You are reading a page from An Essay on Probabilities and their Application to Life Contingencies and Insurance Offices, Augustus de Morgan (1838)
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Term Life Insurance
ON THE RISKS OF LOSS OR GAIN.    9,3
n=10, a=100, b=500; n (n+1) ab=(a+b)= 73333
tot
ü7 3333=270.8; 21+1=101 ; 70,8=373=t
H = 4 very nearly, or about two to three against the proposed event.
Having thus shown the use of the tables at the end of this work, in the solution of complicated questions, I now proceed to the application of the theory to questions involving loss and gain.
CHAPTER V.

ON THE RISKS OF LOSS OR GAIN.
THE proverb which advises us to throw a sprat to catch a whale, shows that mankind consider a chance of a gain to be a benefit for which it is worth while to give up a proportionate certainty. The principle on which depends the determination of the amount which it is safe to hazard, must vary with the circumstances of the person who runs the risk. A man should not hazard his all on any terms ; but in ventures the loss of one of which would not be felt, we may suppose the venturer able to make a large number of the same kind ; in which case, the common notions of mankind, reinforced by the results of theory, tell us that the sum risked must be only such a proportion of the possible gain as the mathematical probability of gaining it is of unity. For instance : suppose I am to receive a shilling if a die, yet to be thrown, give an ace ; in the long run, an ace will occur one time out of six, or I shall lose five times for every time which I gain. I must therefore make one gain compensate the outlay of six ventures, or one sixth of a shilling is what I may give
94    eSSAY ON PROBABILITIES.
for the prospect, one time with another. But is the probability of throwing the ace.
PRINCIPLE. Multiply the sum to be gained by the fraction which expresses the chance of gaining it, and the result is the greatest sum which should be given for the chance.
PROBLEM. I am to gain by the throw of a die as many pounds as there are spots on the face which is thrown, with this exception, that I am to lose as many pounds as there are spots on both the face in question and on that of another die which is thrown at the same time, if the two give doublets. What should I give for the chance in question ?
Call these dice P and Q : then the chance of an ace from P, and something not an ace from Q, is the pro-duct of and or 356 ; and the same for any given throw from P, and not the same from Q. Consequently, the chance of winning, independently of the chance of losing, is worth
A x 1+ 38 x 2 + j x 3 +A x 4 +fib x 5 + so` x 6,
or 6- x 21, or £ 211. But if the chance of winning must be paid for, the chance of losing must be paid. Now the chance of throwing any one pair of doublets is the product of 6 and or 316 : whence the sum I must receive if I pay £ 21 (or not pay out of the £ 2-H, which is the same thing) is

3S x 2+A x4+ fig x 6 +x 8 +yz x 10 +A x 12,
or £ 1. Consequently, I must pay only £ 1 _;. If I were to stand a million of such hazards, paying £11'1 for each, I might expect my ultimate gain or loss to be extremely small, compared with the whole sum risked. If I had besides a very small profit upon each, I might be sure of winning.
We have in the last chapter considered the amount of fluctuation for which there is a given probability : we will now look at the percentage of fluctuation, that is,
ON THE RISKS OF LOSS OR GAIN    95
at the proportion which the fluctuation bears to the whole. It is for want of consideration on this point that many notions prevail which are utterly at variance both with the daily evidence of facts, and the results of exact in
Name any sum of money as a probable gain or loss of a commercial transaction, and we immediately proceed to compare it with the whole capital by which it is to be borne, if lost. A hundred pounds is nothing in one case, because it is only a shilling per cent. on the outlay; in another, the same sum is important, because it is ten pounds per cent. The importance of a gain or loss, then, depends upon the relative, and not on the absolute, value of the sum in question. Similarly, in a large number of transactions of the same sort, the number of them which may go against previous calcuis only important as compared with the whole number in question. The following is the correct prinnamely, that the percentage of fluctuation for which there is a given chance, varies inversely as the square root of the whole number of trials. Suppose, for instance, I find that on a certain hundred risks it is an even chance that one out of twenty goes against calculations made by the preceding methods; then if I take four times as many trials, it is an even chance that one out of forty only will disappoint the calculations ; if nine times as many, one out of sixty, and so on. Thus it appears that the chances may be made to be against any named percentage of fluctuation, however small, by sufficiently multiplying the number of risks. The probable amount of fluctuation increases, but not so fast as the number of risks, in such manner that the proportion of the probable fluctuation to the whole diminishes.
