PROBABILITÉ. Condorcet

Transcription

1 PROBABILITÉ Codorcet Ecyclopédie Méthodique. Mathématiques. Tome II e, Partie (1785) PROBABILITÉ. We will limit ourselves to give here the geeral priciples of the calculus of probabilities, of which oe fids some applicatios i diverse articles. I. 1. The fudametal priciple of this calculus ca be epressed thus. Let A be a evet, & N aother evet cotradictory to the first (that is to say, which, uder the hypothesis, caot eist at the same time); let epress the total umber of the equally possible combiatios, m the oe of the combiatios which give the evet A, m that of the combiatios which give the evet N, m will epress the probability of the evet A, & m that of the evet N. = m + m.. If oe has three evets A, N, N, if is always the total umber of combiatios, m the oe of the combiatios which give A, m the oe of the combiatios which give N, m the oe of the combiatios which give N, the probability of A will be m, that of N will be m m, that of N, & oe will have = m + m + m. It is easy to see that this secod defiitio is a sequel to the first; i effect, here the probability of A is, by this first defiitio, m m, & that of ot havig A, m = m + m ; but, if oe has ot A, the probability of N is m m, & that of N m m : therefore that of N will be, i geeral, m m m = m m, & that of N m m = m. Thus, i the followig of our reflectios o this first priciple, we will cosider oly two evets. 3. If m = 0, = m, & the probability of A is m = 1; but, if o possible combiatio gives the evet N, this evet is therefore impossible, the evet A will arrive therefore ecessarily; thus, 1 will epress the probability of a ecessary evet, or certitude; likewise oe of the two evets A or N arrive ecessarily, & the sum of their probabilities is m + m = 1. This which leads still to the same coclusio. 4. It follows from the same defiitio that, if oe takes ay umber t whatever of successive combiatios of the evets A & N, the probability of each will be epressed by the sequece of terms of the biomial m+m t ; so that the probability to have t times, t the evet A will be mt, that to have t 1 times the evet A, & oe time the evet N t will be t.mt 1 m, that to have the evet A t t times, & the evet N, t times will be t t.t 1...t t t m t t m t. Traslated by Richard J. Pulskamp, Departmet of Mathematics & Computer Sciece, Xavier Uiversity, Ciciati, OH. December 10, 009 1

2 5. We have called probability of a evet, the umber of the equally possible combiatios which give it, divided by the total umber of combiatios which give this evet or the cotradictory evet. Util ow this is a pure defiitio of ame, but oe iteds more: 1. That, if the umber m of the combiatios, which give A, surpasses the umber of the combiatios m which give N, so that the probability m > 1, there is place to believe that the evet A will arrive rather tha to believe that it will ot arrive.. That this motive to believe is so much greater as m is also greater, & approaches uity. 3. That it icreases proportioally i this same ratio. 6. We are goig to demostrate first that these three propositios deped o the 1 st, ad that it suffices eve that it be true for the case where m is much greater tha m, & we will epose what is, i this last case, the ature of the motive to believe that A will arrive rather tha N. First the third propositio depeds o the truth of the first two; we suppose i effect that the evets A or N ca arrive a umber t times; & that t = mp + m p, it follows from the defiitio of the 1st article, that the probabilities that the evet A will arrive t times, & the evet 0 times, the evet A, t 1 times, & the evet N oe time, & thus i sequece will be epressed by the terms of the biomial, so that a term ( m ) t, t ( m ) t 1 m t.t 1... t q q, t.t 1 ( m ) t q ( m ( m ) t ( m will epress the probability that the evet A will arrive t q times, & the evet N q times. This put, it is clear, by cosiderig the formula of the biomial, 1. that the greatest term will be, t.t 1... t m p m p ( m ) t m p ( m ) q ) m p ) = M. The most probable combiatio will be therefore that where the ratio of the umber of evets A to the oe of the evets N, will be the oe of mp to m p, that is to say, the same as the ratio of the probability of the evets.. That the greater will be p, the more it will be probable that the ratio of the umber of the evets A to the oe of the evets N, will deviate little from m m. Ideed, let a fractio t r of t, r beig ay whatsoever, oe will prove, by cosiderig the formula of the biomial, that, if oe takes the umber t r, so much above as below from this mea term M, & if oe divides successively the rest of the terms i the umber of r t r t ito t r parts of r 1 terms each, oe will be able to take t great eough, i order that each of these t r terms the earest to the term M surpasses, i such ratio as oe will wish, the r 1 which correspod to it (See the 4 th part of the Ars Cojectadi of Jacques Beroulli. Besides this demostratio has o difficulty by

