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APPENDIX.
APPENDIX THE FIRST.
ON THE ULTIMATE CHaNCES OF GAIN OR LOSS AT PLAY, WITH A PARTICULAR APPLICATION TO THE GAME OF ROUGE ET NOIR.
THOUGH the first part of the following reasoning is of a mathematical character, I have been induced to insert it by the consideration that the results of page 109. have never yet been introduced into an elementary work, nor even proved to the mathematician except either by incomplete or complicated trains of reasoning. Such being the case, perhaps even a well-informed mathematician might be excused for doubting some of the results of chapter V., and I have therefore digested the following demonstration, that no one who bears such a character may be able to weaken the evidence for the necessity of the pernicious results of gambling which that chapter is intended to afford.
De Moivre was the first who gave a solution of the following problem, and by a method of the most striking ingenuity. But his demonstration has the defect of assuming that one or other of the players must be ruined in the long run. Laplace* and Ampere,—the
* The solution of Laplace gives results for the most part in precisely the same form as those of De Moivre, but, according to Laplace's usual custom, no predecessor is mentioned. Though generally aware that La-place (and too many others, particularly among French writers) was much given to this unworthy species of suppression, I had not any idea of the extent to which it was carried until I compared his solution of the problem of the duration of play, with that of De Moivre. Having been instru
x 2
APPENDIX THe FIRST.
former in his Theorie, &c., the latter in a tract entitled, Considerations sur la Theorie Mathematique du Jeu, Lyons, 1802.—have also solved the problem: both soare of the highest order of difficulty, and cannot be rendered elementary. If my memory be correct, I have seen references to other solutions.
The problem is as follows :—Two players, A and B, the first possessed of m times and the second of n times his stake, play at a game so constituted that it is a to b that A shall win any one game ; required the probability which each has of ruining the other, if the game be indefinitely continued.
I shall first take the case where one of the players, A, is possessed of unlimited means, that is, in which m is infinite. Let Bn, n, represent the probability that B having n counters, shall ruin A who has nt counters. Then, if m be infinite, B„, will after the first game, become either B„+1, „, or B„ _ 1, 00, according as that game is B's or A's; of which the chances are
b    a and —
a+b    a+b
Consequently,
b    a    1
B fl,    a+bBn+l,m 1 a+bBn—1,m.~ which gives


B,t,,=c'~b)n+c”
which, when C' and C" are determined, will represent the probability that B will never be ruined, but will continually gain more and more from A. But the
mental (in my mathematical treatise on Probabilities, in the Encyclopedia Metropolitana) in attributing to Laplace more than his due, having been misled by the suppression, aforesaid, I feel bound to take this opportunity of requesting any reader of that article to consider every thing there given to Laplace as meaning simply that it is to be found in his work, in which, as in the Mecanique Celeste, there is enough originating from himself to make ally reader wonder that one who could so well afford to state what he had taken from others, should have set an example so dangerous to his own claims.
ULTIMATE RESULTS OF PLAY.    111
same equation (1) is equally true if for B,,,„s, we substitute either 1 — B,,, or A,,, „, which two last may be different, for any thing yet proved to the contrary. In fact, the equation (1) is merely the general expression of the condition that n is changed'into n + 1 or n — 1 according as one or the other of two events happens,
b    a
whose chances are — + b and a a b It may be seen
however, immediately, that in the case of n = 0, in which case the proposed contingency becomes an initial impossibility, we must have B = 0, or C'+C"=0. We have then

B,,,C” {1—;b/n}

This result is rational only when a is not greater than b, unless we suppose C" = O. But the necessity for investigating what takes place in this case is saved by observing a very simple relation which exists between B,,,,,,, and Supposing A to have infinite means, it makes no difference in the state of the question if we take any number of stakes m from the stock of A, and suppose that they shall be lost before the rest are touched. Consequently B cannot win indefinitely from A unless he first ruin A's stock of m stakes, and afterwards, be-ginning from n+m stakes, win indefinitely from A's remainder. That is

Yn, co = B,, m " B,, + m, co orBn,mBn,RTBn+m,x
= C" { 1 — l b)n} _C„ f 1 — bin+~nl
bm(bn —an)
bn+m — an +m