Gambling, according to the common notion of the word, means the habit of risking considerable sums in hazardous transactions. The gambler, properly so called, at cards and dice, and the jobber at the Royal Exchange, are called by the same name. Moral considerations
96    eSSAY ON PROBaBILITIeS.
must induce us to regard with an evil eye the person, whatever be the name of his occupation, who thrives by the actual losses of others, in transactions which are not commercially beneficial to the community. I am not prepared to say that stock-jobbers deserve to be included in this class ; it may be that the acquisition or sale of bona fide investments may by their means be rendered more easy to be obtained, and that they themselves may be a useful circulating medium between those who really wish to buy and sell for their own purposes. However this may be, the character of a gambler has no claim, when he is skilful, to the sort of respect which we pay to those who risk. The man of cards and dice, if he be cool and stick to his principles, secures a certainty to himself, and throws all the hazard upon his opponents : taking care never to risk too large a proportion of his means, and thus always enabling himself to take the benefit of the long run, he manages to play upon terms slightly unequal, and, of course, in his own favour. He runs a greater risk in games of mixed skill and chance than in games of pure chance alone. The latter he does not play at, unless he know the chances to be in his own favour ; while in the forit is next to impossible that he should always be more than a match for every opponent. The keeper of a gambling house has the surest game imaginable : the play is so managed that there shall be some chances more for the bank, as it is called, than for those who play against it; care is taken to provide a sum sufficient to stand a considerable succession of losses ; and the preceding principles will enable us to show that no individual re-sources can stand against a large fund thus used. The following problems will illustrate this.
PROBLEM. An indefinite number of successive haare tried of the following kind : —One of the events, A, B, C, &c. must happen at every trial, and each event brings with it a specified gain or loss. What may be expected to be the result of continuing to play a very great number of trials ; or what, in the long
ON THe RISKS OF LOSS OR GAIN.    97
run, may be expected to be the average produce or loss per game ?
RULe. Multiply each gain or loss by the probability of the event on which it depends; compare the total result of the gains with that of the losses: the balance is the average required, and is known by the name of the mathematical expectation. When the balance is nothing, then the play is equal.
For example, a person plays against a bank in the following manner:— Equal stakes of one shilling each are laid down, and if head be thrown twice successively he wins, or if tails be thrown twice he loses. But if head and tail be thrown, then another throw is to be made, by which, if it be head, the player only recovers his stake, but wins nothing: while, if it be tail, he loses his stake. In such problems, the stake is a mere evidence of solvency, with which the mathematical question has nothing to do. Let us suppose, then, that both parties keep their money until it is called for by the result. The events on which the player gains or loses are as follows (H stands for head, T for tail) :
H H gains him a shilling.
T T, H T T, T H T lose him a shilling. H T H and T H H give no result.
The chance of H H is and the same of T T; while for each of the set H T T, THT, HTH, and THH, the chance is A. Hence the value of the prospect of gain isi of one shilling ; that of loss A, + s + , ors of a shilling : consequently the balance against the player is of a shilling each time, and this he would certainly lose in the long run ; that is to say, the number of times he loses would eventually so far exceed the number of times he wins as to make the play cost him three pence per game.