3 employig the theorem of Stirlig & of Mr. Euler o the idefiite products of a very great umber of terms.) If therefore, havig m > m, I judge that the evet m will arrive rather tha m, I will have a probability as great as oe will wish, from a idefiite umber of judgmets, the ratio of the umber of true judgmets, a total umber will deviate very little from m ; therefore, if a greater probability gives me a motive to judge coformably to this probability, the motive must be proportioal to it. The truth of the d propositio that this motive must be stroger, if the probability is greater, depeds yet o the truth of the first. I effect, callig u the probability of A & e that of N, the probability of beig mistake oly p times out of t judgmets, i prooucig that oe will have A will be epressed by u t + t.u t 1 e + t.t 1 u t e + t.t 1... t p + 1 u t p e p = V p Now, if oe supposes to V & to t a costat value, it is easy to see that the greater u will be, the smaller will be p; therefore, by prooucig that the evet A will take place rather tha N, oe will have a equal probability to be mistake so much less ofte as u will be greater, & cosequetly, the greater u will be, the more the motive to judge that the evet A will take place, will have to be strog. It remais therefore to prove oly that a greater probability i favor of a evet A, is a motive to believe that A will take place rather tha N, & for that it suffices to prove that this motive eists whe this probability is very great; i effect, preservig the precedig demostratios, oe will have the probability that A will take place more ofte tha N i t + 1 judgmets, epressed by u + (u e)(ue + 3u e u3 e 3 + t 1.t... t + 1 u t te t ); t 1 which, whe u > e, approaches cotiuously from uity i measure as t icreases: therefore oe will have a probability as great a oe will wish, at least to be mistake i believig that A must arrive more ofte tha N, i a very great umber of judgmets; therefore, if this very great probability is a motive to believe; however little that u eceeds e, oe will have still a motive to judge that A will take place rather tha N. 7. There remais to us therefore ow oly to eamie the ature of the motive which carries us to judge coformably to a very great probability. I suppose that I have i a ur a billio white tickets, & oe black, &, 1. that oe has draw o ticket;. that oe has draw a white ticket, & that it is ot eposed; 3. that oe has draw a black ticket which is ot eposed either. It is clear that, i the three hypotheses, I have the same probability that the ticket is white; however the fact is ucertai i the first case, it is certaily true i the secod, & certaily false i the third. Aother who will have see the tickets would be able to be sure respectig the falsity of a evet of which the truth is for me very probable; there is therefore o real liaiso betwee the probability of a future or ukow evet, & reality. Suppose therefore ow a ur which cotais the black tickets & the white tickets i a ukow ratio, that I have draw a umber t of white tickets without a sigle black, 3

4 & that I seek the probability to brig forth i a followig trial a white ticket rather tha a black. Let be the umber of tickets, & that of the white tickets, the umber of combiatios which give t + 1 white tickets will be t+1, the umber of combiatios t+1 which give t white tickets & oe black, will be t, & the umber of all the possible t+1 combiatios uder the hypothesis, will be cosequetly t+1 + t = t t+1 ; 1 but t is susceptible of all the values from = 0, to = ; therefore t+1, the itegrals t beig take from = to = 0, will epress the probability to have a white ticket. Whece t+1 = t+ t + at+1 + b. t t c. t + 1.t.t 1 t t = t+1 t + 1 at + b t t.t 1.t.3 t 1 c. t t+1 t+.t+1 t + 1 a. = t t + a. t+.t+1 = t + 1 t + t+ 1 a. 1 a t+1 + b t+.t b. t+.t+1.t.3. t+.t+1 + b..3. t+.t + b..3 c t+.t+1.t.t c t+.t+1.t.t 1.t Now it is easy to see that, if we suppose icomparably greater tha t, t + 1 & t + 1 ( t + t + 1 a ) = t + 1 ( t ), will epress the approimate values, the oe greater, the other smaller of this probability. We suppose fially t very great, t+1 t+1 t+ or t+ (1 ) 1 will give us a very great probability. If we eamie at preset what motive we have to believe accordig to this probability, we will fid that it is the same which carries us to believe that a fact arrived steadily will cotiue to arrive agai. But this motive is the oe which makes us admit this geeral priciple to us, that the atural evets are subject to some costat laws, sice we ca base this opiio oly respectig the observatio of the order of the past evets, & o the assumptio that it will cotiue to be the same for future evets. This motive is still the same which makes us believe i the persistece of objects which strike our sese, & cosequetly i the eistece of bodies. It is agai by the same motive that we believe the truth of the demostratios of which we caot embrace the chai by a sigle glimpse; ideed, we caot be sure that a propositio of which we ourselves remember to have followed the demostratio is true, that by log eperiece that our memory does ot deceive us i this case; & that as ofte as we have wished to follow aew the same reasoigs, they have led us to the same results. If therefore oe ecepts the ituitive kowledge of the propositios which our mid ca embrace immediately, all our kowledge respectig ature, all the propositios accordig to which we guide our behavior & all our movemets, & eve as far as the 1 Traslator s ote: The right-had side has bee corrected from t. 4

5 best demostrated mathematical truths, ca ot have for us veritable certitude, & we have o other motive to believe them, tha this tedecy to regard as costat that which has bee costatly observed, that is to say, as a very great probability. Oe sees oly to be bor here differet orders of truths; ideed, i mathematical demostratios, for eample, this motive acts oly i order to make us suppose that the truths which have bee demostrated to us oce, will appear always to us, istead that, i some other kids of kowledge, we have eed of the same motive i order to suppose the reality of the same facts o which we reaso, the o the observed order i these facts, &c. &c. This motive is for us a atural tedecy, which is cofouded eve sometimes with sesatio. It is by virtue of this motive that two me, see at some distaces doubles the oe of the other, appear sesibly equal, although beig see uder some differet agles, the more eteded must appear oe time smaller; it is i effect eperiece which aloe has bee able to mi i our sesatio a secret judgmet which is cofouded with it. It is by virtue of this motive that, if I roll a ball betwee two crossed figers, I sese really two balls, while my reflectio, supported out of some more costat eperieces, forces me to believe that there is oly oe of them. Oe sees, i this last eample, how this tedecy, which seems i itself purely atural, & proportioal perhaps to the force of the impressio of the objects, ca however cede to reaso, but without beig destroyed, & eve without havig o loss of force. We will ot carry further the cosequeces of this observatio, which we believe very importat, & which ca serve to eplicate may pheomea relative to the force of prejudices, to the power of reaso, to liberty, &c. It suffices us here to have show, 1. that the cosequeces which oe draws from the calculus of probabilities, relative to the reality of the objects, are some truths of the same kid as those which are bor of observatios & of the reasoigs which oe makes accordig to them.. That they differ from them oly i this poit, that oe kows the, by the calculus, the value of the motive which carries us to believe, & which oe has the veritable measure of it, istead to cede uiquely to a atural tedecy, which, i may circumstaces, ca deceive us. II. 1. The secod priciple of the calculus of probabilities is this oe. Suppose that we have may evets A, A, A,... of which the probabilities are p, p, p,... & that e, e, e,... represet the values or the effects of these evets, effects or values which oe supposes of the same kid; the mea value of the evet A pe p e p+p +p, as probable will be epressed by p+p +p ; that of the evet A it will be by & that of the evet A p by e p+p +p, & the mea value of the evets or of ay evet whatever which will ecessarily arrive, will be pe+p e +p e p+p +p, the values of e, e, e ca be of differet sigs. Thus, for eample, if I have the probability 1 to wi a écu, the probability 1 4 to wi a half-écu, & the probability 1 4 to lose two écus; the value of the epectatio to wi by the evet A, will be cosequetly 1 écu, & my total epectatio to wi will be écu or 1 8 écu. 5