'Whence, applying the same reasoning to A ,,,,, ,,, we
find that if two players A and B, possessing m and n
x
iv    APPENDIX THE FIRST.
stakes, play at a game which gives a to b in favour of A at each trial, then, in an indefinite number of trials
bm(    bn—an )
B„ „t the chance that B shall ruin A = n+    n + m
>    b    — a
ant bm — am Ant, n the chance that A shall ruin B =     n+m    n + m
b    — a
The sum of these two chances is unity, from which it appears that one or other must be ruined in the long run. These results agree entirely with those of De Moivre, and all the rules in page 109. may be easily deduced from them.
It appears also, that if the conditions of any game, however complicated, can be reduced, in the case where one of the players (A) has unlimited means, to the equation
Bn, oc =13 Bn + 1, co, + a Bn _ 1 , (where a + 13 = 1) ;
then the ultimate results of that game exhibit probabiof precisely the same value as those of a simple game, in which it is a to /3 for A against B.
If, besides cases in which A or B wins, there be cases in which the game is drawn, no alteration is made in the result (though the number of games in which there is a given probability of either party winning must of course be increased.) Let a, /3, and be the chances that A wins, that B wins, and that the game is drawn : then (a +/3 + Z=1)
Bn,,o =$ B. + 1,co +a Bn—1,co +6 B,.,.,    or


13    a    r s    a    `
Bn,cc=1sBn+l,m+1SBn—1,co 1—5+11—S=1J
the same as in a game which must be won or lost, and in which it is a to /3 for A against B.
The following is the problem of the game of rouge et noir, which I shall afterwards proceed to explain.
ULTIMATE RESULTS OF PLAY.    V
A and B play at a game which presents four cases, A, B, D, and T, of which the chances are a, [3, and 0, so that a+ [3 + + 0=1. 'When A happens, the player A wins ; when B happens, the player B wins ; when D happens, the game is unconditionally drawn ; but when T happens, the game is drawn, and in the next game only the player A puts down a stake, and not the player B. If D should follow T, or if T should happen any number of times running, or D, or successions of T and D, still A's stake remains risked, without any from B : nor does B stake again, until the happening of A or B recovers A's stake, or assigns it to B : after which, both parties stake again.
Supposing the means of B to be unlimited, let A,,, represent A's chance of winning indefinitely, immediately before a game in which both are to stake, A having m stakes in his possession: and let AC,/ represent A's chance immediately before a game in which B does not stake. Then, by the preceding method
AmaAm+1+OAm_1 +SAm+BA'm Aim = aAm + /3 Am — 1 +SA'm + BA'm
Eliminate A'm and we have
Am_ a(a+S)     B$A    0(a(a+8/)3z)+B60A (a+p)=+m1+++m—1
in which the sum of the two co-efficients is unity. Hence this game is equivalent in its ultimate chances to a simple game in which it is a (a+ /3) to p3 (a + /3) + 0 /3 for A against B. If a-(3, the last odds are those of 2a to 2a+0.
This game of rouge et noir is described in an uninmanner, and with material omissions, in the later editions of Hoyle, from which work, and from the testimony of persons who have seen it played, I give the best description I can make of it, observing that the most modern method of playing differs in several partifrom that given in the hook referred to.
A number of packs of cards is taken (six, it is said in
s 4
vl    APPeNDIX THe FIRST.
Hoyle) and all the cards are well mixed. Each common card counts for the number of spots on it, and the court cards are each reckoned as tell. A table is divided into two compartments, one called rouge, the other noir, and a player stakes his money in which he pleases. The proprietor of the bank, who risks against all comers, then lays down cards in one compartment until the number of spots exceed 30 ; as soon as this has happened, he proceeds in the same way with the other compartment. The number of spots in each compartment is then be31 and 40, both inclusive, and that compartment wins which has the lower number of spots; so that if, for instance, there should be 37 spots in the rouge, and 32 in the noir, those players who staked upon noir would win from the bank sums equal to their stakes. If the number of spots be the same in both (which is called in Hoyle a refait) the