University of Virginia Library

8. THE DOCTRINE OF PROBABILITIES APPLIED TO GAMBLING.

A DISTINCTION must be made between games of skill and games of chance. The former require application, attention, and a certain degree of ability to insure success in them; while the latter are devoid of all that is rational, and are equally within the reach of the highest and lowest capacity. To be successful in throwing the dice is one of the most fickle achievements of fickle fortune; and therefore the principal game played with them is very properly and emphatically called `Hazard.' It requires, indeed, some exertion of the mental powers, of memory, at least, and a turn for such diversions, to play well many games at cards. Nevertheless, it is often found that those who do so give no further proofs of superior memory and

judgment, whilst persons of superior memory and judgment not unfrequently fail egregiously at the card-table.

The gamester of skill, in games of skill, may at first sight seem to have more advantage than the gamester of chance, in games of chance; and while cards are played merely as an amusement, there is no doubt that a recreation is more rational when it requires some degree of skill than one, like dice, totally devoid of all meaning whatever. But when the pleasure becomes a business, and a matter of mere gain, there is more innocence, perhaps, in a perfect equality of antagonists — which games of chance, fairly played, always secure — than where one party is likely to be an overmatch for the other by his superior knowledge or ability.

Nevertheless, even games of chance may be artfully managed; and the most apparently casual throw of the dice be made subservient to the purposes of chicanery and fraud, as will be shown in the sequel.

In the matter of skill and chance the nature of cards is mixed, — most games having in them both elements of interest, — since the success of the player must depend as much on the chance of the

`deal' as on his skill in playing the game. But even the chance of the deal is liable to be perverted by all the tricks of shuffling and cutting — not to mention how the honourable player may be deceived in a thousand ways by the craft of the sharper, during the playing, of the cards themselves; consequently professed gamblers of all denominations, whether their games be of apparent skill or mere chance, may be confounded together or considered in the same category, as being equally meritorious and equally infamous.

Under the name of the Doctrine of Chances or Probabilities, a very learned science, — much in vogue when lotteries were prevalent, — has been applied to gambling purposes; and in spite of the obvious abstruseness of the science, it is not impossible to give the general reader an idea of its pro-cesses and conclusions.

The probability of an event is greater or less according to the number of chances by which it may happen, compared with the whole number of chances by which it may either happen or fail. Wherefore, if we constitute a fraction whereof the numerator be the number of chances whereby an event may happen, and the denominator the number

of all the chances whereby it may either happen or fail, that fraction will be a proper designation of the probability of happening. Thus, if an event has 3 chances to happen, and 2 to fail, then the fraction 3/5 will fairly represent the probability of its happening, and may be taken to be the measure of it.

The same may be said of the probability of failing, which will likewise be measured by a fraction whose numerator is the number of chances whereby it may fail, and the denominator the whole number of chances both for its happening and failing; thus the probability of the failing of that event which has 2 chances to fail and 3 to happen will be measured by the fraction 2/5.

The fractions which represent the probabilities of happening and failing, being added together, their sum will always be equal to unity, since the sum of their numerators will be equal to their common denominator. Now, it being a certainty that an event will either happen or fail, it follows that certainty, which may be conceived under the notion of an infinitely great degree of probability, is fitly represented by unity.

These things will be easily apprehended if it

be considered that the word probability includes a double idea; first, of the number of chances whereby an event may happen; secondly, of the number of chances whereby it may either happen or fail. If I say that I have three chances to win any sum of money, it is impossible from the bare assertion to judge whether I am likely to obtain it; but if I add that the number of chances either to obtain it or miss it, is five in all, from this will ensue a comparison between the chances that are for and against me, whereby a true judgment will be formed of my probability of success; whence it necessarily follows that it is the comparative magnitude of the number of chances to happen, in respect of the whole number of chances either to happen or to fail, which is the true measure of probability.

