Théorie Analytique des Probabilités

Théorie Analytique des Probabilités Pierre Simon Laplace Book II 5 9. pp. 203 228 5. An urn being supposed to contain the number B of balls, e dra from it a part or the totality, and e ask the probability that the number of balls extracted ill be even. The sum of the cases in hich this number is unity equals evidently B, since each of the balls can equally be extracted. The sum of the cases in hich this number equals is the sum of the combinations of B balls taken to by to, and BÐB Ñ this sum is, by no. $, equal to. The sum of the cases in hich the same number equals $ is the sum of the combinations of balls taken three by three, and BÐB ÑÐB Ñ this sum is $, and thus in sequence. Thus the successive terms of the B development of the function Ð Ñ ill represent all the cases in hich the number of balls extracted is successively $ to B; hence it is easy to conclude that the sum of all the cases relative to the odd numbers is B B B Ð Ñ Ð Ñ or, and that the sum of all the cases relative to the B B B even numbers is Ð Ñ Ð Ñ or. The reunion of these to B sums is the number of all the possible cases; this number is therefore ; thus B the probability that the number of balls extracted ill be even is B, and the B probability that this number ill be odd is B ; there is therefore advantage to ager ith equality on an odd number. If the number B is unknon, and if one knos only that it can not exceed 8, and that this number and all the lesser are equally possible, e ill have the number of all the possible cases relative to the odd numbers by making the sum B of all the values of, from B œ to B œ 8, and it is easy to see that this sum 8 is. We ill likeise have the sum of all the possible cases relative to the even numbers, by summing the function B, from B œ to B œ 8, and e 1

8 find this sum equal to 8 ; the probability of an even number is therefore 8 8 8 then 8 8 2, and that of an odd number is 8 8 2. We suppose no that the urn contains the number B of hite balls, and the same number of black balls; e ask the probability that by draing any even number of balls, e ill bring forth as many hite balls as black balls, all the even numbers being able to be brought forth equally. The number of cases in hich one hite ball of the urn can be combined ith a black ball is evidently BB. The number of cases in hich to hite balls can be BÐB Ñ BÐB Ñ combined ith to black balls is, and thus in sequence. The number of cases in hich e ill bring forth as many hite balls as black balls is therefore the sum of the squares of the terms of the development of the binomial Ð Ñ B, less unity. In order to have this sum, e ill observe that it is equal to a term independent of +, in the development of ˆ B B Ð +Ñ. This function Ð +Ñ B is equal to + B The term independent of +, in its development, is thus the B coefficient of the middle term of the binomial Ð +Ñ à this coefficient is $B Ð$BÑ ; the number of cases in hich e can dra from the urn as many hite balls as black balls is therefore $B Ð$BÑ. The number of all possible cases is the sum of the odd terms in the development of the binomial Ð Ñ B, less the first,or unity. This sum is B B B Ð Ñ Ð Ñ ; the number of possible cases is therefore, hich gives for the expression of the probability sought $B Ð$BÑ B. In the case here B is a great number, this probability is reduced by no. 33 of Book I to, being the semi-circumference of hich 1 is the radius. ÈB1 1 6. We consider a number B B of urns, of hich the first contains : hite balls and ; black balls, the second : hite balls and ; black balls, the third : hite balls and ; black balls, and thus in sequence. We suppose that e dra successively one ball from each urn. It is clear that the number of all the possible cases in the first draing is : ;; in the second draing, each of the cases of the first being able to be combined ith the : ; balls of the second urn, e ill + 2

have Ð: ;ÑÐ: ; Ñ for the number of all the possible cases relative to the first to draings. In the third draing, each of these cases can be combined ith the : ; balls of the third urn; this hich gives Ð: ;ÑÐ: ; ÑÐ: ; Ñ for the number of all the possible cases relative to the three draings, and thus of the rest. This product for the totality of the urns ill be composed of B B factors, and the sum of all the terms of its development in hich the letter :, ith or ithout accent, is repeated B times, and consequently the letter ;, B times, ill express the number of cases in hich e can dra from the urns B hite balls and B black balls. If : :, are equal to :, and if ; ; are equal to ;, the preceding product becomes Ð: ;Ñ B B B B. The term multiplied by : ; in the development of this binomial is ÐB B ÑÐB B ÑâÐB Ñ $ÐB B Ñ $ B : ; B B $ B$ B : B ; B or Thus this quantity expresses the number of cases in hich e can bring forth B hite balls and B black balls. The number of all the possible cases being B B Ð: ;Ñ, the probability to bring forth B hite balls and B black balls is $ÐB B Ñ : ; $B$B Œ Œ : ; : ; : : ; here e must observe that is the probability of draing a hite ball from ; one of the urns, and that : ; is the probability of draing from it a black ball. It is clear that it is perfectly equal to dra B hite balls and B black balls from B B urns hich each contain : hite balls and ; black balls, or one alone of these urns, provided that e replace into the urn the ball extracted at each draing. We consider no a number B B B urns of hich the first contains : hite balls, ; black balls and < red balls, of hich the second contains : hite balls, ; black balls and < red balls, and thus in sequence. We suppose that e dra one ball from each of these urns. The number of all the possible cases ill be the product of the B B B factors, Ð: ; <ÑÐ: ; < ÑÐ: ; < Ñâ The number of cases in hich e ill bring forth B hite balls, black balls and B red balls ill be the sum of all the terms of the development of this product, in hich the letter : ill be repeated B times, the letter ;, B times and the letter <, B B B 3