Every risk of loss must then be compensated by an equal chance of gain, when the play is equal. But it does not follow that equal play means prudent play ;
H
98    ESSAY ON PROBABILITIES.
and for the following reason. Prudence requires that no one should expose himself to great risks of loss, and does not accept it as an excuse that there was a remote chance of enormous gain, or that another had the same chance of the same gain or loss. Again, the matheexpectation is derived from the result which, as can be shown, will be produced in the long run : conno one can prudently play, even with the play in his favour, unless he continue the occupation through such a number of trials that he may reasonably expect an average of all sorts of fortune. And though I have hitherto appeared to speak only of games of chance, yet precisely the same considerations apply to mercantile speculations, and to every species of affair in which no absolute certainty exists. If any possible event enter into the play which, from the nature of the game, cannot often occur, and if a stake be made upon the arrival of that event, proportionate to some enormous benefit which it is agreed that event shall secure, then prudence re-quires that the game shall be very often repeated, or, if that cannot be done, that it shall not be played at all. There is a celebrated case known by the name of the Petersburgh problem, which is one of the most inlessons in this subject, both on account of its paradoxical appearance, and also because very eminent writers have considered it as a sort of stumbling-block, and have endeavoure to evade the conclusion. Con_ dorcet and others have taken a proper view of the subject; while among those who have considered the problem as an anomaly, we may instance D'Alembert.
If p, q, and r be three fractions whose sum is unity, it follows that we may suppose three events, one of which must happen, and neither of the others, and of which the chances are p, q, and r: and similarly of more fractions than three, whose sum is unity ; or of any number of fractions, however great, provided their sum be unity. Let us, then, take an infinite number of fractions whose sum is unity : either of the following series will do.
ON THE RISKS OF LOSS OR GAIN.    99
I `    is !II, &c.    I    t$ Tea 1IIS m3as &c.
73 54' &c.    ra    TI    s$, &c.

Let the game be as follows : — The events which may happen at every trial are E I, of which the chance is ; E 2 ; of which the chance is a ; E3, of which the chance is and so on, ad infinitum. And one of these must occur. The bank engages to give 21. if E should turn up, 41. for E,,, 81. for E3, 161. for E4, and so on, ad infinitum.—What should the player give to the bank for one trial ? Write the several possible gains in a row, and underneath each the chance of its being won, as follows :
12 4 8 16 32 64 128, &c. a    Ys    ~I    ads    r~s &c.
Multiply each gain by the chance of gaining it, and each result is 1 ; consequently the mathematical expectof the player is unity repeated an infinite number of times, or an infinite amount. No sum, then, how-ever great, can compensate the bank for its risk. The Petersburgh problem realises the preceding supposition as follows —A halfpenny is tossed up until a head arrives, which is the event in question. If this happen at the first toss, the player receives 21.; if not till the second, 41. ; if not till the third, 81., and so on. Now, H standing for head and T for tail, the chance of H is 4; of TH, ; of TT H, -';; of TTTH, 116, and so on. But can it be believed, that if I am only to throw until head arrives, and to receive 21., or 41., or 81., &c. ac-cording as this happens at the first, second, third, &c. throw — can it be believed, you will say, that this prosis even worth 1001.; and is it not altogether monto say that an infinite amount of money ought to be given for it?
Firstly, I will advert to a large number of trials which was actually made. Buffon tried 2048 expericuments, or sets of tosses, the results of which were as follows : —In 1061, 11 appeared at the first toss ; in
u 2
100    ESSAY ON PROBABILITIES.
494, at the second ; in 232, at the third ; in 137, at the fourth ; in 56, at the fifth ; in 29, at the sixth ; in 25, at the seventh ; in 8, at the eighth ; and in 6, at the ninth. Let us, then, compute the amount which he would have received if he had bona fide played all these games on the preceding terms.
    
1061    x    2 =    2122    
    494    x    4 =    1976    
    232    x    8 =    1856    
    137    x    16 =    2192    The 2048 games would have
    .56    x    32 =    1792    given    20,1141.    or    nearly
    29    x    64 =    1856    101. per game, one game
    25    x    128 =    3200    with another.
