holding any particular hand is then 635,013,559,600, a very small number. Thus perfect bridge hands cannot be expected very often.

There are some on record, however — which may mean that an excessively large number of bridge hands is being dealt every day.

The probability of holding any five cards named in advance in a poker game is again very small, but not as small as for the named bridge hand. There are 2,598,960 possible combinations of five cards or poker hands, so the probability of being dealt a royal flush is small, but such hands have occurred. The probabilities for the different hands are given in any book of games. Bridge and poker, as games, involve much more than a knowledge of probability, but such a knowledge would improve the chance for success of any thoughtful player.

These illustrations from games of chance are given merely to make clear the meaning of probability. Indeed, the subject grew out of certain problems put to mathematicians by men who spent a great deal of time in ‘gaming’ or ‘play’ — polite names for an aristocratic kind of gambling.

## III

The probability defined above as the ratio of the number of cases of a particular kind to the total number of cases, when treated simply as an a priori calculation, may be called the theoretical or mathematical probability. If, on the other hand, experiments are carried out, the number of favorable cases being counted and divided by the total number of trials, the result is the empirical or experimental probability. If the mathematical probability is calculated, and then by trial the empirical probability is found, the two results may not be exactly the same. If, however, the number of total cases is made very large, the two results, theoretical and empirical, will be found generally to agree very closely indeed. By the term ‘agree’ is meant that the discrepancy between the two results is small compared with the number of events in the experiment. This point is important. Tests with coins, dice, and cards have been made so many times that there is no doubt whatever that the agreement generally is good.

In all experiments, however, the experimenter must be prepared for unexpected results; indeed, as will be seen later, the theory of probability demands departures from what may be called normal happenings. For example, if one were to take five coins and toss them at random, he might find that in spite of the fact that the probability of the occurrence is only 1 to 32, all show the same face. It might not happen in very many throws, but conceivably it might happen on the first trial. A large number of trials is needed, therefore, to find the value of the empirical probability, and conclusions must not be drawn from meagre information. Further, it cannot be emphasized too strongly or too often that absolute randomness must obtain throughout any experiment. There must be no direction given to the coin, die, or card.

In the opinion of the writer, there is a fallacy that has come into writings on probability by those who should know better. It is this: where the probability of an event is very small we must wait a long time for this event to occur, and if we do wait it will certainly occur. Neither idea is correct. The occurrence or non-occurrence of the awaited event is not a matter of time. If it may occur at all, it may occur early or, of course, very late. What has happened in a random tossing of a coin has no influence whatever on what may happen later. A coin has no memory or will, and even if it had either it would be powerless to influence the manner of fall. The improbable, as just pointed out, may happen early and the probable late. If a coin is tossed and a head has appeared five times in succession, not many people would be willing to risk a wager that the next toss will show a head. The probability that the next throw will show a head is, however, 1 to 2, just as before any trial. Another head is as likely as a tail. Many people find this hard to swallow.

A little experience with probability shows that the improbable does happen and the probable need not happen. Everyone can remember the occurrence of events in his life which, before they occur, would have been declared to be highly improbable. Indeed, our lives have much of the improbable in them.

The methods of probability which grew out of games of chance have been extended to a great variety of different fields. Almost every large commercial enterprise, consciously or unconsciously, employs the laws of probability. Industries manufacturing large numbers of small articles cannot inspect every one, but by selecting samples at random can predict fairly accurately how much allowance must be made for defective articles. Here they assume that in a large lot the defective ones will be distributed at random, so that, if a sufficiently large sample is taken, the evidence obtained may be used for appraising the entire product. Sampling to determine the possibilities of a mine must be properly done or the conclusions based on the assays may be far from the truth.

Insurance companies do almost exactly the same thing with human beings instead of inanimate objects. By taking sufficiently large samples and observing the length of life of the individuals in the samples, they build up mortality tables. Then, on the basis of what has happened to one group, they can predict what will happen to another group of the same kind. While the man who takes out his little book to tell you what an insurance policy will cost (if you allow him to get so far) has no information about your future life, he can tell you your expectation of life. All these data about life prospects have been accumulated by a process of sampling, which assumes that one sufficiently large selected group will behave like another large group of the same kind. So that, if an insurance company has a large enough group of clients, the directors may rest assured that, unless conditions of life change, their profits from policyholders are secure.