B times. If all the accented letters : ; are equal to their corresponding non-accented, the preceding product is changed into the trinomial B B B B B B Ð: ; <Ñ. The term of its development, hich has for factor : ; <, is $ÐB B B Ñ $B$B $B : ; B B < B à B B B thus, the number of all the possible cases being Ð: ; <Ñ, the probability to bring forth B hite balls, B black balls and B red balls ill be B B B $ÐB B B Ñ : ; < $B$B$B Œ Œ Œ : ; < : ; < : ; < : ; hence e must observe that : ; <, : ; <, : ; < are the respective probabilities of draing from each urn a hite ball, a black ball and a red ball. We see generally that, if the urns contain each the same number of colors, : being the number of the balls of the first color, ; the one of the balls of the second color, < = those of the balls of the third, the fourth,, B B B B â being the number of urns, the probability to bring forth B balls of the first color, B balls of the second, B of the third, B of the fourth, ill be $ÐB B B B âñ : $B$B$B$B Œ : ; < = â ; < = Œ Œ Œ â : ; < = â : ; < = â : ; < = â < B B B B 7. We determine no the probability to dra from the preceding urns B hite balls, before bringing forth B black balls, or B red balls,. It is clear that, 8 expressing the number of the colors, this must happen at the latest after B B B â 8 draings; because, hen the number of hite balls extracted is equal or less than B, the one of the extracted black balls less than B, the one of the extracted red balls less than B,, the total number of the extracted balls, and consequently the number of draings, is equal or less than B B B â 8 ; e can therefore consider here only B B B â 8 urns. 4

In order to have the number of cases in hich e can bring forth B hite balls at the ÐB Ñ st draing, it is necessary to determine all the cases in hich B hite balls ill be dran at the draing B 3. This number is the term multiplied by : B in the development of the polynomial Ð: ; < âñ B 3, and this term is $ÐB 3 Ñ B 3 : Ð; < âñ $ÐB Ñ$3 by combining it ith the : hite balls of the urn B 3, e ill have a product hich it ill be necessary to multiply by the number of all the possible cases relative to the B B â 8 folloing draings, and this number is e ill have therefore Ð: ; < âñ B B â 8 à ( a) $ÐB 3 Ñ : Ð; < âñ Ð: ; < âñ $ÐB Ñ$3 B 3 B B â 8, for the number of cases in hich the event can happen precisely at the draing B 3. It is necessary hoever to exclude the case in hich ; is raised to the poer B, those in hich < is raised to the poer B, etc.; because in all these cases it has already happened in the draing B 3, either B black balls, or 3 B red balls, or etc. Thus in the development of the polynomial Ð; < âñ, it 0 0 0 is necessary to have regard only to the terms multiplied by ; < = in hich 0 is less than B, 0 is less than B, 0 is less than B, The term multiplied by 0 0 0 ; < = in this development is $ 3 $0$0 $0 ; < = 0 0 0 All the terms that e must consider in the function ( a) are therefore represented by ( b) Ú Ý Û Ý Ü $ÐB 0 0 â Ñ $ÐB Ñ$0$0 : ; < B 0 0 â Ð: ; < âñ B B â 0 0 â 8 because 3 is equal to 0 0 â Thus, by giving, in this last function, to 0 all the hole values from 0 œ! to 0 œ B, to 0 all the values from 0 œ! to 5