    8    x    256 =    2048    
    6    x    512 =    3072    
No person would stake at this game for a single trial, upon the prospect of head being deferred till the ninth throw. Nevertheless, in this instance, it appeared that out of 2048 trials, such a rare occurrence happened often enough to realise more than any other, with one excepIf Buffon had tried a thousand times as many games, the results would not only have given more, but more per game. A larger net would have caught, not only more fish, but more varieties of fish ; and in two millions of sets, we might have expected to have seen cases in which head did not appear till the twentieth throw. Let us turn back to page 43, and inquire, by the rule there given, in how many trials it is 10,000 to 1 that head will be deferred till the twentieth throw. Out of 2'0, or 1,048,576 cases, representing the number of different arrangements which may happen in 20 throws, the arrangement in question is but one; it is then 1,048,575 to 1 against its arrival in any one given trial. Look in the Table opposite to " 10,000 to 1," and we find 921: multiply 1,048,575 by 921, and divide by 100, which gives 9,657,375. It is then more than 10,000 to 1 that head is deferred till the twentieth throw somewhere in ten millions of trials, and more than an even chance that it is found to occur in seventy thousand trials. Thus the reader may readily
ON THE RISKS OF LOSS OR GAIN.    101
conceive that with unlimited license of proceeding in this play, the player might continue until he had realised not only any given sum, but any given sum per game: a result which is indicated by the application of our rule, when it tells us that the mathematical expectation of the player upon a single game is infinite.
The result of all which precedes shows us that great risks should not be run, unless for sums so small that the venturer can afford to repeat them often enough to secure an average. But it should seem as if we were thus told either not to gamble at all, or else to play inWith a little reservation, this is true ; the stake must be lowered, and more games played, instead of risking a large fraction of the whole upon one game. It is better to buy the sixteenth of sixteen different tickets than to stake all upon one ticket ; and this even though it should be better than either not to buy at all. It is more prudent to play twenty games, staking one shilling upon each, than to stake a sovereign upon one game. Lay a proper proportion of the whole capital upon any hazard, and stipulate for as many trials as you please, and it will follow that with any mathemaadvantage, however trifling, in your favour, you must come off a winner. The mistake committed by those who attempt to gamble with professional men, is twofold : firstly, they set out upon unequal terms ; secondly, if the terms were equal, their stakes would be too large a proportion of their means. That the terms are unequal may readily be supposed, and will presently appear. No bank or individual gamester can play on fair terms, without losing as much as he wins in the long run. But even in such a case, the player of superior fortune has a great advantage over his antagonist, unless the stake be very small. If A with twenty guineas engage B with forty, all other things being equal, and if they are to play on until one or other has lost all, it is obviously much more likely that A shall lose his money before B, than the converse. If the play be unequally in B's favour, as well as the
II 3
102    ESSAY ON PROBABILITIES.
largeness of the fund, then it is still more against A in any given succession of games. The truth is, that to a young man who is determined to gamble, whether at one of the private receptacles in London, or the (till lately) recognised saloons of Paris, it is of little con-sequence whether his stakes be high or low, except in this particular, that a longer process of ruination will give him mare chances of seeing his error. The play is against him in both cases, and sooner or later he must be ruined. Nor if his means be ever so great, could he make use of them, against the banks in Paris, at least. Those who conducted the play at the Palais Royal were perfectly aware of the necessity of not staking too much, and limited not only the amount of each stake, but also the number of persons whom they would encugage at once. The consequence was, that though they played with perfect fairness (inasmuch as the inequality which existed in their favour was known to, and recognised by, their opponents), they gained large returns upon their capital, besides paying a considerable duty to the government.
A gambler (meaning a bold venturer, which the term commonly implies) ceases to be such when he makes his stakes bear a proper proportion to his capital, and takes no hazards which are unduly against him. If, then, a government should attempt to discourage the acquisition of great losses and gains, by limiting the number of hazards which an individual should be allowed to take, it might defeat its own object; and this is the case with our law, as it stands at present. In order to prevent individuals from gambling in life-insurance, the legislature has declared that A shall not insure the life of B, unless he have what is called an insurable interest in that life ; that is, unless A have some pecuniary inin B's continuing to live. The insurance offices, for the most part, have virtually, and very wisely, refused to live under this law, by paying all fair claims without questions asked. But supposing that the law were enforced, its effect would be as follows. It is
ON TIM RISKS OF LOSS OR GAIN.    103
tolerably easy to create a bona fide insurable interest on a few lives, while it would be difficult and attended with danger of detection to do the same with many lives. Under the system, then, proposed by the law, it would be easy to gamble, but not easy to carry speculation to the extent which would make it cease to be gambling. Allow the venturer to extend his traffic, and he will soon begin to feel the average, not to his gain but to his loss. For the mathematical advantage is in favour of the insurance offices, which are sure to gain in the long run. If, then, the law had been intended to save the gambler from certain loss, and to make it a real toss up, it would have been rational, considered as means; but if, as I imait was meant to hinder immoral gain, a more futile contrivance can hardly be conceived.