6 I order to demostrate this rule, it suffices to observe that oe ca cosider the evet A havig p possible combiatios which produce it as p evets of the same value each havig oe combiatio; that it will be the same of the other evets, & that thus the rule is reduced to takig for the mea value of a certai umber of equally possible evets, the sum of their values, & to divide it by the umber of these evets.. Thus it is ot at all agaist this rule i itself, but agaist the usage which oe ca make of it that there is raised some objectios of which some have bee isoluble util ow. We suppose two me A & B play together, with the coditio such that A has a probability p to wi a sum s, & B a probability p to wi the same sum, & that p + p = 1; the mea value of the epectatio of A will be, by the precedig rule, equal to ps, & that of B equal to p s. If ow this sum has ought to be furished by A & B at the begiig of the game, & if oe demads i what proportio they must furish it i order to play at a equal game, either with respect to the other player, or i a arbitrary maer; oe will respod that each must give a sum equal to the value of their epectatios, that A cosequetly must give ps & B, p s. What does oe ited ow by a equal game? 1. this is ot that the lot of the players is the same as before the game, sice, eve by supposig p = p & the lot of the players absolutely similar, each before the game, is 1 s, & that accordig to the game the oe will certaily have s, & the other 0.. It is ot that the state of the two players is similar; because player A, after the game, will have a sum p s of gai, or a sum ps of loss, & player B a sum ps of gai, or p s of loss, so that their state is essetially differet, ecept whe oe has p = p. 3. Oe uderstads therefore that the players havig agreed to chage the state by puttig ito the game, & before, after the game, to have a state differet from the first, & differet also for each player, oe has proportioed their stakes i a way that it is either betwee their state before playig, & their state after the game, or betwee the state of them after the game, the greatest equality that the ature of the thigs ca permit. Now here this equality ca be established oly by supposig the game repeated a great umber of times; & the oe ca require, 1. that the most probable case is precisely the oe which chages othig i the state of the two players;. that the probabilities to wi or to lose for A as for B approaches more ad more to be equal to 1, i measure as the umber of trials is multiplied. 3. Fially, let oe have, uder the same hypothesis, a probability always icreasig, that the loss of A, or that of B, will ot eceed, either a fied sum, or a proportioal part of the total stake, if the precedig coditio ca ot be eecuted. 4. Let therefore, 1. tp + p be the umber of trials, the most probable evet or the greatest term of the series p + p tp+p will be t.p + p.tp + p 1... tp + 1 p tp p tp tp Now let be the wager of A, & s that of B, tps t will epress that A will have wo; therefore, makig tps t = 0, oe will have = ps for the value of the stake of A. 6

7 . I supposig the same law established, it is clear that, i p + p tp+p all the combiatios where the epoet of p is greater tha tp, will give a positive sum for the gai of A, & that all the terms where this epoet is below, will give a positive sum for the gai of B; ow, the more t will be great, the more the sum of these terms which are favorable to A, & that of the terms which are favorable to B, will approach the oe & the other to the quatity 1 ; i a maer that, supposig t always icreasig, if p > p, the sum of the terms favorable to A will be first greater tha 1, & will dimiish i brigig them together, istead that, if p < p, the sum of the same terms will be first smaller tha 1, & will icrease i brig them together. 3. If, i the sequece of terms of the formula p + p tp+p, oe takes all those where the epoets of p are above t.p a, whatever be a, i measure as t will become great, the sum of these terms will approach to uity to which it will become equal, if t is ifiity. Now let this epoet be t.p a, the loss of A will be tps ts.p a = ats; therefore the more oe will multiply the trials, the more A will have a cotiually icreasig probability of ot losig beyod tas, that is to say, beyod a fractio a p of his total stake tps, a beig a quatity as small as oe will wish, provided that it is fiite: &, sice the term where the power of p is p tp a, is the limit to where it is ecessary to go i order that the total sum of the precedig terms ca approach idefiitely to uity, i measure as t icreases (a beig always a quatity as small as oe will wish, but fiite), oe sees that this same coditio to have a probability always icreasig to ot lose beyod a certai sum, ca take place oly for a sum proportioal to ts. Oe sees therefore here ot oly that the established law put betwee the state of the players before or after the game, & betwee their respective states, the greatest equality possible, or the sole oe which ca be compatible with the differece of these states, but oe sees at the same time that with ay other, oe caot fulfill the same coditios. 5. That which we have said o the moey destied to form the sum s, that A & B have a uequal epectatio to wi, will be applied equally to the case where it will be ecessary to partitio the sum, the game beig supposed stopped, & their epectatios beig p & p if it had bee cotiued; &, i geeral, to the case where oe buys, for a give sum, the epectatio of aother sum, or else, where oe divides amog may persos a sum to which they have some more or less probable claims. 6. I the free agreemets, as the game, oe sees that this law must have bee established fially that there is o advatage to play or to ot play, to choose rather the lot of A tha the oe of B, & that oe is ot absolutely determied by some particular social covetios. This is very early that which arrives i commerce, where a commo price of commodities is established i a maer as such day, for eample, a septier de bled, a ell of such material equivalet to four ouces of silver, & that cosequetly the preferece give by certai persos to oe of these thigs over the other, will keep to their eeds or to their particular wishes, without that oe ca say, i geeral, that oe is preferable to the other. If the covetio is forced, the oe must adopt the same law, sice it is that i Traslator s ote: This is a uit of poor lad. 7