To find the probability of throwing an ace in two throws with a single die. The probability of throwing an ace the first time is 1/6; whereof 1/6 is the first part of the probability required. If the ace be missed the first time, still it may be thrown on the second; but the probability of missing it the first time is 5/6, and the probability of throwing it the second time is 1/6; therefore the probability of miss

ing it the first time and throwing it the second, is 5/6 x 1/6 =5/36 and this is the second part of the probability required, and therefore the probability required is in all 1/6 + 5/36 =11/36.

To this case is analogous a question commonly proposed about throwing with two dice either six or seven in two throws, which will be easily solved, provided it be known that seven has 6 chances to come up, and six 5 chances, and that the whole number of chances in two dice is 36; for the number of chances for throwing six or seven 11, it follows that the probability of throwing either chance the first time is 11/36, but if both are missed the first time, still either may be thrown the second time; but the probability of missing both the first time is 25/36, and the probability of throwing either of them on the second is 11/36; therefore the probability of missing both of them the first time, and throwing either of them the second time, is 25/36 x 11/36 =275/1296, and therefore the probability required is 11/36 + 275/1296 =671/1296, and the probability of the contrary is 625/1296.

Among the many mistakes that are committed about chances, one of the most common and least suspected was that which related to lotteries. Thus,

supposing a lottery wherein the proportion of the blanks to the prizes was as five to one, it was very natural to conclude that, therefore, five tickets were requisite for the chance of a prize; and yet it is demonstrable that four tickets were more than sufficient for that purpose. In like manner, supposing a lottery in which the proportion of the blanks to the prize is as thirty-nine to one (as was the lottery of 1710), it may be proved that in twenty-eight tickets a prize is as likely to be taken as not, which, though it may contradict the common notions, is nevertheless grounded upon infallible demonstrations. When the Play of the Royal Oak was in use, some persons who lost considerably by it, had their losses chiefly occasioned by an argument of which they could not perceive the fallacy. The odds against any particular point of the ball were one and thirty to one, which entitled the adventurers, in case they were winners, to have thirty-two stakes returned, including their own; instead of which, as they had but twenty-eight, it was very plain that, on the single account of the disadvantage of the play, they lost one-eighth part of all the money played for. But the master of the ball maintained that they had no reason to complain, since he would undertake that any particular point of the ball should come up in two and twenty throws; of this he would offer to lay a wager, and actually laid it when required. The seeming contradiction between the odds of one and thirty to one, and twenty-two throws for any chance to come up, so perplexed the adventurers that they began to think the advantage was on their side, and so they went on playing and continued to lose.

The doctrine of chances tends to explode the long-standing superstition that there is in play such a thing as luck, good or bad. If by saying that a man has good luck, nothing more were meant than that he has been generally a gainer at play, the expression might be allowed as very proper in a short way of speaking; but if the word `good luck' be understood to signify a certain predominant quality, so inherent in a man that he must win whenever he plays, or at least win oftener than lose, it may be denied that there is any such thing in nature. The asserters of luck maintain that sometimes they have been very lucky, and at other times they have had a prodigious run of bad luck against them, which whilst it continued obliged

them to be very cautious in engaging with the fortunate. They asked how they could lose fifteen games running if bad luck had not prevailed strangely against them. But it is quite certain that although the odds against losing so many times together be very great, namely, 32,767 to 1, — yet the possibility of it is not destroyed by the greatness of the odds, there being one chance in 32,768 that it may so happen; therefore it follows that the succession of lost games was still possible, without the intervention of bad luck. The accident of losing fifteen games is no more to be imputed to bad luck than the winning, with one single ticket, the highest prize in a lottery of 32,768 tickets is to be imputed to good luck, since the chances in both cases are perfectly equal. But if it be said that luck has been concerned in the latter case, the answer will be easy; for let us suppose luck not existing, or at least let us suppose its influence to be suspended, — yet the highest prize must fall into some hand or other, not as luck (for, by the hypo-thesis, that has been laid aside), but from the mere necessity of its falling somewhere.