The accumulations of data regarding past events are known as statistics, and statistical calculations using these data are based on the probability that a proper sample gives a true picture of the entire situation and that other events, or whatever the data represent, will run true to form. For example, coffin manufacturers as well as clothing manufacturers are interested in the distribution of heights of the population they serve. Transportation companies must know in advance something about the probable demands on their equipment, and therefore collect the necessary data. H. C. Levinson in his interesting book, Your Chance to Win, says that a firm in New York employing several hundred delivery men has accumulated data regarding the number of times its employees have been bitten by dogs, and by what kind of dogs. They found the Airedale the most intelligent, as ‘he bites after the parcels are delivered.’ Telephone companies have elaborate tables by which the demands on their lines may be predicted. Thus statistics are gathered by all sorts of enterprises so that the future trends may be predicted. It is always the probable future only which is indicated, for unexpected conditions may arise which upset the calculations. The tables, however, are indispensable in modern business.

A properly aimed gun with good ammunition is surprisingly accurate. For example, the quick-firing 18-pounder will place its shots so closely that at 2000 yards one half of them fall with a deviation of less than 1½ feet. How these shots will be distributed on the target is entirely a matter of chance. Some will lie close to the point of aim, some farther away, and the distribution may be calculated with some accuracy — what proportion will fall, say, inside a six-inch circle, and then what proportion between this circle and another circle one foot in diameter. Slight variations in the ammunition, the weight, and the shape of the projectiles all have their small effects. Unless the gun is moved between shots, the shots will fall within a very small area, so the feeling that shells never enter the same shell-hole may be only a superstition. In Marry at’s Peter Simple, one of the midshipmen on the Diomede informs Peter ‘that he always made it a rule, upon the first cannon-ball coming through the ship’s side, to put his head into the hole which it had made, as by a calculation made by Professor Inman, the odds were 32,647, and some decimals to boot, that another shot would not come in at the same hole.’ Of course the probability that the second or any other shot will st rike the same hole is very small, but so is the probability that a shot will strike any designated place.

To determine whether or not a roulette wheel is true, a great many trials are made in order to find out if certain numbers come up more often than they should. This is a statistical test. A similar method is possible to determine the fairness of dice. The dice in a set of 24 were suspect, and to determine whether or not they were true 36 casts of all at a time were made. If the dice were true, each face should appear approximately 144 times. Fives appeared 184 times and fours only 87 times. There was obviously something more than chance operating here, and the dice were certainly not fair. Another set of 12 was tried with 250 casts, with the result that sixes appeared 568 times when they should appear only 500 times. These dice, too, were probably unfair. They were well made, however, and, as with most modern dice, the markings were put on by means of a little coloring matter in small holes. These holes apparently displace the centre of gravity enough to affect the result of a cast, for certain critical cases. This raises a question as to the fairness of dice in general.

Dr. Rhine of Duke University has attracted considerable attention by his claim to the discovery of what he calls extrasensory perception. It will be recalled that he uses a pack of twentyfive cards made up of five sets of five cards each. The five varieties have simple designs or patterns easy of identification. The cards are shuffled and placed face-down in a pile on the table; then the subject is asked to name the cards. In some experiments the top card is removed after each trial, in others the subject is expected to name the cards in order down through the pack.

The probability of a correct random guess is 1 to 5. If the subject knows that one or more trials have been successes or failures, the probability is modified. The results of the trials are recorded and form a statistical table from which Dr. Rhine concludes that the number of correct results is greater than random guesses could possibly yield.

There are many writers who have observed the work and believe that Dr. Rhine has made his case. Others equally competent believe that the results are merely in accord with the probability theory. Continued experiment should reveal whether or not we have here a genuine case of extrasensory perception. The accumulation of data by different observers would give the statistical material on which a final judgment could be based. The subject is of sufficient importance to justify a complete investigation.