8 which it is most probable that there will result a smaller sum of ijustices from a great umber of distributios made by virtue of this law. See ABSENS. But there results from this that we just said, a remarkable differece betwee these two cases. I effect, i the secod where the agreemet is forced, the law must always be followed; but i the first, if the kid of equality that this law established does ot appear sufficiet, there must result from it that, as little as oe acts with prudece, oe will ot wish at all to form the agreemet. I the first case, oe decides accordig to the law because oe ca oly cosider the total mass of similar agreemets, & to seek to do so that there results from it the least possible iequality. I the secod, if oe wishes to act with prudece, if the object is importat, oe must led oeself to the agreemet oly as much as oe ca evisage the possibility to establish betwee the two parties a sufficiet equality. 7. This put, cosider two players, of whom oe A has a epectatio e to wi, & a risk 1 e to lose; & the other B a epectatio 1 e to wi, & a risk e to lose, & let the stake of A be to the stake of B as e is to 1 e; so that i wiig A will wi 1 e times, & that i losig he will lose e times a certai sum regarded as uity. If e < 1 e i measure as the umber of trials will be multiplied, the probability that A will have to wi will approach 1, but it will always remai below, & e ca be small eough i order that, eve for a great umber of trials, this probability is still quite iferior to 1, while the probability to lose that the same player would have, would always be quite above 1. I the same case, the probability to ot lose above a th part of the total stake, will icrease i favor of A whatever be ; but if is very small, it will be ecessary to suppose the game cotiued a very great umber of times i order that this probability becomes great eough. It is ecessary to observe et that, for the same player A, the probability to ot wi beyod a certai portio of his total stake, icreases at the same time as that to ot lose beyod the same proportio. It is likewise of it i all the cases which oe could choose, so that i geeral the oe of the two players who have the least probability, wis i the combiatio of the greatest umber of trials o the side of the epectatio of ot losig, & loses as much to the epectatio to wi much while to the cotrary the oe who has a great probability loses from the epectatio to wi, & at the same time is eposed to a smaller risk to lose much. 8. This maer to cosider the law which we eamie, & which cosists i regardig the value which results from it for the epectatios & for the risks, as a proper mea value to restore the greatest possibility equality betwee those who echage betwee them a certai value & a ucertai epectatio, or two ucertai epectatios, &c. has seemed to us to be able to make vaish most of the difficulties which this rule has appeared to preset i its applicatio. We are goig to eamie here some & we will begi with those which the famous problem of Petersburg presets. I this problem, oe supposes that a player A must give to a player B a coi if he brigs forth tails o the first toss, two if he brigs it forth o the secod, four if he brigs it forth o the third, & thus i sequece; & oe demads what is the value of the epectatio of B, or what sum he must give to A i order to play a equal game. The 8

9 rule of the calculus gives this sum equal to = 1 ( ) = A coclusio which appears so much more absurd, as this stake of B beig supposed greater tha ay give quatity; oe ca have a probability as great as oe will wish, that B will lose i this agreemet. But oe ca observe 1. that the case which becomes the most probable, by supposig that oe cotiues the game, the oe where there is either loss or gai, ca ot take place here, at least if oe does ot suppose the game repeated a ifiite umber of times.. That the probability of B to wi, will o loger approach to be equal to 1, & cosequetly to be equal to the probability of loss, but by supposig also the game repeated a ifiite umber of times; it begis eve to be fiite oly at this term. 3. That the probability to lose oly a certai part of the total stake as we have see should icrease with the umber of trials, is fiite for B oly by supposig ifiite the umber of times that the game is repeated, & that i this case this part of the stake is ecessarily still a ifiite quatity. Oe sees therefore that the priciple o which we have said that the geeral rule must be fouded, the oe to put the greatest possible equality betwee two essetially differet states, ca have o place here, sice this equality would require that oe embraces the combiatio of a ifiite umber of games, so that the limit which, i the ordiary problems is a ifiite umber of games is ecessarily here a ifiity of the secod order. It is therefore ot the rule which is i default, but the applicatio of the rule to a case which oe presets as real, & which however ca ot be, sice it supposes the reality of a ifiite sum, of a ifiite umber of trials i each game, & of a ifiite umber of games. Thus, the problem must be cosidered ot as a real case, but as the limit of the real questios of the same kid as oe ca have i view. This eplicatio however is ot yet satisfactory. Ideed, oe has remarked, with reaso, that the rule would appear to be i default eve whe oe would limit the umber of possible trials, because the sum that B must give to A uder this hypothesis i order to play at a equal game, is yet such, if the umber of trials is i the least great, that ay reasoable ma would risk to give it. Noetheless i most of the solutios give to this questio, oe is cofied to say that it was ecessary to limit the umber of trials, either because beyod a certai umber, it was ecessary to regard the probability as too small, or because it was ecessary that this umber was such that the wealth of A, or the sum that he reserves to this game, suffices to pay that which he must to B, if tails does ot arrive at the last trial. Such is therefore ow the case which remais to us to eamie, the oe where the umber of trials is fied, & where the sum which B must give, & the probability which he has to wi beig fiite, the problem becomes a real problem. We suppose that each game is limited to trials, & that oe pays 1 if tails arrives the first trial, if it arrives the secod, 4 if it arrives the third, 8 if it arrives the fourth... 9