Among the many curious results of these inquiries according to the doctrine of chances, is

the prodigious advantage which the repetition of odds will amount to. Thus, `supposing I play with an adversary who allows me the odds of 43 to 40, and agrees with me to play till 100 stakes are won or lost on either side, on condition that I give him an equivalent for the gain I am entitled to by the advantage of my odds; — the question is, what I am to give him, supposing we play at a guinea a stake? The answer is 99 guineas and above 18 shillings,[52] which will seem almost incredible, considering the smallness of the odds — 43 to 40. Now let the odds be in any proportion, and let the number of stakes played for be never so great, yet one general conclusion will include all the possible cases, and the application of it to numbers may be worked out in less than a minute's time.'[53] [52] The guinea was worth 21s. 6d. when the work quoted was written. [53] De Moivre, Doctrine of Chances.

The possible combinations of cards in a hand as dealt out by chance are truly wonderful. It has been established by calculation that a player at Whist may hold above 635 thousand millions of various hands! So that, continually varied, at 50 deals per evening, for 313 evenings, or 15,650

hands per annum, he might be above 40 millions of years before he would have the same hand again!

The chance is equal, in dealing cards, that every hand will have seven trumps in two deals, or seven trumps between two partners, and also four court cards in every deal. It is also certain on an average of hands, that nothing can be more super-stitious and absurd than the prevailing notions about luck or ill-luck. Four persons, constantly playing at Whist during a long voyage, were frequently winners and losers to a large amount, but as frequently at `quits;' and at the end of the voyage, after the last game, one of them was minus only one franc!

The chance of having a particular card out of 13 is 13/52, or 1 to 4, and the chance of holding any two cards is 1/4 of 1/4 or 1/16. The chances of a game are generally inversely as the number got by each, or as the number to be got to complete each game.

The chances against holding seven trumps are 160 to 1; against six, it is 26 to 1; against five, 6 to 1; and against four nearly 2 to 1. It is 8 to 1 against holding any two particular cards.

Similar calculations have been made respecting the probabilities with dice. There are 36 chances upon two dice.

It is an even chance that you throw 8. It is 35 to 1 against throwing any particular doublets, and 6 to 1 against any doublets at all. It is 17 to 1 against throwing any two desired numbers. It is 4 to 9 against throwing a single number with either of the dice, so as to hit a blot and enter. Against hitting with the amount of two dice, the chances against 7, 8, and 9 are 5 to 1; against 10 are 11 to 1; against 11 are 17 to 1; and against sixes, 35 to 1.

The probabilities of throwing required totals with two dice, depend on the number of ways in which the totals can be made up by the dice; — 2, 3, 11, or 12 can only be made up one way each, and therefore the chance is but 1/36; — 4, 5, 9, 10 may be made up two ways, or 1/8; — 6, 7, 8 three ways, or 1/12. The chance of doublets is 1/36, the chance of particular doublets 1/216.

The method was largely applied to lotteries, cock-fighting, and horse-racing. It may be asked how it is possible to calculate the odds in horse-racing, when perhaps the jockeys in a great mea

sure know before they start which is to win?

In answer to this a question may be proposed: — Suppose I toss up a half-penny, and you are to guess whether it will be head or tail — must it not be allowed that you have an equal chance to win as to lose? Or, if I hide a half-penny under a hat, and I know what it is, have you not as good a chance to guess right, as if it were tossed up? My knowing it to be head can be no hindrance to you, as long as you have liberty of choosing either head or tail. In spite of this reasoning, there are people who build so much upon their own opinion, that should their favourite horse happen to be beaten, they will have it to be owing to some fraud. The following fact is mentioned as a `paradox.'