0 œ B, and thus in sequence, the sum of all these terms ill express the number of cases in hich the proposed event can happen in B B â 8 draings. It is necessary to divide this sum by the number of all the possible B B B â 8 cases, that is to say by Ð: ; < âñ. If e designate by : the probability of draing a hite ball from any one of the urns, by ; that of draing from it a black ball, by < that of draing a red ball,, e ill have : ; < : œ ; œ < œ âà : ; < â : ; < â : ; < â the function ( b), divided by Ð: ; < âñ B B B â 8 $ÐB 0 0 â Ñ $ÐB Ñ$0$0 : B ; 0 < 0 â, ill become thus The sum of the terms hich e ill obtain by giving to 0 all the values from 0 œ! to 0 œ B, to 0 all the values from 0 œ! to 0 œ B, ill be the sought probability to bring forth B hite balls before B black balls, or B red balls, or, etc. We can, after this analysis, determine the lot of a number 8 of players A, B, C,, of hom : ; < represent the respective skills, that is to say their probabilities to in a trial hen, in order to in the game, there lacks B trials to player A, B trials to player B, B trials to player C, and thus in sequence; because it is clear that, relatively to player A, this reverts to determine the probability to bring forth B hite balls before B black balls, or B red balls,, by draing successively a ball from a number B B B â 8 of urns hich contain each : hite balls, ; black balls, < red balls, â, : ; < being respectively equal to the numerators of the fractions : ; < reduced to the same denominator. 8. The preceding problem can be resolved in a quite simple manner by the analysis of the generating functions. We name C BB B the probability of player A to in the game. At the folloing trial, this probability is changed into C B BB, if A ins this trial, and the probability for this is :. The same probability is changed into C BB B, if the trial is on by player B, and the probability for this is ; ; it is changed into CBBB if the trial is on by player C, and the probability for this is <, and thus in sequence; e have therefore the equation in the partial differences 6

BB B B B B BB B BB B C œ : C ; C < C â Let? be a function of > > > such that CBBB is the coefficient of B B B > > > in its development; the preceding equation in the partial differences ill give, by passing from the coefficients to the generating functions,? œ?ð: > ; > < > âñ hence e deduce consequently, œ : > ; > < > âà : œ > ; > < > â this hich gives??: > œ Ð ; > < > âñ œ?: B B B B Ú BÐ; > < > âñ Ý BÐB Ñ Ð; > < > âñ Û BÐB ÑÐB Ñ Ð; > < > Ý $ Ü â B B? > B BB B $ âñ No the coefficient of >! > > in is C, and the same coefficient in B 6 6 any term of the last member of the preceding equation, such as 5?: > >, is B 5: C!B > B > à the quantity C!B > B > is equal to unity, since then player A lacks no coup. Moreover, it is necessary to reject all the values of C!B > B > in hich 6 is equal or greater than B, 6 is equal or greater than B, and thus in sequence, because these terms are not able to be given by the equation in the partial differences, the game being finite, hen any one of the players B, C, have no more coups to play; it is necessary therefore to consider in the last member of the preceding equation only the poers of > less? than B, only the poers of > less than B,. The preceding expression of > B ill give thus, by passing again from the generating functions to the coefficients, à 7

C œ : BB > B > B Ú BÐ; > < > âñ Ý BÐB Ñ Ð; > < > âñ Û BÐB ÑÐB Ñ Ð; > < > âñ Ý $ Ü â $ à provided that e reject the terms in hich the poer of ; surpasses B, those in hich the poer of < surpasses B, etc. The second member of this equation is developed in one sequence of terms contained in the general formula $ÐB 0 0 â Ñ $ÐB Ñ$0$0 : B ; 0 < 0 â The sum of these terms relative to all the values of 0 from 0 null to 0 œ B, to all the values of 0 from 0 null to 0 œ B,, ill be the probability C BB > B >, this hich is conformed to that hich precedes. In the case of to players A and B, e ill have, for the probability of player A, BÐB Ñ BÐB ÑÐB ÑâÐB B Ñ B B : B; ; â ; $ÐB Ñ By changing : into ; and B into B, and reciprocally, e ill have B BÐB Ñ B ÐB ÑÐB ÑâÐB B Ñ B ; B : : â : $ÐB Ñ for the probability that player B ill in the game. The sum of these to expressions must be equal to unity, this hich e see evidently by giving them the folloing forms. The first expression can, by No. 37 of Book I, is transformed into this one Ú B B ; ÐB B ÑÐB B Ñ; Ý â B B : : : Û B ÐB B ÑâÐB Ñ ; Ý Ü $âðb Ñ : B à 8