## IV

Many years ago a French mathematician, Buffon, invented a problem since known by his name. If a series of equidistant parallel straight lines are ruled on a plane horizontal surface, and a short rod or needle, whose length is less than the distance between the lines, is thrown on to the plane at random, what is the probability that the rod will come to rest lying across one of the lines? If the distance between the lines is a and the length of the needle is c, the theoretical probability is easily found to be P = 2a/πc. The appearance of the number 7r gives great interest to the problem.1 For if a large number of trials are made we may calculate the empirical proba-

At the Century of Progress World’s Fair at Chicago a few years ago, an automatic device for carrying out the experiment and for recording the results was shown. This device made possible the accumulation of a large amount of data, so that a fairly accurate result for π could be obtained. There is, however, a limit beyond which it is not profitable to carry such experiments. It will be seen that the calculation of P involves knowledge of the distance between the lines and the length of the rod. Neither of these distances can be measured with sufficient accuracy to warrant the securing of, say, 1,000,000 observations. As soon as the number of observations reaches the point where the empirical value of the probability can be found with greater accuracy than the theoretical value, it is useless to go further

An understanding of the laws of probability is so important in modern physics that the writer felt that an experiment which would bring these laws home to students would be of great value, so the work was begun about ten years ago, and has been continued to the point where some important results have appeared.

The ‘Queen’s experiment’ consists in dropping steel balls at random on to a horizontal steel plate pierced with holes bored in a regular array, to see whether or not they pass through a hole. These balls, the ordinary steel balls used in bearings, are extraordinarily accurate in both shape and size. The holes in the plate are bored with specially made tools and are very uniform in size, and they are so accurately spaced that all parts of the plate are alike. Of course the holes are larger than the balls — indeed, more than twice as large. They are staggered so that they form a network in triangular arrays resembling the top of a honeycomb. It is easy to calculate the probability that a ball falling vertically on to the plate will pass through a hole without contact with the plate. This theoretical probability involves merely the ratio of the free area to the total area of a properly chosen part of the plate.2 The value is 0.3555 ± .0005.

As in Buffon’s problem, there is a limit to the accuracy with which measurements may be made, hence the small uncertainty in the value.

With the assistance of students, members of the staff, and employees in the laboratory, 500,000 observations have been made — that is, this is the number of balls that have been dropped with recorded results. Of this number, 177,785 passed through holes without contact with the plate. The empirical probability is therefore 0.35557, which is very close to the theoretical value. The plate and dropping tube are supported on arms so that the plate may be moved through a distance of about six inches and the dropper about three inches. Both the plate and the dropper are moved at random after each ball is dropped, so that the half million shots are independent, random events.

It was to be expected that the two values of the probability would be very close together, since we have a very large number of events. That they were so close was gratifying but is not of great significance, as such verifications of theory are fairly common. Nor is the fact that π may be calculated from these results of great importance, although it adds greatly to the interest of the work.

If we substitute the empirical value of P in the formula, the value of π is 3.145. This result is about as good as may be expected, since the measurements of radii and distances are not perfect, as already explained. If, however, the record were studied, a place might be found where the value of P for that part of the record would yield a more accurate value of π. For example, the number of free passages of the balls in the first 275,000 observations is 97,697. The value of P for this run is 0.35525, and the resulting value of π is 3.14167. This method obviously is unfair, because in a probability experiment we have no right to make such a selection. Randomness is the essence of probability.

So in the experiment the probability that one ball will pass through a hole is 0.3555; of two in a row is (0.3555)2 or 0.1263; and of n in a row is (0.3555)“. It can be seen, then, that if n is a large number the probability of that number of successive passages through the holes becomes very small. In the experiment the number of events — that is, balls dropped — is very large, so the number of times that a series of any length may be expected to occur is the probability of the series multiplied by the total number of balls dropped. Thus in a series of 500,000 we may expect that there will be 63,215 cases where two balls in succession will pass through holes; 22,477 cases of three in succession; 7992 of four. There were actually 62,559, 24,482, and 8579 respectively for runs of two, three, and four. The agreement happens to be better for certain longer runs. For example, eights, nines, tens, and elevens should appear 128, 45, 16, and 6 times respectively. They actually appeared 146, 46, 16, and 8 times respectively. The probability that there will be thirteen successive passages is about one in a million.