It happened at Malden, in Essex, in the year 1738, that three horses (and no more than three) started for a £10 plate, and they were all three distanced the first heat, according to the common rules in horse-racing, without any quibble or equivocation; and the following was the solution: — The first horse ran on the inside of the post; the second wanted weight; and the third fell and broke a fore-leg.[54] [54] Cheany's Horse-racing Book.

In horse-racing the expectation of an event is considered as the present value, or worth, of whatsoever sum or thing is depending on the happening of that event. Therefore if the expectation on an event be divided by the value of the thing expected, on the happening of that event, the quotient will be the probability of happening.

Example I. Suppose two horses, A and B, to start for £50, and there are even bets on both sides; it is evident that the present value or worth of each of their expectations will be £25, and the probabilities 25/50 or 1/2. For, if they had agreed to divide the prize between them, according as the bets should be at the time of their starting, they would each of them be entitled to £25; but if A had been thought so much superior to B that the bets had been 3 to 2 in his favour, then the real value of A's expectation would have been £30, and that of B's only £20, and their several probabilities 30/50 and 20/50.

Example II. Let us suppose three horses to start for a sweepstake, namely, A, B, and C, and that the odds are 8 to 6 A against B, and 6 to 4 B against C — what are the odds — A against C, and the field against A? Answer: — 2 to 1 A against

C, and 10 to 8, or 5 to 4 the field against A. For

A's expectation is 8

B's expectation is 6

C's expectation is 4

— —

18 But if the bets had been 7 to 4 A against B; and even money B against C, then the odds would have been 8 to 7 the field against A, as shown in the following scheme: —

7 A

4 B

4 C

— —

15

But as this is the basis upon which all the rest depends, another example or two may be required to make it as plain as possible.

Example III. Suppose the same three as before, and the common bets 7 to 4 A against B; 21 to 20 (or `gold to silver') B against C; we must state it thus: — 7 guineas to 4 A against B; and 4 guineas to £4, B against C; which being reduced into shillings, the scheme will stand as follows: —

147 A's expectation.

81 B's expectation.

80 C's expectation.

— —

311

By which it will be 164 to 147 the field against A, (something more than 39 to 35). Now, if we compare this with the last example, we may conclude it to be right; for if it had been 40 to 35, then it would have been 8 to 7, exactly as in the last example. But, as some persons may be at a loss to know why the numbers 39 and 35 are selected, it is requisite to show the same by means of the Sliding Rule. Set 164 upon the line A to 147 upon the slider B, and then look along till you see two whole numbers which stand exactly one against the other (or as near as you can come), which, in this case, you find to be 39 on A, standing against 35 on the slider B (very nearly). But as 164/311 and 147/311 are in the lowest terms, there are no less numbers, in the same proportion, as 164 to 147, — 39 and 35 being the nearest, but not quite exact.

Example IV. There are four horses to start for a sweepstake, namely, A, B, C, D, and they are supposed to be as equally matched as possible. Now, Mr Sly has laid 10 guineas A against C, and also 10 guineas A against D. Likewise Mr Rider has laid 10 guineas A against C, and also 10 guineas B against D. After which Mr Dice

laid Mr Sly 10 guineas to 4 that he will not win both his bets. Secondly, he laid Mr Rider 10 guineas to 4 that he will not win both his bets.

Now, we wish to know what Mr Dice's advantage or disadvantage is, in laying these two last-mentioned wagers.

First, the probability of Mr Sly's winning both his bets is 1/3 of 14 guineas; and Mr Dice's expectation is 2/3 of 14 guineas, or £9 16s., which being deducted from his own stake (10 guineas), there remains 14s., which is his disadvantage in that bet.

Secondly, Mr Rider's expectation of winning his two bets is 1/4, and, therefore, Mr Dice's expectation of the 14 guineas, is 3/4, or £11 0s. 6d., from which deduct 10 guineas (his own stake), and there remains 10s. 6d., his advantage in this bet, — which being deducted from 14s. (his disadvantage in the other), there remains 3s 6d., his disadvantage in paying both these bets.