game is drawn, and the parties withdraw, diminish, or augment their stakes at pleasure, for a new game : except only when the number of spots in both compartments is 31 (called in Hoyle a refait trente et nu), in which case the bank is allowed to with-draw its stakes, and those of the players, whatever their compartment may be, are impounded (placed en prison). In the next game (now called an apres), the impounded stakes are played for, the players choosing their compartments as before : should the bank win it takes the stake, should the bank lose the player recovers his stake. Should a second refait trente et un occur, or a drawn game, the stakes still remain impounded, and are not released until a gain or loss arrives. In the meanwhile new stakes may be put down, before the fate of the old ones is decided.
The chances of this game depend, in a slight degree, upon the number of packs of cards which are mixed together. When, however, there are as many packs as six, it is very nearly # indeed the same thing as if the
The only ways in which 31, for example, could be obtained from an unnumber of packs, and which could not equally well be obtained from six packs, are those in which more than 24 aces occur. Now the proba-
ULTIMATE ReSULTS OF PLaY.    Vii
number of packs were unlimited. The following tables, computed upon the latter supposition, will represent, with more than sufficient accuracy, the chances of the several arrivals. The first three columns of the first table exthe chance of arrival of each number among the several sum-totals which precede the arrival of thirty-one or more. Thus opposite to 4 we see 0961, which is the chance that the beginning of the sequence drawn shall show one of the following sets of cards (1 standing for ace, &c.) :
1, 1, 1, 1 12, 1, 1 I2, 2 1, 1, 2    Ill 1, 3,    4
1, 2, 1    3, 1
    1    0769    11    1200    21    '1398    31    1481
    2    0828    12    1247    22    1424    32    1379
    3    0892    13    '1293    23    1449    33    1275
    4    0961    14    'i340    24    '1472    34    '1169
    5    '1035    15    '1386    25    1493    35    '1060
    6    1114    16    1432    26    '1511    36    '0950
    7    1200    17    1476    27    '1527    37    '0838
    3    '1292    18    1519    28    1541    38    '0723
    9    1392    19    1559    29    1552    39    '0607
    10    3806    20    '2129    30    '1683    40    '0518
0219 34 0137 37 0190 35 I '0112 38 0163 36 '0090 39
0070    40
0052    0037 D. G.
31 32 33
'0027 0878
The fourth column contains all the chances of those points which lie between 31 and 40, that is to say, the chance of each being the first which arrives. Thus 1060 is the chance that 35 will appear, and that it will be the first which appears above 30. The second table (containing the squares of the numbers in the last column of the first) shows the chance of each refait, that of the
bility that out of 31 cards drawn at hazard, 24 or more shall be aces, is altogether beneath consideration. It is less than one out of a million of million of millions.
Viii    APPeNDIX THe FIRST.
refait* trente et un being 0219. Opposite to D. G. is the sum of the chances of all the retails except the first, which sum is the chance of a drawn game.
If then we say that 021 is the chance of a refait trente et un and .087 that of a drawn game, there remains 892 for the chance that either the bank or the player must win ; which chances being equal, give 446 for the player, and the same for the bank (exclusive of the benefit of the apres). Returning then to the result in page v., we find a= /3 = 446, 8 = 021, and 2a to 2a
8 is '892 to 913, or 892 to 913. Consequently; —
At the game of rouge et noir, as now played, the chances of ultimate ruin to the bank or the player are the same as they would be at a simple game, which must be either won or lost at each throw, and in which the bank has 913 chances of winning, where the player has 892.
The bank, as noticed in page 110., is playing against the whole public, or against a player with unlimited means. Taking 892 to 913, or more correctly 8903 to 9122, and applying the rule in page 110., the followtable results, which must be thus used. Opposite to 30 we find 4824, which is the chance of the ultimate ruin of a bank which risks one-thirtieth of its means at every game : —
    