and the second can be transformed into this one Ú B B : ÐB B ÑÐB B Ñ: Ý â B B ; ; ; Û B ÐB B ÑâÐB Ñ : Ý Ü $âðb Ñ ; B à The sum of these expressions is the development of the binomial B B Ð: ; Ñ and consequently it is equal to unity, because, A or B must in each trial, the sum : ; of their probabilities for this is unity. The problem hich e just resolved is the one hich e name the problem of points in the Analysis of chances. The chevalier de Méré proposed it to Pascal, ith some other problems on the game of dice. To players of hom the skills are equal have put into the game the same sum; they must play until one of them has beat a given number of times his adversary; but they agree to quit the game, hen there lacks yet B points to the first player in order to attain this given number, and hen there lacks B points to the second player. We demand in hat ay they must share the sum put into the game. Such is the problem that Pascal resolved by means of his arithmetic triangle. He proposed it to Fermat ho gave the solution to it by ay of combinations, this hich caused beteen these to great geometers a discussion, to the continuation of hich Pascal recognized the goodness of the method of Fermat, for any number of players. Unhappily e have only one part of their correspondence, in hich e see the first elements of the theory of probabilities and their application to one of the most curious problems of this theory. The problem proposed by Pascal to Fermat reverts to determine the respective probabilities of the players in order to in the game; because it is clear that the stake must be shared beteen the players proportionally to their probabilities. These probabilities are the same as those of to players A and B, ho must attain a given number of points, B being the number of those hich player A lacks, and B being the number of those hich player B lacks, by imagining an urn containing to balls of hich one is hite and the other black, both carrying the no. 1, the hite ball being for player A, and the black ball for player B. We dra successively one of these balls, and e return it into the urn after each draing. By naming C BB the probability that player A ill attain, the first, the given number of points, or, that hich reverts to the same, that he ill have B points before B has B, e ill have 9

CBB œ CB B CBB à because, if the ball that e extract is hite, CBB is changed into CB B, and if the ball extracted is black, CBB is changed into CBB, and the probability of each of these events is ; e have therefore the preceding equation. The generating function of C BB in this equation in the partial differences is, by No. 20 of Book I, Q > > Q being an arbitrary function of >. In order to determine it, e ill observe that C!! can not have place, since the game ceases hen one or the other of the variables B and B is null; Q must therefore have for factor >. Moreover C!B is unity, hatever be B, the probability of player A is changing then into certitude; 3 > no the generating function of unity is generally >, because the coefficients of the poers of > in the development of this function are all equal to unity; in the present case, C!B being able to have place hen B is either 1, or 2, or 3, etc., 3 > must be equal to unity; the generating function of C!B is therefore equal to > ;! this is the coefficient of > in the development of the generating function of CBB or in e have therefore this hich gives Q > > à Q > > œ > Q œ consequently the generating function of > Ð > Ñ Ð > Ñ C BB is 10

> Ð > Ñ Ð >. ÑÐ > > Ñ By developing it ith respect to the poers of >, e have $ > > > > â > > Ð > Ñ $ Ð > Ñ$ The coefficient of > B in this series is > B Ð > ÑÐ > Ñ à B C is therefore the coefficient of > BB > Ð > ÑÐ > Ñ œ B in this last quantity; no e have B BÐB Ñ $ BÐB ÑÐB ÑâÐB B Ñ B > B> > â > â B $ÐB Ñ > By reducing into series the denominator of this last fraction and multiplying the numerator by this series, e see that the coefficient of > B in this product is that hich this numerator becomes hen e make > œ ; e have therefore Ú B BÐB Ñ BÐB ÑÐB Ñ â Ý $ $ CBB œ Û Ý BÐB ÑâÐB B Ñ $ÐB Ñ B Ü à a result conformed to that hich precedes. We imagine presently that there is in the urn a hite ball carrying the no. 1, and to black balls, of hich one carries the no. 1, and the other carries the no. 2, the hite ball being favorable to A, and the black balls to his adversary, each ball diminishing by its value the number of points hich lack to the player to hich it is favorable. C BB being alays the probability that player A ill attain first the given number, e ill have the equation in the partial differences 11

CBB œ CB B CBB CBB à $ $ $ because, in the folloing draing, if the hite balls exits, CBB becomes CB B; if the black ball numbered 1 exits, CBB becomes CBB, and if the black ball numbered 2 exits, CBB becomes CBB, and the probability of each of these events is $ The generating function of C BB is Q > > > $ $ $ Q being an arbitrary function of >, and in the present case is equal to so that the generating function of > Ð > > Ñ > $ $ C BB is > Ð $ > $ > Ñ Ð > ÑÐ > > > Ñ $ $ $ The coefficient of > B in the development of this function is > $ B > Ð > > > Ñ B $ $ $ and there results from this that e come to say that the coefficient of development of this last quantity is equal to Ú $ Ý B> Ð > Ñ BÐB Ñ > Ð > Ñ > $ $ Û $ B $ Ý BÐB ÑÐB Ñ > Ð > Ñ Ü â $ $ $ à à > B in the by rejecting from the development in this series all the poers of > superior to B >, and supposing in this that e conserve > œ, this ill be the expression of C BB. It is easy to translate this process into formulae. Thus, by supposing B even and equal to <, e find 12