These examples may suffice to show the working of the system; regular tables exist adapted to all cases; and there can be no doubt that those who have realized large fortunes by horse-racing managed to do so by uniformly acting on some such principles, as well as by availing themselves of

such `valuable information' as may be secured, before events come off, by those who make horse-racing their business.

The same system was applied, and with still greater precision, to Cock-fighting, to Lotteries, Raffles, Backgammon, Cribbage, Put, All Fours, and Whist, showing all the chances of holding any particular card or cards. Thus, it is 2 to 1 that your partner has not one certain card; 17 to 2 that he has not two certain cards; 31 to 26 that he has not one of them only; and 32 to 25 (or 5 to 4) that he has one or both — that is, when two cards are in question. It is 31 to 1 that he has three certain cards; 7 to 2 that he has not two; 7 to 6 that he has not one; 13 to 6 that he has either one or two; 5 to 2 that he has one, two, or three cards; that is, when three cards are in question.

With regard to the dealer and his partner, it is 57,798 to 7176 (better than 8 to 1) that they are not four by honours; it is 32,527 to 32,448 (or about an even bet) that they are not two by honours; it is 36,924 to 25,350 (or 11 to 7 nearly) that the honours count; it is 42,237 to 22,737 (or 15 to 8 nearly) that the dealer is nothing by honours.[55] [55] Proctor, The Sportsman's Sure Guide. Lond. A.D. 1733.

Such is a general sketch of the large subject included under the term of the calculation of probabilities, which comprises not only the chances of games of hazard, insurances, lotteries, &., but also the determination of future events from observations made relative to events of the same nature. This subject of inquiry dates only from the 17th century, and occupied the minds of Pascal, Huygens, Fermot, Bernouilli, Laplace, Fourier, Lacroix, Poisson, De Moivre; and in more modern times, Cournot, Quételet, and Professor De Morgan.

In the matter of betting, or in estimating the `odds' in betting, of course an acquaintance with the method must be of some service, and there can be no doubt that professional gamesters endeavoured to master the subject.

M. Robert-Houdin, in his amusing work, Les Tricheries des Grecs devoilées, has propounded some gaming axioms which are at least curious and interesting; they are presented as those of a professional gambler and cheat.

1. `Every game of chance presents two kinds of chances which are very distinct, — namely, those relating to the person interested, that is, the player; and those inherent

in the combinations of the game.'

In the former there is what must be called, for the want of a better name, `good luck' or `bad luck,' that is, some mysterious cause which at times gives the play a `run' of good or bad luck; in the latter there is the entire doctrine of `probabilities' aforesaid, which, according to M. Houdin's gaming hero, may be completely discarded for the following axiom: —

2. `If chance can bring into the game all possible combinations, there are, nevertheless, certain limits at which it seems to stop. Such, for instance, as a certain number turning up ten times in succession at Roulette. This is possible, but it has never happened.'

Nevertheless a most remarkable fact is on record. In 1813, a Mr Ogden betted 1000 guineas to one guinea, that calling seven as the main, the caster would not throw that number ten times successively. Wonderful to relate! the caster threw seven nine times following. Thereupon Mr Ogden offered him 470 guineas to be off the bet — which he refused. The caster took the box again and threw nine, — and so Mr Ogden won his guinea![56] In this case there

seems to have been no suspicion whatever of unfair dice being used. [56] Seymour Harcourt, The Gaming Calendar.

3. `In a game of chance, the oftener the same combination has occurred in succession, the nearer we are to the certainty that it will not recur at the next cast or turn up. This is the most elementary of the theories on probabilities; it is termed the maturity of the chances.'

`Hence,' according to this great authority, `a player must come to the table not only "in luck,'' but he must not risk his money excepting at the instant prescribed by the rules of the maturity of the chances.'