10 1 if it arrives the th, & if it does ot arrive at all. The probabilities will be & the stake of B must be 1, 1 4, 1 8, 1 1, 1, = + 1, & we will fid first that B will begi to wi whe tails will arrive at a trial p, such that p 1 > + 1, or < p. If = p, the there is either loss or gai; but i the same case , or 1 1 p 1, epresses the probability that B has to p 1 lose. Suppose, for eample, p = 4 & = p = 14; the stake of B will be 8. We will have 7 8 for the probability that A will wi, 1 16 for the probability that he will have either loss or gai, & 1 16 for this that B will wi. At the same time because = 14, it will be possible that he wis 16376, i truth the probability of this gai will be oly O his side A will have a probability 16 of o loss, but he would wi oly 7 i the most favorable case, & could lose up to Oe sees therefore that there is a very great iequality betwee the positios of A & of B, by cosiderig oly a sigle trial, & that ot oly there are some circumstaces where either oe or the other must wish to coset to chage the state where they are before the game agaist the oe which results from this agreemet, but that this must take place early geerally. If oe cosiders et a sequece of games, the oe will seek to determie, either the sum regarded as uity, or the umber of trials, i a way 1. that the probability to wi for A & for B approaches to equality,. that oe has a great eough probability that either A or B i a umber m of games will lose beyod a value which is a give proportio with m. The umber of games must the be determied by the coditio to be such that it has a probability early equal to uity, or to certitude that the loss that A ca have will ot eceed at all his wealth, or the sum that oe believes that he will wish or is able to place ito the game. The same assumptio of certitude is the oly rigorous, it is the sole way that B is ot here at disadvatage. We take, for eample, a simpler case, the oe where of 100 tickets, B chooses 1 of them, & gives 1 to A, o the coditio that if this ticket arrives, A will give to him 100, & we suppose that oe plays 00 trials, the probability that A will wi will be epressed by ,000, that he will either lose, or gai by ,000, ad that B will wi by 100,000, the probabilities to wi for A & for B will be therefore here very early as 5 to 4, & cosequetly already eighborig to 35 equality. I the same case, the probability for A to ot lose beyod 500 will be 100,000, a risk already very small. 3 Oe sees therefore that provided that B has the epectatio to be able to play 00 trials, there is established i the game a sort of equality. It is true that the established 3 Traslator s ote: Codorcet has made a error here. Let X have a biomial distributio with parameters = 00 ad p =.01. Pr(B wis 100(k )) = Pr(X = k) for k = 0, 1, For A to lose more tha 500 requires B to wi more tha 500, amely, Pr(X 8) = Furthermore, each of the remaiig probabilities i this ad the followig paragraph are slightly i error. 10

11 law ca take place oly by supposig that if A would lose 00 times, he would pay , or 0,000; but eve though he would ot have them, as the probability that A will ot lose above 10,000, for eample, is the early equal to uity, & that i the other very rare cases, B would wi always 10,000; it is easy to see that eve though A would pay oly this sum, B would still coset to play this game, where he ca epect to wi 10,000 by riskig oly 00. B uder this hypothesis would keep besides a probability 3, ,000 to wi agaist a probability 40, ,000 to wi 100 agaist a probability 7, ,000 or more agaist a probability 13, ,000 18,145 to lose, a probability 100,000 to wi 00 14,188 to lose 100, & a probability 100,000 to lose 00. Thus, eve though B would ot have the absolute certitude that A would pay all the possible loss, his state i regard to A would coserve still a sufficiet equality. 9. It is ecessary however here to cosider two quite distict cases, the oe where, for eample, the two hudred trials above form a liked game, so that if A & B agree oe time to play it, they are egaged to cotiue the umber of trials; & uder this hypothesis, the state of each player, & the kid of equality which subsists betwee them, & which ca be regarded as sufficiet, is epressed as we just said it. But if A & B coserve the liberty to make at each trial the same covetio, there is moreover a observatio to make: sice it is by cosiderig at each time the system of future trials that A & B are determied to play, there results from it that they must regulate the stake regarded as uity, i a maer that at each trial they ca evisio as possible the umber of trials ecessary i order to establish a sufficiet equality, that is to say that it is ecessary that the wealth of each of the two, or the sum that oe has cause to believe that he would wish to risk, ca suffice to this umber of trials; thus i order to coserve the ecessary equality, the stake must chage after a certai umber of trials. I some of the possible combiatios, that is to say, i those where the wealth of oe of the two players is arrived to a value which obliges to this chage, if oe makes to eter this dimiutio of the stake i the calculus, oe will see that there must result from it ecessarily the possibility to play a much greater umber of trials; whece there must result also betwee the players a greater equality; because this kid of equality cosists i this that if oe cosiders the sequece of future trials, oe has a probability early equal for each of the players, to lose or to wi, & a very great probability that the loss or the gai of ay of the two, will ot eceed a very small part of the total stake: ow i this case, the first coditio holds as i the precedig, & the part of the total stake ca eve be, uder this last hypothesis, regarded as a give quatity. 10. The maer i which we have cosidered the established rule, ca eplicate also two cotradictory pheomea which are themselves preseted i the applicatios of this rule to some real cases. It happes equally, & let a reasoable ma A refuse to give a sum b for the probability to wi a sum a > b, & also let a reasoable ma B coset to give a sum b for the probability to wi a sum a < b. The first case takes place whe b is a cosiderable sum with respect to the state of the fortue of A, either i itself, & whe is very small. The secod takes place to the cotrary particularly whe b is a very small sum, & whe is a etremely small quatity. I the first case, although, if the game were supposed to be repeated a very great 11

