To the Church-Wardens, Overseers, and each respectable inhabitant in the Parish."
A card was enclosed, as follows:-- "****
Gaming House Keeper, and **** **** to The Honourable House of Commons No. 7 and 8 **** St, St James"s."
This circular was sent to Stockdale, the publisher, in 1820, who published it with the names in asterisks suppressed. It was evidently intended to expose some doings in high places.
CHAPTER VIII. 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 princ.i.p.al 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 amus.e.m.e.nt, 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 processes 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 const.i.tute 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 a.s.sertion 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/ 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 missing 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 a.n.a.logous 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 ent.i.tled 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 superst.i.tion 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 a.s.serters 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 hypothesis, 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 repet.i.tion 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 ent.i.tled 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 superst.i.tious 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, c.o.c.k-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 measure 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 Ess.e.x, in the year 1738, that three horses (and no more than three) started for a L10 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 L50, and there are even bets on both sides; it is evident that the present value or worth of each of their expectations will be L25, 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 ent.i.tled to L25; 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 L30, and that of B"s only L20, 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 L4, 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