Founded on this theory we have the following precepts for gamesters: —

1. `For gaming, prefer Roulette, because it pre-sents several ways of staking your money[57] — which permits the study of several. [57] `Pair, impair, passe, manque, and the 38 numbers of the Roulette, besides the different combinations of position' and `maturities' together.

2. `A player should approach the gaming table perfectly calm and cool — just as a merchant or tradesman in treaty about any affair. If he gets into a passion, it is all over with prudence, all over

with good luck — for the demon of bad luck invariably pursues a passionate player.

3. `Every man who finds a pleasure in playing runs the risk of losing.

4. `A prudent player, before undertaking anything, should put himself to the test to discover if he is "in vein'' — in luck. In all doubt, you should abstain.'

I remember a curious incident in my childhood, which seems much to the point of this axiom. A magnificent gold watch and chain were given towards the building of a church, and my mother took three chances, which were at a very high figure, the watch and chain being valued at more than £100. One of these chances was entered in my name, one in my brother's, and the third in my mother's. I had to throw for her as well as myself. My brother threw an insignificant figure; for myself I did the same; but, oddly enough, I refused to throw for my mother on finding that I had lost my chance, saying that I should wait a little longer — rather a curious piece of prudence for a child of thirteen. The raffle was with three dice; the majority of the chances had been thrown, and 34 was the highest. After declining to throw

I went on throwing the dice for amusement, and was surprised to find that every throw was better than the one I had in the raffle. I thereupon said — `Now I'll throw for mamma.' I threw thirty-six, which won the watch! My mother had been a large subscriber to the building of the church, and the priest said that my winning the watch for her was quite providential. According to M. Houdin's authority, however, it seems that I only got into `vein' — but how I came to pause and defer throwing the last chance, has always puzzled me respecting this incident of my childhood, which made too great an impression ever to be effaced.

5. `There are persons who are constantly pursued by bad luck. To such I say — never play.

6. `Stubborness at play is ruin.

7. `Remember that Fortune does not like people to be overjoyed at her favours, and that she prepares bitter deceptions for the imprudent, who are intoxicated by success.'

Such are the chief axioms of a most experienced gamester, and M. Houdin sums up the whole into the following: —

8. `Before risking your money at play, you must deeply study your "vein'' and the different

probabilities of the game — termed the maturity of the chances.'

M. Robert-Houdin got all this precious information from a gamester named Raymond. It appears that the first meeting between him and this man was at a subscription-ball, where the sharper managed to fleece him and others to a considerable amount, contriving a dexterous escape when detected. Houdin afterwards fell in with him at Spa, where he found him in the greatest poverty, and lent him a small sum — to practise his grand theories as just explained — but which he lost — whereupon Houdin advised him `to take up a less dangerous occupation.' He then appears to have revealed to Houdin the entertaining particulars which form the bulk of his book, so dramatically written. A year afterwards Houdin unexpectedly fell in with him again; but this time the fellow was transformed into what he called `a demi-millionnaire,' having succeeded to a large fortune by the death of his brother, who died intestate. According to Houdin the following was the man's declaration at the auspicious meeting: — `I have,' said Raymond, `completely renounced gaming. I am rich enough, and care no longer for fortune. And

yet,' he added proudly, `if I now cared for the thing, how I could break those bloated banks in their pride, and what a glorious vengeance I could take of bad luck and its inflexible agents! But my heart is too full of my happiness to allow the smallest place for the desire of vengeance.'

A very proper speech, unquestionably, and rendered still more edifying by M. Houdin's assurance that Raymond, at his death three years after, bequeathed the whole of his fortune to various charitable institutions at Paris.

With regard to the man's gaming theories, however, it may be just as well to consider the fact, that very many clever people, after contriving fine systems and schemes for ruining gaming banks, have, as M. Houdin reminds us, only succeeded in ruining themselves and those who conformed to their precepts. Et s'il est un joueur qui vive de son pain, On en voit tous les jours mille mourir de faim. `If one player there be that can live by his gain, There are thousands that starve and strive ever in vain!'