12 umber of times, it was favorable to A, however he will refuse to play it; 1. because he ca ot cotiue it a great eough umber of times;. because for a sigle trial he has a very great probability to lose his stake, & by hypothesis, to make a loss which icoveieces him, or which deprives him of agreeable ejoymets. I the secod, B agrees to play, because the small sum b is a very moderate sum of which he does ot regret the loss, & of which the epectatio to wi the cosiderable sum b, egages him to epose himself, eve with disadvatage, to this loss regarded as light: this is here the case of the lotteries. There are some games where the stregth of the players is ot equal, & where oe gives advatage to a baker; as the baker is obliged to play a very cosiderable game, which requires some advaces, & eposes to the possibility of eormous losses; which besides he is subject for the stakes, to be submitted, with certai limits, at the will of the puters; & that fially if he would have o advatage, he would have, especially whe the umber of puters is great, & whe they play very early the same game, a very great probability to make oly very little loss & gai: it has appeared ecessary to accord to him a advatage which gave to him a assurace to wi at legth; & the puters have coseted to buy at this price the pleasure to play, & to coduct their game at their whim up to a certai poit. 11. Oe has observed that amog the games which deped altogether o chace & o good play, the oes had oly a very short duratio, while others coserved their vogue a very logtime: oe of the causes of this differece, is the way to combie i these games, the ifluece of chace & of good play, so that the differece i stregth of the players, whe it is small, alters ot at all sesibly i the two or three games which oe wishes to play i a day, the equality of the probability to wi, as they could have amog them of some equal players. If oe gives too much to chace, oe takes off to these games a great part of their pleasure; if the chace iflueces too little, the differece of stregth becomes too sesible, it humiliates the self-respect. We will ote fially, that i the eterprises where me epose themselves to a loss i view of a profit, it is ecessary that the profit be greater tha the oe which follows the geeral rule, it establishes equality: ideed, as i geeral oe is delivered ot at all as i the game, by the appeal of the pleasure to play, or as i the lotteries, by the epectatio to wi much with a small stake, oe ca have motive to risk, oly a advatage which, by evisioig a series of similar risks, produces a assurace great eough to wi, & a probability early equal to certitude of o loss at all beyod a certai part of the stake. These reflectios have appeared to us proper to accommodate the rule established i the calculus of probabilities, with the setimet & with the behavior of reasoable & prudet me, i most of the cases where this rule would appear at first glace to be cotrary to it. III. 1. Util here we have regarded the umber of combiatios which give each evet as determied & kow. We are goig ow to suppose this umber ukow & variable, so that it has o loger a determied probability of the evets, but oly a mea probability accordig to which oe ca determie that of their productio. 1

13 . Suppose, for eample, that oe has a ur cotaiig some black balls & some white balls, that oe has draw white & m black balls, & that oe demads what is the probability to draw p white balls & q black balls. Suppose moreover that the umber of these balls is ifiite, so that the ratio of the white balls, to the total umber, ca have all the values from 1 to 0. Let be this ukow ratio, the probability to draw first white balls, & m blacks, & et p whites & q blacks, will be m m p + q... p q m+p 1 p+q ; & that to draw first white balls, & m blacks, & et p + q white or black balls, will be m m 1 m ( + 1 ) p+q = m m.1 m, where ca have all the values from 1 to 0; therefore m p + q... p m q +p 1 m+q d will epress the sum of the combiatios which give the evet demaded, & m m 1 m d the sum of all the possible combiatios; the itegrals beig take from 1 to 0, the probability will be therefore p + q... p q = p + q... p q +p 1 m+q d 1 m d p.m m + q + m m + p + q If > m, & if oe demads the probability that, i the sequece of evets, the umber of white balls will surpass that of the blacks, by a determied quatity, oe will fid, 1. that this probability ca ever approach idefiitely to 1;. that, followig the hypotheses of plurality, it ca, after havig bee icreasig, become decreasig; 3. that after a certai term, it will cotiue idefiitely to approach the fuctio 1 m d 1 1 m d > 1 the formula 1 m 1 d idicatig that the itegral is take oly from = 1 to = 1. This formula idicates the value of the probability, whe the umber of drawigs is supposed ifiite. 13

14 I the case of m <, the same probability is epressed by the same formula; but the oe has 1 m d 1 1 m d < 1. Likewise 1 m p p+q d 1 m d. The itegral of the formula above beig take from = 1, to = p+q, will epress the probability that, i the sequece, supposed ifiity of drawigs, the umber of the white balls will be to that of the black balls i a ratio greater tha that of p to q, a probability that is greater or lesser tha 1, accordig as oe will have +m >< p p+q. Fially p beig greater tha p, the probability that the ratio of the umber of A to that of N, will be betwee the limits p p+q & p p+q, will be epressed by 1 m p p+q d 1 m d 1 m d 5. If oe applies these formulas to the atural evets of which the order has bee kow by observatio, oe will fid, 1. that, if the questio is of a costat observed order, oe will ever have a probability 1, or the certitude that it will cotiue to be i perpetuity, whatever be the umber of observatios;. that, if oe demads that there is oly a fractio 1 a of the total umber which deviate from this order, the probability will be epressed by 1 ( ) a 1 m+1, a a probability so much greater, as a is smaller, & m greater. 3. That, if there is a questio oly of absolute or proportioal plurality, observed betwee the evets, the probability that this plurality will cotiue idefiitely, will be epressed by a+b 1 a 1 d a+b 1 a d p p+q p i the first case, ca 1 a c c+1 d ca 1 a d ; i the secod; therefore the first is so much greater as a & b are great, & the secod so much greater as a is great; fially that which it will be betwee is c c + 1 < c c + 1 & c c + 1 > c c + 1, ca 1 a c c+1 d ca 1 a d ca 1 a d c c+1 14