According to theory, then, if a million balls are dropped, there should be one case where thirteen balls pass freely. As a matter of fact, a run of thirteen came in the nineteenth thousand. Again, a run of twelve might be expected once in 500,000 trials, but such a run was recorded in the forty-ninth thousand. These results are important, as they show very clearly that one does not always have to wait a long time for the improbable to happen. What may happen at all may happen at any time. All thousands are alike, and a run of twelve or a thirteen or an even longer run might have happened at any time, early or late. On the other hand, neither of these phenomena need have appeared at all.

The other point brought out in the experiment is the distribution of the numbers of free passages of the balls when the observations are grouped in series of definite length. The record is kept so that the results appear in groups of 100 trials each; thus for this number the point may be studied easily. The entire record was examined, and the numbers of times each number of passages in 100 trials occurred were tabulated. For example, the smallest number in 100 trials was 21 and the largest 54. Each of these numbers appeared only once. Then between 21 and 54 the number of times that 22, 23, 24, and so forth, appeared was recorded. Next,

NUMBER OF OCCURRENCES

10 200 300 400 500 600

points representing all of these numbers — 21, 22, 23, up to 54 — were laid off horizontally on graph paper. Then, vertically over these points, other points showing the number of times that each value appeared were laid out and a smooth curve drawn as nearly as possible through these points. This curve shows graphically the distribution of these numbers.

It will be noticed, as might be expected, that the largest numbers of passages in 100 trials are 35 and 30. This is because the probability of a passage is 0.3555. From 35 and 36 in either direction the numbers of times that each number appears decrease rapidly. The further one goes from the most probable value in 100, the smaller is the number, until the end of the experimental results is reached. This curve is a close approximation to the probability curve and is very important. Wherever statistical data are collected, such a curve may be drawn. In many cases it is exactly like the curve shown; in some the shape departs from the symmetrical form.

We are able, then, to see that in the human experiment with two billion cases the normal life may well be represented by the highest point on the curve, where the greatest number of people are involved. As we go away from the peak in either direction, we have the less probable life which concerns relatively fewer people. Very far from the middle of the curve is the highly improbable life which comes to a very few. On the one side is good fortune, with health, beauty, pleasant surroundings, recognition, while on the other are suffering, malformation, idiocy, obscurity, early extinction. Good or bad luck is not all a matter of intelligence or industry, but may be due in part to causes which may be entirely beyond control and perhaps unknown, but which are in accordance with statistical laws. No one would be so foolish as to attempt to apply them to any individual at any particular time.

In a universe governed by laws which are only partially understood and sometimes, no doubt, not known even approximately, where the simplest phenomena are very complex, as they are in the operations of human beings, it is hopeless to attempt as yet to bring all into an orderly, theoretical arrangement for study. We have seen that when the number of events becomes very large we may pursue statistical methods with great success; we may rely on probability.

1. π, of course, is the ratio of the circumference of a circle to the diameter, and appears in this problem since the probability sought involves the angular positions of the rod about its centre, π is 3.1416 to four places of decimals. — AUTHOR bility, substitute this for P, and calculate the value of ¶- from the formula. The experiment has been carried out by a number of observers, and the value of π is found to be very close to the true value.
2. Since the plate is made up of a series of triangles, the free area is the area of the parts of the circles enclosed in any triangle whose vertices lie at. the centres of the circles. These circles are really the circles whose common radius is the difference between the radii of the actual holes and the balls. The total area is the area of the triangle itself. The expression for the probability is p = π(R-r)/a2 3. A microphone attached to the plate makes contact between a ball and the plate evident by a fairly loud sound. — AUTHOR