< BÐB Ñ BÐB ÑâÐB < Ñ CBB œ B â $ B Œ Œ $ $ $ < $ BÐB ÑâÐB <Ñ Ð< Ñ< Ð< Ñ< Ð< Ñ â $Ð< Ñ$ B < $< BÐB ÑâÐB < Ñ Ð< ÑÐ< Ñ Ð< Ñ â $Ð< Ñ$ B < $Ð< Ñ â BÐB ÑâÐB <Ñ $Ð< Ñ$ B < If e suppose B odd and equal to <, e ill have < BÐB Ñ BÐB ÑâÐB < Ñ CBB œ B â $ B Œ Œ $ $ $ < $ BÐB ÑâÐB <Ñ Ð< Ñ< Ð< Ñ<$ Ð< Ñ â $Ð< Ñ$ B < $Ð< Ñ BÐB ÑâÐB < Ñ Ð< ÑÐ< Ñ Ð< ÑÐ< Ñ& Ð< Ñ â $Ð< Ñ$ B < $Ð< Ñ â BÐB ÑâÐB < Ñ $Ð< Ñ$ B < Thus, in the case of B œ and B œ &, e have C œ & $&! (* We imagine further that there are in the urn to distinguished hite balls, as the to black balls, by the nos. 1 and 2; the probability of player A ill be given by the equation in the partial differences CBB œ CB B CB B CBB CB B The generating function of C BB is then, by No. 20 of Book I, 13

Q R> > > > > Q and R being to arbitrary functions of >. In order to determine them, e ill observe that C!B is alays equal to unity, and that it is necessary to exclude in Q the null poer of > ; e have therefore Q œ > Ð > > Ñ > In order to determine R, e seek the generating function of C B. If e observe that C!B is equal to unity, and that, player A having no more need of a point, he ins the game, either that he brings forth the hite ball numbered 1 or the hite ball numbered 2, the preceding equation in the partial differences ill give CB œ CB CB We suppose C œ C à e ill have B B CB œ CB CB The generating function of this equation is 7 8> > > 7 and 8 being to constants. In order to determine them, e ill observe that C! œ!, and that consequently C! œ, this hich gives 7 œ. The generating function of C B is therefore 8> > > We have next evidently C œ, this hich gives C œ à C is the coefficient of > in the development of the preceding function, and this coefficient is 8 ; e have therefore 8 œ, or 8 œ The generating function of unity is >, because here all the poers of > can be admitted; e have thus 14

> > > or > > Ð > ÑÐ > > Ñ for the generating function of CB This same function is the coefficient of > in the development of the generating function of C BB, a function hich, by that hich precedes, is this coefficient is by equating it to e ill have The generating function of > > Ð > > Ñ R> > > > > > R Ð > ÑÐ > > Ñ > > > Ð > ÑÐ > > Ñ C BB R œ is thus If e develop into series the function > > > Ð > > Ñ >> > Ð > > > > Ñ ( ) à à e ill have > Ð > > Ñ >> > Ð > > > > Ñ > ( ) 15

Ð >Ñ>> Ú $ $ $ > Ð > Ñ > Ð > Ñ > Ð > Ñ â >Ð >Ñ $ $ $ > Ð > Ñ > Ð > Ñ > Ð > Ñ â Ý ) Û > Ð >Ñ $ $ $& $ $ > Ð > Ñ > Ð > Ñ > Ð > Ñ â $ $ $ $ > Ð >Ñ & &' $ $ > Ð > $ Ñ > Ð > Ñ > Ð > Ñ â $ $ Ý Ü â à B If e reject in this series all the poers of > other than > and all the poers of > B superior to >, and if in that hich remains e make > œ, > œ, e ill have the expression of CBB hen B is equal or greater than unity; hen B is null, e have C!B œ. It is easy to translate this process into formulae, as e have made for the preceding case. We name DBB the probability of player B; the generating function of DBB ill be that hich the generating function of C BB becomes hen e change in it > into >, and reciprocally, this hich gives, for this function, >Ð > > Ñ >> ( >) Ð > > > > Ñ By adding the to generating functions, their sum is reduced to in hich the coefficient of > > >> > > Ð >ÑÐ > Ñ B B > > is unity; thus e have C D œ BB this hich is clear besides, since the game must be necessarily on by one of the players. BB 9. We imagine in an urn < balls marked ith the n 1, < balls marked ith n 2, < balls marked ith n 3, and so on in sequence to the n 8. These balls being ell mixed in the urn, one dras them successively; one requires the probability 16