15 If oe wishes oly a proportioal plurality which is ot smaller tha that of c to 1, the the probability will be ca 1 a c c+1 d ca 1 a d. 4. Such are the very arrow limits of the kid of probability which we ca have respectig the order of the future evets, & the costacy of the observed laws i ature, at least uder the two hypotheses; 1. that the probability of the successive evets is always the same;. i this where oe cosiders a class of evets as isolated, & the order that oe observes as idepedet of that which is observed i some other evets. Suppose, for eample, that, i oe same city, oe has observed that there are bor more boys tha girls; that this observatio has bee cofirmed durig a great eough umber of years without that there have bee great variatios i the proportio of these umbers, & without that oe suspected ay cosiderable chage i the costitutio of the climate, or of the ihabitats, the oe could reasoably regard as costat the probability that there will be bor a boy rather tha a girl; &, as oe does ot kow a priori if, i the laws of ature which determie this productio, there is oe of them by virtue of which there must eist a costat superiority i favor of oe of the sees. The two suppositios above ca be regarded as legitimate. Thus, let a + b be the umber of boys, & a the umber of girls. a+b 1 a d 1 a+b 1 a d will epress the probability that, all remaiig i the same state, there will be bor i a idefiite time more boys tha girls, or that there is a physical & real cause of this superiority of umber. See below o. 10 & the followig, & the article VÉRITÉ. 5. Oe has ofte applied the ordiary calculus of the probabilities to some questios of the public ecoomy where the questio of paymets of which the period or the duratio could deped o this calculus. Oe could suppose the that the observed ratios i the past evets would be rigorously the same i the future evets; a hypothesis more or less ear to the truth, accordig as the umber of these past evets is great, & as the oe of the future evets is small, but which is eteded i a very sesible maer i the case where the ratio of the umber of the firsts to that of the secods is ot very great. It ca therefore be useful to apply to these same questios the method which we just eposed. We are goig to give some very simple eamples, but sufficiet to make sesible the maer to resolve the questios of this kid. 6. Let be, 1. a ma aged years, & let, out of p me of the same age, oe have observed that there are p who have lived q years, + q is here the last term of the life of the huma kid p who have lived q 1 years... p q who have lived oe year oly, & let oe demad the life auity which it is ecessary to give to this ma for a sum 1, the rate of iterest beig supposed give. Let c be the value which, placed at this iterest, is worth 1 at the ed of oe year, a the life auity, it is clear that the value of 15

16 this pesio, for the oe who lived oe year, will be ca, a.c + c for the oe who lived two, ac 1 cq 1 c for the oe who lived q. Let fially be the ukow probability of livig q years, that of livig q 1 years... q that of livig oe year oly, the mea value of the pesio will be epressed by a.c 1 cq 1 c p +1 p qp q + ac. 1 cq 1 1 c p p +1 qp q + a.c. p p qp q +1 The whole divided by p p qp q. Cosequetly, i order to have the mea value of all the values of, it will be ecessary to itegrate separately the umerator & the deomiator a umber q 1 times, after havig made q = 1 q 1, & havig take the itegrals from q 1 = 1 q to q 1 = 0, from q = 1 q 3, to q = 0... from = 1 to = 0. Now all these operatios beig eecuted, this formula becomes ( a c.p q c + c.p q c. 1 cq 1 1 c p + q p c. 1 cq 1 c p + 1 By puttig back p i place of p +p +p q 1 +p q, oe will deduce from it therefore a = 1.p + q c.p q c + c.p q c. 1 cq 1 1 c p c. 1 cq 1 c p + 1 istead that, followig the ordiary method, oe would have had a = 1.p c.p q + c + c.p q 1 c. 1 cq 1 1 c p + c. 1 cq 1 c p that is to say, the same formula, if oe suppose ifiite the umbers p, p,... whece oe sees that, as here + q beig the greatest umber of years of huma life, ca ot be a very great umber, if the oe of the observatios is to the cotrary a very great umber, the error of the ordiary method will be slightly sesible. 7. I this method, as i the ordiary method, if oe supposes r persos of the same age, ledig each a sum 1, & if oe demads the pesio a which it is ecessary to pay to each, i order that the borrower gives oly the iterest supposed here give, oe will have the same value of a as above; thus, however great that r be, there will be othig to chage i the value of a, oly the probability that the borrower has to ot pay i reality, accordig to the future evets, a iterest either sesibly above, or sesibly below, will icrease less uder the actual hypothesis tha uder the ordiary hypothesis. 8. If oe cosiders the pesios o all heads, oe ca follow accordig to the followig method; oe will take, as above, for each age the value a of the life auity, sice oe will take i the tables of ivestmets i life auities of all heads, the umber ) 16