It must be remembered that, in the long run, events will happen in proportion to the chances of their happencuing in a single trial. We see this result in Buffon's trial of the Petersburgh problem, for which I write down the numbers of cases as they did arise, and unas they would have arisen, one time with an-other, if a great many series of 2048 trials each had been made.
5048 1061 494 232 137 56 29 25 8 6 2048 1024 512 256 128 64 32 16 8 4.

This rule, however, will only apply when so many cases are taken as will produce a great many of every event to which reference is made. For instance, in 2048 trials, one time with another, we can only expect the deferment of head till the seventh throw 16 times ; the result gave 25 times, or 56 per cent. more than the probable average. But the occurrence of head at the first throw, which, one set with another, would have occurred 1024 times, did really occur 1061 times, or 3 per cent. too many times. If we remember that in the long run, and on 2048 trials, we might expect two
4
104    ESSAY ON PROBABILITIES.
sets in which head should not appear till the tenth throw, and one in which no such thing should take place till the eleventh, and if we calculate the total amount which would have been realised had the average case occurred, we shall find it to be x:11 per game. In the experiment in question, it would have produced . 10. In precisely the same way, sets each consisting of 2n games would have realised ,fin per game (2048 is the eleventh power of 2).
' I now come to the estimation of the chances of fluctuation in loss or gain, meaning by fluctuation any departure from that general average to which the results of more and more trials will continually approach. It has been assumed in what precedes, that the proporwhich the fluctuation will bear to the whole will diminish without limit as the number of speculations increase. The following problems are easy deductions from those in the last chapter.
PROBLeM. It is known to be a to b for A against B.
A is an event which brings a loss or gain of g pounds ;
B is another event which brings a loss or gain of h pounds. What is the general average of such trials; and what is the chance that in n times a + b trials, the result as to loss or gain shall differ from the general average by not more than v pounds.
RULE. Find g times a, and h times b, and if g.and h be both gains or both losses, take their sum ; but it one be a gain and the other a loss, take the difference, counting it gain or loss, according as the term which contained the gain or the loss was the greater. Multiply the result by n, which gives the most probable total result (call this M). The general average is the n (a + b)th part of this ; or, more simply, the (a + b)th part of the balance of g times a and h times b. Take the differof g and h, if of the same name, or their sum, it of opposite names, and by it divide v. Take one more than twice the quotient. Having found this result, divide it by a square root immediately to be described, and let the quotient be t. Then the value of H in
ON THe RISKS OF LOSS OR GAIN.    105
Table I. is the probability that the resulting gain or loss shall lie between M + v and M — v pounds.
If it be absolutely known that the chances are as a to b for A against B, then, as in page 81, the square root is that of the product of 8, n, a, and b, divided by a + b. But if all that is known be that in a + b previous speculations, a gave A, and b gave B, then, as in page 92, the square root is that of the product 8, n, n + 1, a, and b, divided by a -}- b.
EXaMPLE. It has been observed, that of 100 specu70 yielded a profit of ,,t 20 each, and the remainder a loss of £ 25 each. What is the probability that in 150 more such speculations the total result shall not differ by more than 100 from its most probable amount?


a = 70, b = 30, n = 111, g = ,£ 20 gained, h = £ 25 lost,
v=£100.
g times a = 1400 gain l
h times b = 750 loss s j    6'1 general average again.
650 gain x l3 = 975 probable total.
g; = 20 gain l ,. = 2.2222. 2.2222 x 2 + 1 = 5.4444
h = 25 loss "5
Add 45.