that there ill come forth at least one of these balls at the rank 1 indicated by its label 2, or that there ill come forth of them at least to, or at least three, etc. We seek first the probability that there ill come forth at least one of them. For this, e ill observe that each ball can come forth at its rank only in the first 8 draings; one can therefore here set aside the folloing draings; no the total number of balls being <8, the number of their combinations 8 by 8, by having regard for the order that they observe among themselves, is, by that hich precedes, <8Ð<8 ÑÐ<8 ÑâÐ<8 8 Ñà this is therefore the number of all possible cases in the first 8 draings. We consider one of the balls marked ith the n 1, and e suppose that it comes forth at its rank, or the first. The number of combinations of the <8 other balls taken 8 by 8 ill be Ð<8 ÑÐ<8 ÑâÐ<8 8 Ñà this is the number of cases relative to the assumption that e just made, and, as this assumption can be applied to < balls marked ith n 1, one ill have <Ð<8 ÑÐ<8 ÑâÐ<8 8 Ñ for the number of cases relative to the hypothesis that one of the balls marked ith the n 1 ill come forth at its rank. The same result takes place for the hypothesis that any one of the 8 other kinds of balls ill come forth at the rank indicated by its label. By adding therefore all the results relative to these diverse hypotheses, one ill have ( a ) <8Ð<8 ÑÐ<8 ÑâÐ<8 8 Ñà for the number of cases in hich one ball at least ill come forth at its rank, provided hoever that one removes from them the cases hich are repeated. In order to determine these cases, e consider one of the balls of the n 1, coming forth first, and one of the balls of the n 2, coming forth second. This 1 Translator's note: This means that a ball marked ith 1 ill be dran first, a ball marked ith 2 ill be dran second, and so on. In other ords, balls ill be dran consecutively by number. 2 Translator's note : The ord here is numéro, number. Hoever, this refers to the use of a number as a label. In order to distinguish it from nombre, number or quantity, I choose to render it as such. 17

case is contained tice in the preceding number; for it is contained one time in the number of the cases relative to the assumption that one of the balls labeled 3 1 ill come forth at its rank, and a second time in the number of cases relative to the assumption that one of the balls labeled 2 ill come forth at its rank; and, as this extends to any to balls coming forth at their rank, one sees that it is necessary to subtract from the number of the cases preceding the number of all the cases in hich to balls come forth at their rank. 8Ð8 Ñ The number of combinations of to balls of different labels is < ; for the number of the labels being 8, their combinations to by to are in number 8Ð8 Ñ, and in each of these combinations one can combine the < balls marked ith one of the labels ith the < balls marked ith the other label. The number of combinations of the <8 balls remaining, taken 8 by 8, by having regard for the order that they observe among themselves, is Ð<8 ÑÐ<8 $ÑâÐ<8 8 Ñà thus the number of cases relative to the assumption that to balls come forth at their rank is 8Ð8 Ñ < Ð<8 ÑÐ<8 $ÑâÐ<8 8 Ñà subtracting from it the number ( a), one ill have ( a' ) Ú<8Ð<8 ÑÐ<8 ÑâÐ<8 8 Ñ Û 8Ð8 Ñ Ü < Ð<8 ÑÐ<8 $ÑâÐ<8 8 Ñ for the number of all the cases in hich one ball at least ill come forth at its rank, provided that one subtracts again from this function the repeated cases, and that one adds to them those hich are lacking. These cases are those in hich three balls come forth at their rank. By naming 5 this number, it is repeated three times in the first term of the function ( a' ); for it can result, in this term, from three assumptions of each of the three balls coming forth at its rank. The number 5 is likeise contained three times in the second term of the function; for it can result from each of the assumptions relative to any to of the three balls coming forth at their rank. Thus, this second 3 : The ord is, numbered. I have chosen to render it as such Translator's note numérotées for the same reason as above. 18

term being affected ith the sign, the number 5 in not found in the function ( a' ); it is necessary therefore to add it to ( a' ) in order that it contain all the cases in hich one ball at least comes forth at its rank. The number of combinations of 8Ð8 ÑÐ8 Ñ 8 labels taken three by three is $, and, as one can combine the < balls of one of these labels of each combination ith the < balls of the second label and ith the < balls of the third label, one ill have the total number of combinations 8Ð8 ÑÐ8 Ñ in hich three balls come forth at their rank, by multiplying $ < $ by Ð<8 $ÑÐ<8 ÑâÐ<8 8 Ñ a number hich expresses that of the combinations of the <8 $ balls remaining, taken 8 $ by 8 $, by having regard for the order that they observe among themselves. If one adds this product to the function ( a' ), one ill have ( a ) Ú <8Ð<8 ÑÐ<8 ÑâÐ<8 8 Ñ Ý 8Ð8 Ñ Û < Ð<8 ÑÐ<8 $ÑâÐ<8 8 Ñ Ý 8Ð8 ÑÐ8 Ñ $ Ü < Ð<8 $ÑÐ<8 ÑâÐ<8 8 Ñ $ This function expresses the number of all cases in hich one ball at least comes forth at its rank, provided that one subtracts from it again the repeated cases. These cases are those in hich four balls come forth at their rank. By applying here the preceding reasonings, one ill see that it is necessary again to subtract from the function ( a ) the term 8Ð8 ÑÐ8 ÑÐ8 $Ñ < Ð<8 ÑÐ<8 &ÑâÐ<8 8 Ñ $ By continuing thus, one ill have, for the expression of the cases in hich one ball at least comes forth at its rank, ( A) Ú <8Ð<8 ÑÐ<8 ÑâÐ<8 8 Ñ 8Ð8 Ñ < Ð<8 ÑÐ<8 $ÑâÐ<8 8 Ñ Ý 8Ð8 ÑÐ8 Ñ Û $ < Ð<8 $ÑÐ<8 ÑâÐ<8 8 Ñ $ 8Ð8 ÑÐ8 ÑÐ8 $Ñ < Ð<8 ÑÐ<8 &ÑâÐ<8 8 Ñ Ý $ Ü â 19