17 c 1 c ot of me of each age, but of the uities of a sum placed o the heads of each age, let r, r,... r q be these umbers, a, a,... a q the correspodig pesios. The value a of the commo pesio will be epressed by c.r a.r + 1 a q.r q + 1 r + r + r q + q This maer to calculate the rate respectig the pesios is ot more complicated, & it is more eact tha the ordiary method. It will be applied equally to the pesios o two heads, & eve with more facility tha the ordiary method. We have here cosidered oly etire years; but the way to make the fractios of years eter ito the calculatio, as for the paymet of pesios, or eve as for the ages, has aother difficulty oly to require some slightly loger calculatios. 9. We will choose here for a secod eample the evaluatio of a evetual claim; suppose therefore a sigle claim, & let p, p,... p epress the values of this claim, such as they have bee observed durig a certai umber of years, let the claim p have bee paid i b years, the claim p i b years, the claim p i b years, & let,,... q be the probabilities that the followig year oe must pay p, p,... or p. It is clear that, settig 1 1, istead of the total value of the claim for a ifiite umber of years will be epressed by b (1 r 1 )b (p + p + p.1 1 ) d d d 1 b b (1 1 )b d d d 1 = c b + 1.p + b + 1.p + b + 1.p 1 c b + b b + Followig the ordiary method to take some meas, this value would be c b p + b p b p 1 c b + b b whece oe sees that these two values caot be regarded as very differet, at least whe the b are very early equal or very great, with respect to, or that fially the p are also very early equal. I order to apply this method to the claim due o a sigle good, it would be ecessary to divide this total sum i the ratio of this good to the oe of the total mass of the goods which have produced the claims p, p,... p ; &, if oe has take it i order to form the mass of the goods of the same ature, & before early to be eposed to the same evets as the oe for which oe seeks the value of the claim, this method ca be regarded as sufficietly eact. 10. We have regarded util here the probability as beig costatly the same i the sequece of evets of a like ature. There are some cases where this assumptio ca appear gratuitous. Suppose ow that these evets ca be idepedet from oe aother, as to be subject to preserve the same probability, & we preserve always here the deomiatios of the secod kid, if the probability is costat, that to have the 17

18 +1 evet A, after havig obtaied A, times, & N, m times will be +m+1 ; but, if the evets are idepedet, this same probability will be epressed by d, the itegral beig take from = 1 to = 0, that is to say, by 1. But the probability to have A, times & N, m times uder the first hypothesis, is + m m 1 m d & uder the secod it is + m m d m 1 d m.m These two probabilities will be therefore as m+1 to 1 +m & cosequetly the mea probability A will be m+1.m m m+1 m.m m m This hypothesis ca be applied to some questios, for eample, if oe has cast a coi + m times if oe has heads m times & tails times, > m, & if oe demads the probability to brig forth heads: this probability will be epressed by the precedig formula, if I have o other reaso besides to suppose that the reaso for which I have had heads more ofte tha tails, holds to this that the coi is costructed i a maer to fall more ofte o this side, rather tha to regard this superiority as produced by the maer of castig the coi, which is without ifluece o the followig trials. Oe sees fially that there are some cases where however great that m & be, & although has the the superiority to m, oe has o probability which leads to suppose a costat law. 11. Now cosider more closely the ature of the first hypothesis, we will fid first that that it is legitimate oly i two cases: 1. whe the probability of each evet is always the same, as whe oe draws some black or white balls always from oe same ur;. whe drawig them from differet urs, oe supposes that these urs have bee repleished by takig some balls i a commo mass, where they were i a certai ratio. I the first case, it is the probability itself which is costat; & i the secod, it is oly the mea probability. For eample, suppose a sequece of packs of cards i umber r + s, that from r of these packs oe has draw red & m black cards, & that oe demads the probability to draw p red & q black cards of p + q = s followig packs, oe ca suppose either that oe kows that i each of these packs there are the same umber of red cards & of black cards, or else that these packs have all bee formed by drawig them at radom from oe same pile of cards, i this case the probabilities are the same i each pack, & oe could epress this by havig p red cards, & q by the precedig formula; but if oe kows ot i advace the reality of oe of these two hypotheses, or that of the cotrary hypothesis, that is to say, that where there is o liaiso betwee the probability, relative to each of the piles of cards, & that oe demads the epressio of the same probability, oe will take the followig method, let r + s = t &,,... t, t differet 18

19 probabilities of the evet A & makig + t + + t = X & t t t = X, X will epress the mea probability of A, X the mea probability of N & s.s 1... p q X +p X m+q d d... X X m d d... the probability to have p, A & q, N after havig had, A & m, N, the itegrals beig take successively for each from 1 to zero. It is easy to see, by eamiig this formula, that if > m oe will have a greater probability i favor of A tha i favor of N, that this probability will be so much greater as m & will be greater, & that if oe seeks the probability i the idefiite sequece of future evets, the umber of A will surpass that of N, oe will have a probability so much greater i favor of A, as will surpass m, & that these umbers will be greater, although this probability is much smaller tha if oe had supposed the same for all the evets. 1. I all precedig formulas we have supposed the probabilities the same. I some order that the evets succeeded themselves, or that which reverts to the same, we have supposed whether variable or costat the value of these probabilities was idepedet of the order i which the evets succeeded each other. But this hypothesis, far from beig admitted ecessary i every case, appears, i may, cotrary to that which the simple reaso seems to idicate. Suppose ideed that out of oe sequece of twety thousad successive acts 4 A or N, the umber of A has surpassed the oe of N by 300, & that oe demads the probability that i two hudred followig acts the umber of A will surpass the umber of N, it is easy to see that oe must aturally regard this coclusio as more probable, if i oe hudred sequeces of two hudred evets each the umber of A surpasses that of N, tha if after havig surpassed by much i the first oes, this differece moreover was dimiished successively, i a maer that, i the followig sequeces, the plurality has bee i favor of N. Oe must therefore i geeral, & if oe has ot a priori some reaso to adopt aother hypothesis, to regard the probability ot oly as depedet o the evets, but also as depedet o the order that they follow amog them. For this oe will desigate the probabilities of A & N, for the successive evets by A B & thus i sequece: oe will take the sequece of products of these probabilities correspodig to those of the observed evets, & of those which oe supposes must succeed 4 Traslator s ote: Codorcet usually uses the word évéeme for a evet. However, here he itroduces the word fait for the same purpose. To distiguish the two here, the latter is redered as act. 4 19