10    7843    110    0690    210    '0061
    20    6151    120    0541    220    0048
    30    4824    130    0425    230    0037
    40    3783    140    '033.3    240    0029
    50    2967    150    0261    250    0023
    60    2327    160    0205    260    0018
    70    1825    170    0161    270    0014
    80    1431    180    0126    280    0011
    90    1122    190    0099    300    0007
    100    '0980    200    0077    400    '0001
The editor of Hoyle says, or implies, that the chance of the arrival of 31 is one-eighth, or 125. This is, no doubt, a conclusion drawn from obThe table in Hoyle, exhibiting the odds (page 147., which refers to 141., edition of 1814;, is altogether erroneous.
ULTIMATe RESULTS OF PLAY.    1X
Hitherto all our results seem in favour of the bank ; that is, tending to show that its advantage is not so great as is commonly supposed. No person, granting a bank permission to exist, would grudge it such an advantage as would make it 49 to 1 against its being ruined by the possible fluctuation attendant upon an unlimited duration of play. This chance of being ruined, namely 02, appears from the table to be exceeded, unless the bank possess 160 times the sum risked at each game: if this were 1001., the bank would need a capital of 16,0001. But I must now request attention to the other side of the question ; first, considering the bank against the public; and, secondly, the bank against an individual player. One of the most important features in this game (which springs from the old game of Faro, as did the last from the still older game of Basset*) is, that the bank does not risk the whole sum it lays down, but only the difference between those sums which the caprice of the players obliges it to stake on rouge and on noir. If 20 players have each staked a guinea, 12 on rouge and 8 on noir, and if rouge win, the bank loses 12 guineas and gains 8, and consequently did not risk more than four guineas. It is impossible to say what chance there is of the bank having to risk a given sum in such a case, as this depends on the will of the players. When the cards have several times decided for rouge, those players who think the run is not finished will stake on that colour, while others who think differently will stake on noir. I am wholly without the means of saying what average exists, but I should incline to think it very unlikely that the bank really risks more than one fourth of its deposits.' But the advantage which it derives from the refait trente et
 An assertion of the editor of Hoyle, which is true as to the principle of the game — namely, that besides equal chances for the bank and the player, there are chances for a drawn game, and a case in which the hank has a direct advantage amounting to half the stake—but the details are very different. Both games are described in De Moivre.
t The bank is evidently (its chance of the aprls excepted) merely the means of equalising the sums staked on the two colours.
X    APPENDIX THE FIRST.
tin, and its consequences, is gained on the whole of the opponent stakes. The following is the method of estithe mathematical advantage of the bank:—Before a common game, the prospects of the bank lie entirely in the chance of that game being followed by an apres, since, in all other respects, the chances of gain and loss are the same. After a refait trente et fin, on whichever colour any player may choose to risk his impounded stake, the bank has the chance a of winning that stake, and none of losing. But besides this, the bank has all the chances of a second refait (or 0+ or 1—2a), for having another trial of the same kind : if then x express the fraction of a pound, which such a chance of 11. is worth, we have
x=a + (1—2a) ,r, or x=.
The mathematical advantage of the bank is therefore, the chance 0, or '0219, of being put in possession of the worth of half the stakes ; or 0111. of all the sums it deposits. This is 11. 2s. per cent. * per deposit ; which, to those who know the rapidity with which the risks succeed one another, will appear to yield, in the course of the year, an ample return, not merely to the deposits, but to the sums which are reserved for security against fluctuation. It is probably 41 per cent.upon every real risk ; and the return in the course of a year may be easily guessed at. Imagine 100 different games, played on each of 100 different evenings, the sum risked by the collective players on each game being 501. The total de-posits of the bank would be 500,0001., on which 11. 2s. per cent, is 55001. The capital required to make this spemuch more safe than any mercantile adventure, would not be larger than its probable return in one year.
Some time after this was written I chanced to find the following sentence in the lately published Theory of Probabilities of M. Poisson.
Dans les jeux publics de Paris l'avantage a chaque coup est peu consider-able : au jell de trente-cf-gnarantc par exemple, it est un peu au-dessous de .0I1 de chaque [Wise. ' oyez cur les chances de ce jell, le metnoire que j'ai insere clans le journal de M. Gergonne, tome xvi. eumero b, Decembre, 1825." I have not seen this memoir; the accordance of the result with my own, shows that I hay described the game correctly, as it was played in Paris ; of which, from paucity of information, I was by no means sum.
ULTIMATE RESULTS OF PLAY.    xi
If any persons, aware that the preceding calculations are new, should imagine that there must be some miscalculation in a result which shows that cent. per cent. on the necessary capital might be gained three times in a year, I reply, that the chance of a refait trente et un, as given by the editor of Hoyle, produces a result of nearly as surprising a character. For 0219 read 0156, or -A, and the 55001. above-mentioned becomes a trifle less than 40001. The preceding results, or either of them, being admitted, it might be sup-posed hardly necessary to dwell upon the ruin which must necessarily result to individual players against a, bank which has so strong a chance of success against its united antagonists. But so strangely are opinions formed upon this subject, that it is not uncommon to find persons who think they are in possession of a specific by which they must infal
libly win. The last table given will show the chances which any single player has of ruining the bank, and of being ruined himself, as follows : —If the player stake one mth part of his means * at each throw, and the bank one nth part, from unity subtract the number in the table opposite to n and to n+ m, and the first result divided by the second shoes the chance that the bank will ruin the player. Suppose, for example, that the player risks 1-10th and the bank 1-160th of their respective resources. Then opposite to 160 and 160 ± 10, we find 0205 and 0161 ; which, being subtracted from 1, give 9795 and 9839,
whence is the chance for the bank; or it is 9795 to 44, or 223 to 1, that such a player will be ruined. Even if both the player and the bank stake 1cu160th parts of their several funds, the bank will still have 98 or 49 to 1 in its favour against that one player.
The last column of the table in page vii, shows that the bankers at rouge et noir, by making the apres de
We must not here consider what the bank stakes against the individual player, but the whole sum which it risks.
xii    APPENDIX THE FIRST.
pend on the refait trente et un have chosen the most favourable out of the ten cases. The following table will show the effect of substituting any other refait ; the first column pointing out the refait in question; the second, the simple game to which it is equivalent in the chances of ultimate ruin ; the third, the benefice of the bank upon every 1001. deposited :
    
