Analytic Theory of Probabilities

Transcription

1 Aalytic Theory of Probabilities PS Laplace Book II Chapter II, 4 pp A lottery beig composed of umbered tickets of which r exit at each drawig, oe requires the probability that after i drawigs all the tickets will exit We ame z,q the umber of cases i which, after i drawigs, the totality of the tickets,, 3, q will exit It is clear that this umber is equal to the umber z,q of cases i which the tickets,, 3,, q exit, less the umber of cases i which, these tickets beig brought out, the ticket q is ot draw ow this last umber is evidetly the same as the oe of the cases i which the tickets,, 3,, q would be extracted, if oe removed the ticket q from the tickets of the lottery, ad this umber is z,q oe has therefore i) z,q = z,q z,q Now the umber of all possible cases i a sigle drawig beig ) ) r+) 3r, the oe of all possible cases i i drawigs is [ ] i ) ) r + ) 3 r The umber of all the cases i which the ticket will ot exit i these i drawigs is the umber of all possible cases, whe oe subtracts this ticket from the tickets i the lottery, ad this umber is [ ] i ) ) r) 3 r the umber of cases i which the ticket will exit i i drawigs is therefore [ ] i [ ] i ) ) r + ) ) ) r), or 3 r [ ] i ) ) r) 3 r 3 r Traslated by Richard J Pulskamp, Departmet of Mathematics & Computer Sciece, Xavier Uiversity, Ciciati, OH September 4, 00

2 this is the value of z, This put, equatio i) will give, by makig successively q =, q = 3,, [ ] i ) 3) r ) z, =, 3 r [ ] i 3) 4) r ) z,3 = 3, 3 r ad geerally [ ] i q) q ) r q + ) z,q = q 3 r Thus the probability that the tickets,, 3, q will exit i i drawigs beig equal to z,q divided by the umber of all possible cases, it will be q [ q) q ) r q + )] i [ ) ) r + )] i If oe makes i this expressio q =, oe will have, s beig here the variable which must be supposed ull i the result, [ss ) s r + )] i [ ) r + )] i for the expressio of the probability that all the tickets of the lottery will exit i i drawigs If ad i are very great umbers, oe will have, by the formulas of No 40 of Book I, the value of this probability by meas of a highly coverget series We suppose, for example, that he brigs forth oly oe ticket at each drawig the precedig probability becomes s i i Let us propose to determie the umber i of drawigs i which this probability is k, ad i beig very great umbers By followig the aalysis of the sectio cited, oe will determie first a by the equatio this which gives 0 = i + a s ca c a, { } a = i + c a + s sc a +s Pages 60-65

3 Oe has ext, by No 40 of Book I, whe c a is a very small quatity of the order i, as that takes place i the preset questio, oe has, I say, to the quatities early of order i, s beig supposed ull i the result of the calculatio, s i i = i i+ ) i+ c a i c a ) i i+ c a Now oe has, to the quatities early of the order i, ) i+ i = c i + by supposig ext c a = z, oe has c a ) = c i )z + i ) z moreover, the equatio which determies a gives whece oe deduces i + a = i + )z, c a i = c iz z) oe will have therefore, to the quatities early of order i, s i i = c z + i + z + i ) z I order to determie z, we take up agai the equatio a = i + i + c a oe will have, by formula p) of No of Book II of the Mécaique céleste, z = c a = q + i + q + 3 i+ ) q beig supposed equal to c i+ This value of z gives c z = c q [ i + )q ] ) q i+ 3 q 4 +, 3 cosequetly, s i i = c q + i + q + i + ) q 3

4 By equatig this quatity to the fractio k, oe will have ow oe has q = log k + i + + i + ) log k i + = log q oe will have therefore very early, for the expressio of the umber i of drawigs, accordig to which the probability that all the tickets will come forth is k, i = log log log k) + log k) + log k oe must observe that all these logarithms are hyperbolic We suppose the lottery composed of 0000 tickets, or = 0000, ad k =, this formula gives i = 95767, 4 for the expressio of the umber of drawigs, i which oe ca wager oe agaist oe, that the te thousad tickets of the lottery will exit it is therefore a little less tha oe agaist oe to wager that they will come forth i drawigs, ad a little more tha oe agaist oe to wager that they will exit i drawigs Oe will determie by a similar aalysis the umber of drawigs i which oe is able to wager oe agaist oe that all the tickets of the lottery of Frace will exit This lottery is, as oe kows, composed of 90 tickets of which five exit at each drawig The probability that all the tickets will exit i i drawigs is the, by that which precedes, [s s )s )s 3)s 4)] i [ ) ) 3) 4)] i, beig here equal to 90, ad s previously beig supposed ull i the result of the calculatio If oe makes s = s, this fuctio becomes where, by developig i series, [ss )s 4)] i { )[ ) ][ ) 4]} i, s 5i 5i s 5i + ) ) 5i [ ] 5i + ) +, s previously beig supposed equal to i the result of the calculatio Oe has, by No 40 of Book I, by eglectig the terms of order i ad supposig c a very small of order i, s 5i ) 5i = 5i+ a ) 5i 5i 5i+ ) 5i c )a 5i c a ) ) 5i + 5i a c a 5i c a ), 4