a series being continued as far at it can be. In this function, each combination is not repeated: thus the combination of = balls coming forth at their rank is found here only one time; for this combination is contained = times in the first term of the function, since it can result from each of the = balls coming forth at its rank; it is subtracted times in the second term, since it can result from to by to =Ð= Ñ =Ð= ÑÐ= Ñ $ combinations of the = balls coming forth at their rank; it is added times in the third term, since it can result from the combinations of = letters taken three by three, and so in sequence; it is therefore, in the function (A), contained a number of times equal to =Ð= Ñ =Ð= ÑÐ= Ñ = â $ and consequently equal to Ð Ñ =, or to unity. By dividing the function (A) by the number <8Ð<8 ÑÐ<8 ÑâÐ<8 8 Ñ of all possible cases, one ill have, for the expression of the probability that one ball at least ill come forth at its rank, (B) Ú Ð8 Ñ< Ð8 ÑÐ8 Ñ< Ý Ð<8 Ñ $Ð<8 ÑÐ<8 Ñ Û $ Ð8 ÑÐ8 ÑÐ8 $Ñ< Ý â Ü $Ð<8 ÑÐ<8 ÑÐ<8 $Ñ We seek no the probability that = balls at least ill come forth at their rank. The number of cases in hich = balls come forth a their rank is, by that hich precedes, ( b) 8Ð8 ÑÐ8 ÑâÐ8 = Ñ = < Ð<8 =ÑÐ<8 = ÑâÐ<8 8 Ñ $ = provided that one subtracts from this function the cases hich are repeated. These cases are those in hich = balls come forth at their rank, for they can result, in the function, from = balls taken = by = ; these cases are therefore repeated = times in this function; consequently it is necessary to subtract them = times. No the number of cases in hich = balls come forth at their rank is 8Ð8 ÑÐ8 ÑâÐ8 =Ñ = < Ð<8 = ÑÐ<8 = ÑâÐ<8 8 Ñ $Ð= Ñ 20

By multiplying it by = and subtracting it from the function ( b), one ill have Ú 8Ð8 ÑÐ8 ÑâÐ8 = Ñ Ý = < Ð<8 =ÑÐ<8 = ÑâÐ<8 8 Ñ $ = ( b' ) Û Ý =Ð8 =Ñ< Ü Ð= ÑÐ<8 =Ñ In this function, many cases are again repeated, namely, those in hich = balls come forth at their rank; for they result, in the first term, from = balls coming forth at their rank and taken = by = ; they result, in the second term, from = balls coming forth at their rank and taken = by =, and moreover multiplied by the factor =, by hich one has multiplied the second term. They are therefore contained in this function the number of times Ð= ÑÐ= Ñ =Ð= Ñà thus it is necessary to multiply by unity, less this number of times, the number of cases in hich = balls come forth at their rank. This last number is 8Ð8 ÑÐ8 ÑâÐ8 = Ñ = < Ð<8 = ÑÐ<8 = $ÑâÐ<8 8 Ñà $Ð= Ñ the product in question ill be therefore 8Ð8 ÑÐ8 ÑâÐ8 = Ñ = =Ð= Ñ < Ð<8 = ÑâÐ<8 8 Ñ $Ð= Ñ By adding it to the function ( b' ), one ill have Ú 8Ð8 ÑÐ8 ÑâÐ8 = Ñ = < Ð<8 =ÑÐ<8 = ÑâÐ<8 8 Ñ Ý $ = Ú = Ð8 =Ñ< ( b Û) Ý Ð= Ñ Ð<8 =Ñ Û Ý Ý Ü Ü = Ð8 =ÑÐ8 = Ñ< = Ð<8 =ÑÐ<8 = Ñ à This is the number of all possible cases in hich = balls come forth at their rank, provided that one subtracts from it again the cases hich are repeated. By continuing to reason so, and by dividing the final function by the number of all possible cases, one ill have, for the expression of the probability that = balls at 21