8xn x n+lxaxb=8x13x2,'ix70x30=63,000.


68,000    5.4444
1~ = 630, 4/630 = 25.100,        - _ 217 = t.
25.1
Table I., ift=217,H=241.
Hence it is more than 3 to 1 against the result lying within the given limits.
PROBLeM. All things remaining as in the last prowhat is the amount of departure from the prototal for being within which there is the given odds p to q?
RULE. Turn p divided by p + q into a decimal fraction, and find it in the column H of 'fable I., taking out the corresponding value of t. Multiply t by
106    ESSAY ON PROBABILITIES.
the square root above mentioned, and having subtracted-1, divide the remainder by 2. Multiply the quotient by the difference or sum of g and h, according as they are of the same or different names, and the product is the answer required.
Example. In the preceding example, within what departure from £ 975 is it 10 to 1 that the result shall be contained?
p = 10, q = 1, p = (p + q) = 9091, 9091 x 25.1 = 22.8184. 3 (22.8184 — 1) = 10.9092 10.9092 x 45 = 490.91.
It is, then, 10 to 1 that the balance of 150 speculations shall lie between 975 + 491 and 975 — 491 pounds, or 14661. and 4841. Even such a case shows the effect of multitude in diminishing risks. The possible extremes of the problem (or the result of the problem itself, if we supposed one speculation instead of 150) are a gain of 30001., and a loss of 37501.
I will now add an example which will tend to show the ultimate effect of gambling against a bank with a slight mathematical advantage in its favour. Suppose the game is such, that at each trial it is 30 to 29 * that the bank shall win, the stake on both sides being one sovereign. Here a + b is 59, and making n a whole number for convenience of calculation, let n = 50, or let 2950 games be tried. The bank has each time a mathematical advantage (page 97) of - - f, or Iv of a sovereign, and will, in the long run, realise .£ 50 upon 2950 games. What are the chances in favour of the individual fluctuation of this one set of 2950 games leaving the bank without any profit, and with more or less loss ? To apply the preceding rule, we must first ask what are the chances that the departure from the probable total of 501. shall not exceed 501. ; that is, that the bank shall realise between 01. and 1001. Here we have


This supposition is more in favour of the player than is often the case.
ON THE RISKS OF LOSS OR GAIN.    107
a = 30, b = 29, n = 50, g=€l gained h= 1 lost v=.e50.
g times a = 30 gain 1    1
h times b = 29 loss } 59 = ,f5g general average of gain. 1 gain x 50 = £ 50 probable total.
g=.e l gain }
50=25;25x2+1=51.
1
h = ,e loss
Add £2.
In this case the probability is absolutely given. The square root is therefore the first one mentioned. 8 x n x a x b= 8 x 50 x 30 x 29 = 348,000
848000    51
59 = 58988, ü5898.3 = 76.800, 76.8 = -664 = t. Table I., if t = 664, H = 652.
Consequently -o5 5(,a is the chance that the bank shall not lose, but shall gain something less than 100; and consequently the chance that the bank shall either lose, or gain more than . 100 is -p64Ga On account of the nearness of 30 and 29, the results last mentioned are nearly equally probable, and it is near enough for our present purpose, to say that -r167046 is the chance of the bank gaining more than £ 100, the supposition being against us ; for it is more likely that the bank should gain more than £ 100 than that it should lose. Hence it follows, that the chance of the bank gaining on 2950 games is more than 1=4), or about five to one. If such be the case with a bank much less unfairly constituted than is often the case, against a player who can not only command 2950, but who has the prudence to deter-mine that he will only play 2950 games, at a stake of .e, 1 for each game, what must he the chances against those who risk a larger proportion of their means at more unequal play, with a determination to win—that is, to go on till they are ruined ?
