LOTTERY. 



tained from caufes conneAed with the lottery fyftem, it is 

 doubtful whether any real advantage is derived from this 

 fouroe ; and if even the whole of the above was a real fav- 

 ing, the evils attending it are fuch as to lead us to hope, 

 that minifters will find fome other means of raifing an equi- 

 valent) founded upon more liberal prmciples, and lefs dan- 

 gi^rous to the morals and happinefs of the people. 



Having faid thus much with regard to the general policy 

 of lotteries, we (hall conclude the prefent article by an invef- 

 tigation of the theory of lotteries, as it is connefted with the 

 doArine ofchances. 



Prob. I. 



Any number of things being given, as a, i, c, d, e, f; 

 to find the probability that in taking three of them, as they 

 happen, they (hall be any three propofed, as a, b, c. 



Firft, the probability of taking either a or i, or c, will be 

 .jths, and fuppofing one of them, as a, to be taken, then the 

 probability of taking either b or c will be ^ths. Again, let 

 either of them be taken, fuppoi'e i ; the probability of taking 

 s in the third place will be |th ; wherefore the probability 

 of taking the three thmgs propofed, viz a, b, c, will be 



3 2 I I 



65 4 30* 



Olherwife, we might confider what number af combinations 

 of fix things can be formed by taking three at a time ; and 

 out of this number there is obvioudy only one combination 

 that anfwers the conditions of the problem propofed ; and 

 there are, therefore, fo many chances to one againit the fuc- 

 cefs of tW' trial. 



Now, the number of fuch combinations is exprefled by 



6 x- 5 X 4 



3x2 X I 



And, confequenlly, the chance of drawing the fpecified things 



a, 6, c, is ,3th, as before. 



Corollary. — Univerfally, the number of combinations that 



can be formed, of n things taking^, at a time, is exprefled 



by 



n__{ n- l) (>» - 2) («- 3) .. . (n - />) 

 p (p ~l) (p-2) {p- i) 1 



and confequently, the reciprocal of this fraftion will be the 

 probability of fuccefs in any cafe that may arife. 



fpecified ones q, we rauft divide the firft of the foDowing 

 feries by the fecond, viz. 



10. 



pip- 1) (/>- 2) (p- 3) 



(P-I) 



? If- i) (?- 2) (?- 3) « 



2 " (" - I) (" -^) (" -3) •••("- 7 ^ 



9 (?- (?-*)(?- 3) I 



that is, the propofed chance will be exprefled by the fracUoii 



pip- -i) { p--%) {p-i) . . . {p -q) 

 n (n - I) (o - 2) (n - 3) . . . (n - y)" 



Prob. III. 



To find what probability there is, that in taking at random 

 feven counters out of twelve, whereof four are white, and 

 eight black, there fliall be at lead three white ones. 



I. Find the chance for taking three wliite out of four, 

 which will be 



4x3 X 2 



3x2x1 



= 4- 



2. The number of chances for taking four black out of 

 eight is, on the fame principle, found to be 



8x7x6x7 



: i = 70. 



4^3x2x1 



And, therefore, the chances of both fucceeding is 4 x 70 



= 280. 



But by the queftion, he may hold four white and three 



black, becaufe it is only limited that three white be taken, 



and not that there (hould be three white and no more. 



3. How the number of chances for taking four white out 

 of four is one. 



4. The number of chances f»r taking three black out of 

 eight is 



8x7x6 _ 

 3x2x1" ~ ^ 

 And the produft of thefe two is 56 x T = 56, there- 

 fore the whole number by which the event may fucceed, is 

 280 + 56 xr 336. 



•5. But the whole number of combinations that can be 

 formed out of twelve things, taking feven at a time, is 



12. II . 10 . 9 . 8 . 7 . 6 



— = 792 : 



therefore 



336 

 792 



Prob. II. 



Let the fame fix things be propofed as above, to deter 

 mine the probability, that in drawing four of them, the the event will ha 

 three fpecified ones, as a, b, c, fhall be taken. 



Firft, the number of combinations that can be formed of 



6. J 



I 



. 4. 3 . z 



will cxprefs the probability that 

 14 _ 19 



fix things, taking three at a time, is ^- ^ = 20 • 



3-2.1 



and the number of combinations that can be formed out of 



four things, taking three at a time, is =; 4. 



3x2x1^ 



Whence it follows that out of the twenty combinations of 

 threes which may happen, four of them will be in hand ; 

 and, therefore, the probability of taking the three fpecified 



things under the condition of the problem, is — = _ . 



20 ^ 



And hence, generally, to determine the probability, that in 

 drawing out of a given number of tickets n, any propofed 

 aumber «, there (haH be found amoncrll them any number of 

 Voi„ XXI. 



ppen, and confequently i — — = — ^ 



33 33» 

 IS the probability of its failing ; that is, the odds againft three 

 white counters being drawn, are as 19 to 14. 



Corollary Let a be the number of white counters, b the 



number of black, n the whole number = a + A ; c the num- 

 ber of counters to be taken out of the number n : alfo, let 

 * reprefent the number of white counters that are to be 

 found precifely in c. Then the number of chances for tak- 

 ing none of the white, or one of the white, or two of the 

 white, and no more ; or three of the white and no more ; or 

 four of the white and no more, &c. will be exprefl"ed as fol- 

 lows : 



{" 

 {' 



a — 3 



5 

 *- 2 



} 

 } 



> 3 



The number of terms in which a enters being equal to the 

 3 H number 