least ill come forth at their rank, ( C) Ú = Ð8 ÑÐ8 ÑâÐ8 = Ñ< $=Ð<8 ÑÐ<8 ÑâÐ<8 = Ñ Ý Ú Û = Ð8 =Ñ< = Ð8 =ÑÐ8 = Ñ< Ý = <8 = = Ð<8 =ÑÐ<8 = Ñ Û $ Ý = Ð8 =ÑÐ8 = ÑÐ8 = Ñ< Ý â Ü Ü = $ $Ð<8 =ÑÐ<8 = ÑÐ<8 = Ñ à One ill have the probability that none of the balls ill come forth at its rank by subtracting the formula (B) from unity, and one ill find, for its expression, Ð$<8Ñ 8<Ò$Ð<8 ÑÓ $ <8 One has, by n 334 of Book I, hatever be 3, 8Ð8 Ñ $3 œ ( B 3.B- B < Ò$Ð<8 ÑÓ â the integral 5 being taken from B null to B infinity. The preceding expression can therefore be put under this form ( o) ' B.BÐB <Ñ - ' B <8.B- B <8 8 8 B We suppose the number <8 of balls in the urn very great; then, by applying to the preceding integrals the method of n 246 of Book I, one ill find very nearly for the integral of the numerator, 8 È <8 1 \ ˆ < \ \ - È8\ 8Ð< ÑÐ\ <Ñ \ being the value of B hich renders a maximum the function <8 8 8 B B ÐB <Ñ - The equation relative to this maximum gives for \ the to values 4 pages 128 137. 5 : The constant denotes, the base of the natural logarithm. Translator's note - / 6 pages 94 96. 22

È <8 < < Ð8 Ñ <8 \ œ One can consider here only the greatest of these values hich is, to the quantities nearly of the order, equal to 8 <8 <8 8 à then the integral of the numerator of the function ( o) becomes nearly È <8 <8 Ð<8Ñ - ˆ 8 1 È 8 <. ÉÐ< ÑÐ Ñ The integral of the denominator of the same function is, by n 33, quite nearly, the function ( o) becomes thus One can put it under the form È Ð<8Ñ 8 1 <8 <8 8 - à ˆ È 8 < ÉÐ< ÑÐ Ñ ˆ 8 8 8 8 <8 <8 ÉÐ Ñ <8 being supposed a very great number, this function is reduced quite nearly to this very simple form Œ 8 8 This is therefore the expression approached more and more by the probability that none of the balls of the urn ill come forth at its rank, hen there is a great number of balls. The hyperbolic logarithm of this expression being 8 â 8 $8 one sees that it alays goes on increasing in measure as 8 increases; that it is à 23

null, hen 8 œ, and that it becomes -, hen 8 is infinity, - being alays the number of hich the hyperbolic logarithm is unity. We imagine no a number 3 of urns each containing the number 8 of balls, all of different colors, and that one dras successively all the balls in each urn. One can, by the preceding reasonings, determine the probability that one or more balls of the same color ill come forth at the same rank in the 3 draings. In reality, e suppose that the ranks of the colors are settled after the complete draing of the first urn, and e consider first the first color; e suppose that it comes forth the first in the draings of the 3 other urns. The total number of combinations of the 8 other colors in each urn is, by having regard for their situation among themselves, $Ð8 Ñà thus the total number of these 3 combinations relative to 3 urns is Ò$Ð8 ÑÓ à this is the number of cases in hich the first color is dran the first altogether from all these urns, and, as there are 8 colors, one ill have 8Ò$Ð8 ÑÓ 3 for the number of cases in hich one color at least ill arrive at its rank in the draings from the 3 urns. But there are in this number some repeated cases; thus the case here to colors arrive at their rank in these draings are contained tice in this number; it is necessary therefore to subtract them from it. The number of these cases is, by that hich precedes, 8Ð8 Ñ 3 Ò$Ð8 ÑÓ à by subtracting it from the preceding number, one ill have the function 8Ò$Ð8 ÑÓ 8Ð8 Ñ 3 3 Ò$Ð8 ÑÓ But this function contains itself some repeated cases. By continuing to exclude from them as one has made above relatively to a single urn, by dividing next the final function by the number of all possible cases, and hich is here Ð$8Ñ 3, one ill have, for the probability that one of the 8 colors at least ill come forth at its rank in the 3 draings hich follo the first, â 83 Ò8Ð8 ÑÓ 3 $Ò8Ð8 ÑÐ8 ÑÓ3 an expression in hich it is necessary to take as many terms as there are units in 24

8. This expression is therefore the probability that at least one of the colors ill come forth at the same rank in the draings of 3 urns. 25