The inequality of means is an important consideration in calculating the chances of two antagonist gameIf two persons, with equal means and equal chances, play for equal stakes, it is an even chance
108    eSSAY ON PROBaBILITIES.
whether A shall ruin B, or B shall ruin A ; but mat one or other will ultimately be ruined is certain. Sup-pose each party to have a hundred guineas, the stake being one guinea, and suppose two millions of games are to be played. The most probable individual case is that each shall win a million of games ; but if the fluctuation amount to 100 in favour of either, the other is ruined. Now, page 81, the probability that the number of games won by A shall lie between a million +100 and a million — 100 is  112, which is therefore the chance that neither player shall be ruined. Consequently, it is about nine to one that one player or other is ruined or more than ruined in two millions of games. And the chance is even greater than this : for the preceding method of treating the problem supposes the players not to balance their account till two million of games have been actually played, so that one player or the other may have been repeatedly playing on credit. The same rule may be easily applied to any inequality of play, the fortunes of the players being equal; and the result is, 1. that ultimate ruin to one or other player is certain ; 2. that, if the stake be a sufficiently small fraction of the player's income, the number of games which must be played to render probable the ruin of either may be made as large as we please. There are but two conditions under which gambling can be prudently followed as an amusement—small stakes and equal play. In games of pure chance it is possible to obtain the latter, and almost impossible in games of mixed skill and chance. Unfortunately, the stimulus of gambling, a combination of suspense and hope of large gain, cannot be obtained upon any terms which prudence would sanction.
When two players, of unequal fortunes, play together for the same stake, however equal the play may be, the larger fortune has an unfair advantage. To estimate the amount of the disadvantage, proceed as follows.
PROBLEM. Two players, A and B, having funds of m and n times their stake, play a game, at which it is a to b that A wins, or b to a that B wins. — What is the
ON THE RISKS OF LOSS OR GAIN.    109
chance that in the long run B will ruin A, and that A will ruin B ?
RULE I. If a and b be equal, it is m to n that A will ruin B, and n to m that B will ruin A.
RULE II. If a and b be unequal, let M represent the difference of the mth powers of a and of b, multiplied by the nth power of a, and let N represent the difference of the nth powers of a and of b, multiplied by the mth power of b. Then it is M to N that A ruins B, or N to M that B ruins A.
N.B.—If m and n be considerable, it is almost imto apply the above rule without the aid of logarithms. When b is many times a, take the nth power of a for M, and the nth power of b for N.
RULe III. When m and n are equal, it is as the mth power of b to the mth power of a that B will ruin A.
RULE IV. If the means of both players be unlimited, then it is certain the player who has odds in his favour on a single game will, in time, gain any sum, however great; if the means of the stronger player be unlimited, then it is certain he must at last ruin the weaker.
RULE V. If the means of the weaker player be unlimited, and those of the stronger limited, the chance that the latter will in time win any sum, however great, from the former, is as follows. Let B be the stronger player (that is, b greater than a), and let him begin with It times his stake, while A has unlimited means: then it is the excess of the nth power of b over that of a to the nth power of a that B gains any sum, in the long run, from A, and vice versa for A ruining B.
EXAMPLe. A has 51. and B 31., and they play at a game for which it is three to two that B shall win any one game.—What are the chances for the ultimate success of each player ?
a=2,b=3,m=5,n=3
M or (35 — 25) x 23 is (243 — 32) x 8 or 1688
N or (33 — 23) x 33 is (27 — 8) x 243 or 4617: It is therefore 4617 to 1688 that B shall ruin A.
A gaming bank must be considered as a player of
110    ESSAY ON PROBABILITIES.
limited means, playing against all who choose to enter, that is, playing against unlimited means. It is, therefore, essential to its existence that some matheadvantage should be allowed, even more than is necessary to reproduce the expenses of its maWhat I have hitherto said on the subject refers to the relation between the bank and the indiplayer against it . but considering the former as the antagonist of all who choose to play, it absolutely requires the protection of a mathematical advantage. But having this advantage, it must, in the long run, ruin its individual opponents; so that bankruptcy to itself, or degradation and suicide to its customers, are the initial conditions of its existence. But since the banks flourish, it is plain that whatever advantage is necessary to their continuance, is really obtained by them ; and I shall now inquire how much this advantage must be in several cases.