5 a beig give by the equatio a = 5i + ) c a ) ) ) + c a Oe has thus, by eglectig the terms of order i, s 5i ) 5i = ) 5i + c a c a ) 5i c a ) c 5i+)c a 0ic a ow oe has ) 5i + c a = c 0ic a, c a ) 5i = c 5ic a + 5i c a ) 5i 5i = c + ) 5i + 0i oe will have therefore, to the quatities early of order i, s 5i ) 5i = c a ) ) 5i 5i 5i + 0i + a c a ) 0i ), c a + 5i c a + a c a 0i By substitutig for a its value ad observig that i is very little differet from i the preset case, as oe will see hereafter, oe has, very early, ) a c a 0i = 5i + ) c a I keep, for greater exactitude, the term c a ), although of order i, because of the size of its factor oe will have therefore s 5i ) 5i = c a ) [ + 5i + 6 c a + 5i ] ) c a If oe chages i this equatio 5i ito 5i, oe will have that of s 5i ) 5i but the value of a will o loger be the same Let a be this ew value, oe will have this which gives, very early, Now oe has a = 5i ) ) c a ), ) + c a a = a c a = c a c a, 5

6 whece oe deduces, by eglectig the quatities of order i, c a ) = c a ) cosequetly oe has, by eglectig the quatities of order i, s 5i ) 5i = c a ) Oe will have therefore, to the quatities early of order i, [ss )s 4)] i [ ) ) 3) 4)] i = c a ) [ + 5i + 6 c a + 5i ) c a This quatity must, by the coditio of the problem, be equal to, this which gives [ c a 5i + 6 = ) c a 5i ] c a, whece oe deduces c a = ) [ + 5i + 6 ) cosequetly oe has, by hyperbolic logarithms, ) 5i + 6 a = log ) ow oe has, to the quatities early of order i, oe will have therefore i = 5 a = 5i + ) [ 6 0i ] ) log By substitutig for its value 90, oe fids i = 85, 53, ] + 5i ] c a 5i c a ) so that it is a little less tha oe to oe to wager that all the tickets will exit i 85 drawigs, ad a little more tha oe to oe to wager that they will exit i 86 drawigs 6

7 s i i A quite simple ad very approximate way to obtai the value of i is to suppose, or the series ) i + equal to the developmet ) i + [ of the biomial ) ) ) i, ) i ) ] i I reality the two series have the first two terms equal respectively Their third terms are also, very early, equal amog themselves for oe has quite early ) i equal to ) i I reality, their hyperbolic logarithms are, by i eglectig the terms of order, both equal to i Oe will see i the same way that the fourth terms, the fifth, are very little differet, whe ad i are very great umbers but the differece icreases without ceasig i measure as the terms move away from the first, which must i the ed produce i them a evidet <differece> betwee the series themselves I order to estimate it, we determie the value of i cocluded from the equality of the two series By equatig to k [ the biomial ) ] i, oe will have ) log k i = log ), these logarithms may be, at will, hyperbolic or tabulated Let k = z We will have, by takig the hyperbolic logarithms of each member of this equatio this which gives, very early, z log k = log z) = z + +, z = log k log k ) oe will have therefore, by hyperbolic logarithms, ) log = log z = log log k log log k k, Oe has ext log = The precedig expressio for i becomes i this way, very early, i = log log log k) ) + log k 7

8 the excess of the value foud previously for i over this oe is log k log log log k) this excess becomes ifiite, whe is ifiite but a very great umber is ecessary i order to reder it very evidet ad i the case of = 0000 ad of k =, it is still oly three uits If oe cosiders likewise the developmet ) i 5 + [ of the expressio [s s )s )s 3)s 4)] i [ ) ) 3) 4)], as i 5 ) i ] the oe of the biomial, oe will have, i order to determie the umber i of trials i which oe ca wager oe agaist oe that all the tickets will exit, the equatio this which gives [ ) ] i 5 = log i = ) log ) 5 These logarithms ca be tabulated By makig = 90, oe fids i = 85, 04, this which differs very little from the value i = 85, 53 that we have foud above 8