LOTTERY. 



number^ ; and the number of terms in which b enters being 

 equal to the number c — p. 



But the number of all the chances for taking a certain 

 number c, of counters out of the number n, is expreffed 



by 



— I 



2 



— X 



» — 2 



— X 



&C. 



1 2 3 



to be continued to as many terms as there are units in c. 



If the numbers n and c were large, fuch as n — 40000, 

 and c = 8000, the foregoing method would feem imprac- 

 ticable, on account of the great number of terms to be taken 

 in both fcrics, whereof the firit is to be divided by the fe- 

 cond ; though if thofe terms were aftually fet down, a 

 great many of them being common divifors might be ex- 

 punged out of both ferics ; for which reafon it will be con- 

 venient to ufe the following theorem, which is a contraction 

 of that method, and which will be chiefly of ufe when the 

 white counters are but few. Let, therefore, n be the num- 

 ber of all the counters, rt the number of white, b the number 

 of black, c the number of counters to be taken out of the 

 number ?(, p the number of white that are to be taken pre- 

 cifely, then making n — c — d. The probability of tak- 

 ing precifely the number /> of white counters will be as fol- 

 lows : ^•/'z. making 



c.{c-^) {c - 2) (.-3) &c. =C 

 d.{d- I) {d~ 2) {d-3) &c. = D 



the three particular benefits, which will be found to be 

 32000 X .:ii099X 31998 ^ 6£ ^^ ^^.^^ ^^.^^ ^^^^ 



40000 X 39999 X 39998 125 



trafted from unity, gives a remainder, i — — -=--—, 



(hewing the probability required ; and therefore the odds 

 againft taking any of three particular benefits will be 64 

 to 61 nearly. 



PnoB. V. 



To find how many tickets ought to be taken in a lottery 

 confiliing of 40000, among which are throe particular be- 

 nefits, to make it as probable that one or more of thefe 

 three may be taken as not. Let the number of tickets 

 requifite to be taken be =. x, and the probabihty of not 



taking any of the particular benefits will be X 



a 

 - X 

 I 



a — I 

 X 



2 



n (n - l) (n - 

 the probability = 



- 3 



&c. = A 



3 4 



2) («-3) &c. = N 

 D X A 



N 



where it is to be obferved, that the firft and third fcries 

 contain as many terms as there are units in p ; the fecond as 

 many as there are in a — p ; the fourth as many as there are 

 in a. 



Let us now apply thefe refults in tlie folution of the fol- 

 lowing problems. 



Prob. IV. 



In a lottery confiding of 40000 tickets, among which are 

 three particular benefits, what is the probability that taking 

 8000 of them, one or more of the particular benefits fliall be 

 among them. Subiluute 8000, 40000, 32000, 3, and i 

 refpeftively for c, n, d, a, and/), in that problem ; and the 

 probability of taking precifely one of the three particular 

 Sooo X 32000 y 31999 X 3 _ 

 40000 X 39999 < 39998 



benefits will appear to be 



n — X — I 



X 



— ; but this probability is equal to 



n ^- I n — 2 



■i, fince by hypothefis, the probability of taking one or more 



of them is equal to ^ ; whence we (hall have the equation 



n — X n — X — I n — x — 2 , r .irt 



X X = h from- the folu- 



n n — I n — 2 



tion of which x will be found nearly equal to 8252. The 

 terms of this equation, M. De Moivre obferves, may be 

 confidered as being in geometric progredion ; fince the fac- 

 tors both of the numerator and denominator are few and in 

 arithmetic progreffion, and their difference very ^mall in 

 refpcdl of n : and, therefore, tlie cube uf the middle term 

 may be fuppofed equal to the product of the multiplication 



n — X " I 



of thefe terms ; whence will arife the equation, , =: 



n— ]| 



\ ; or, neglefting the unit botli in the numerator and denomi- 



\' 



nator. 



or n (1 — •§ <*/ 4) : but n — 

 0.2063 ; therefore x = 8252 



T^, and s, confequently, : 

 40000, and 1 



2 ^/4 = 



Prob. VI. 



To determine accurately, in a lottery of looooo tickets, 

 whereof 9COCO are blanks, and loooo are benefit?, what the 

 odds are of taking or not talking a benefit, in any number of 

 tickets affigned. Let the number be 6 ; and it will appear, 

 by the above cited problem, that the number of chances for 

 taking no prize in 6 tickets, making a =: lOOOO, b :=; 90000, 



^nearly. If ^ = 2, the probability of taking precifely two 

 125 



. , . /-^ -11 u 8000 X 7999 X 32000 X 3 

 of the particular benefits will be — — — LL22. 2 ^ 



^ 40000 K 39999 X 3999** 



= — ?^ nearly. If * = 3, the probability of taking all the 

 125 



. , , r. •„ 1 8000 X 7999 X 7998 



three particular benefits will be •- : — - = 



*■ '^ 4000Q X 39999 X 39998 



-i-. And the probability of taking one or more of the 

 125 



. , , . .,, . 48 -t- 12 -I- I 61 

 three particular benefits wiU be — — ^= very 



nearly. Thefe three operations might have been contrafted 

 tato one by inquiring the probability of not taking any of 



4 



f = 6, / = 



•11 I 900C0 

 n =: loocoo, will be -^ X 



89999 



89998 



19997 

 4 



5 



89995 



whole number of chances will be 



and that the 

 X '5999° ^ 



99908 ^ 99997. ^ 

 4 



£9996 ^ 9929J , tj,,„ d,,id; th^ 



3450 

 firft number of chances by the fecond, by means of loga- 

 rithms, the quotient will be 0.53143, the probability re- 

 quired : this decimal traflion being fubtrafted'from unity, 

 the remainder 0.46857 fiiews the probability of taking one 

 prize or more in fix tickets ; wherefore the odds againfl: 

 taking any prize in fix tickets will be 53143 to 46857. If 

 the number of tickets be feven, then carrying each number 



of 