EXAMPLE (Rule V.) It is 30 to 29 for the bank upon each game, and the bank stakes the tenth part of its means at every game.—What are the chances of its perpetual continuance? (b = 30, a = 29, n = 10).


30j0 = 590,190,000,000,000 2910 = 420,707,233,300,201
169,782,766,699, 799.
Answer. About 170 to 421, or such a bank would not be likely to last; that is, in the long run, only 1 70 out of 421 such banks would avoid ruin.
PROBLeM. What is the mathematical advantage which a bank must have, in order that its permanent continuance may have k to I in its favour ; the supposition being that the bank stakes the nth part of its means at every game?
RULE. The odds in favour of the bank, on a single game, must be the nth root of 1 + lc to 1. Thus if, judging by the experience of the Parisian* banks, we say
t These banks were open to the public and to the municipal police. Of the gaming-houses in London, those who know them must speak. The
ON THe RISKS OF LOSS OR GAIN.    111
that the permanency of each had 100 to 1 in its favour, we are entitled to conclude that (the tenth root of 101 being 1.59) the games played were each not less than 3 to 2 in favour of the bank, if they staked the 10th of their resources at each throw, or 11 to 10 if they stake one fiftieth.
PROBLEM. The odds in favour of the bank being b to a, required the greatest proportion of the fund which may be staked at one game, in order to insure the chance k to 1 for the permanency of the establishment.
RULe. Divide the logarithm of one more than k by the excess of the logarithm of b over that of a : the quotient is the denominator of the fraction required, 1 being the numerator. Suppose, for instance, that the odds for the bank, on single games, are 30 to 29, then, if 99 to 1 be required to be the chance that the bank shall continue to exist, divide the logarithm of 99 + 1 (which is 2) by the excess of the logarithm of 30 (or 1'47712) over the logarithm of 29 (or P46240), and 2 _ 01472 gives a fraction more than 135 ; whence the 135th part of the capital is the highest which should be staked.
A merchant, who engages in speculations which must produce a fixed loss or a fixed gain, and who offers to deal with any one in such a manner, is precisely in the position of a bank such as is above described. The reason why neither party need be ruined in this instance is that the produce of the earth and sea is an unlimited fund, upon which the merchant draws. Trade by itself would tend to ruin the many, and accumulate all the stakes in the hands of a few, and the theory of probabilities would enable us to foretell that continual approximation to-wards the extremes of wealth and poverty which commercial countries always present. We have seen that the poorer player must, to maintain his ground, have a
protection and encouragement which legal regulation of gambling-houses would appear to give to gambling in general, is a good reason for the state of our own law ; otherwise, there can be no doubt that much particular evil would be prevented.by allowing a regulated system.
112    ESSAY ON PROBABILITIES.
mathematical advantage in his favour. Now, it is the nature of free trade that whatever mathematical advantage can be gained at all, is more accessible to the rich speculator than to the poor one. Consequently, the richer player, for that reason, can make himself the stronger player. If, then, a certain number of persons were to play upon a fixed total of stakes, equally divided at the commencement, with the condition that every stake won should enable the winner to make his next throw with somewhat more (no matter how little) of mathematical advantage than he had before, it is certain that, in the long run, the whole of the stakes would be in the hands of some one of the players. But, in the actual state of things, there is always an accession of new stakes and new players, so that the original players are contending against an unlimited fund. If the conaugmentation of stakes and players be not suffito counterbalance the tendency to extremes, a wise government would throw the burthen of taxation more upon the rich and less upon the poor. The matheadvantage of wealth would be taxed, as well as its power of procuring luxuries. Such a result never can be expected until the public mind is better informed upon the subject of which this work treats.
CHAPTER VI

ON COMMON NOTIONS WITH REGARD TO PROBaBILITY.
Those who have not considered this subject with parattention, seldom fail to think that there must be more or less of fallacy in the attempt to connect its principles with its results. Some, indeed, of the latter are strange and new, and are used as arguments against the validity of the theory. I propose in this chapter to