31    10,000    to    10,246    £    s.    d.
            1    2    0
    32        10,213    0 19    0
    33        10,183    0 16    6
    34        10,154    0 13    6
    35        10,126    0 11    0
    36        10,101    0    9    0
    37        10,079    0    7    0
    38        10,058    0    5    0
    39        10,042    0    4    0
    40        10,030    0    3    0
The above is a graduated scale of poisons, each one being slower in its operation than the preceding ; the first, or quickest of all, being that which is used at present. Of all the illegal games, none that I know of is less likely to lead to ruin than rouge et noir ; anti, the results of this investigation give a sufficient notion of the state of the case between the banker and his dupes.
The first table, in page vii., is calculated in the folmanner : — The chance of any given card at a given point is lt; for every number which a card can give, excepting 10, the chance of which is 13, on account of the value given to the court cards. Let x be a number greater than 10, and let V, be the chance of arriving at that number in the laying down of the cards. Then, if (Vi) signify the event of which the chance is V, and if by (a) (b), we mean the consecutive happening of the two events whose chances are a and b, it follows that (Vr) when it happens, must happen in one of the following ways : --
ULTIMATE RESULTS OF FLAY.    Xiii
( Vx_, ) (19) or (Vx_2 ) (19) . .. or ( Vx-9 ) (1!9) or ( Vx—10 ) (T9)
or Vr = r9 ( Vx— 1 +    + ... + Vx–9 ) + T9 Vx–10
from which it is readily found that
nVx = 79 (Vx + 4 e. Vx_10    )
The first eleven values of Vx are thus determined: V1 is evidently 73; V, is i9 V1 + - or T9 . i3; Va is T3 V1 + i9 V Y + 79 or 11-3 ('T,*3)=; and so on up to V10 which is
r ' 3 (V, + V o + .. + V 9) + i3 or r9 (14)9 + is V11 is(V10 +V9 ...+V,)+r'3V1 or
(13)10+ f f i2—T'.3.
The rest were then calculated by the preceding forfor oV= to six places of decimals (by which the accuracy of four was insured) as far as V31 inclusive. The remainder were calculated by those which preceded, leaving out the terms which the necessary distinction already mentioned requires to be omitted.
It may perhaps appear to some that a part of the preceding reasoning is inapplicable, since it only calcuthe chances of ruin in an indefinite succession of games, whereas any practicable number of games, though great, may not involve the same chances as an infinite number. The objection is valid in principle, but the correction which is rendered necessary by it is not worth consideration, if any large number of games be in ques
The following are the only cases in which a simple approximate rule can be given, connected with finite numbers of games : —
PROBLEM. Both parties have the same number of stakes (which should not be less than 20), say a, and the play is equal, or either has an even chance of winany one game. 'What is the chance that one or other shall have been ruined before x games have been played ? (x being a large number).
xiv    APPENDIX THE FIRST.
RULE. Divide 30 times x by 56 times the square of a, and from the quotient subtract 1049. If the result be the common logarithm of x, then — 1 to 1 are the odds in favour of the event.
EXAMPLE. The number of stakes is 45, and the play equal : what is the chance that one or other is ruined before 1520 games have been played ?
a = 45, x = 1520, 56 x a x a=113400
45600
x x 30 = 45600, 113400 = '4021


4021— 1049 = 2972 = logarithm of 1.982.
Answer, 982 to 1, or 982 to 1000,
PROBLEM. The stakes being equal, and also the play, as before, what is the number of games in which it is n to 1 that one party or the other will have been ruined ?
RULE. To the common logarithm of one more than is add 1049 : multiply the result by 56 times the square of the number of stakes, and divide by 30, which gives the number required, very nearly.
EXAMPLE. Both parties have 50 stakes: in what numof games is it 10 to 1 that one or other will be ruined? a = 50, n = 10,n+1=11, log. 11=1.0414
1.0414 + 1049 = 1.1463, 56 xaxa = 140000
160482
1.1463 x 140000 = 160482:    30 = 5349
Answer: in about 5349 games.
To find the number of games in which it is an even chance that one or other will be -ruined, from three-fourths of the square of the number of stakes, subtract its hundredth part. Thus, if both parties have 40 stakes, then 40 x .10 being 1600, three-fourths of which is 1200, from 1200 subtract 12, which gives 1188 for the number of games (very nearly) in which it is an even chance that one or other will be ruined.
If a player with a stakes play with one of unlimited means, the chances being the same for both, it is an even chance that he is ruined in a number of games