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 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
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/ 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 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 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 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 superstitious 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 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 Essex, 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 entitled 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
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 L9 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 L11 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, &c., 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, Quetelet, 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
devoilees, 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 presents 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 L100. 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!'