Chances. 



CHANCE 





and b respectively ; and that the common denomina- 

 tor of ail the fractions will be (4~i) 3< 



For instance, the probability in three trials cf the 

 happening of 



The thra*CYcnt5, ~! f-r^rrs Prob. 24, 

 Only two of them, 

 Only one, of them, 

 Neither of them, 



Prob. 25. 

 : Prpb,26. 

 Cur. to Prob.' 



The sum of -these four probabilities is evidently 

 uaity, as.it ought to be. 



36,. Supposing a to express the number of chances 

 for the happening of an event, and b 'for the chances 

 of its failing, then it is evident from the foregoing 

 problems, that every question that can possib-ly.be 

 proposed respecting the happening or failing of any 

 number, of such events in in trials, will be answered 

 by means of one or more terms of the developement 

 of (a -{-), as a numerator, and the whole expression, 

 as a denominator. In particular, 



1st, The probability of the happening of the ?a 



events, will be 



The probability of the happening of m 1 of the 

 events, will be 



m a n ~ * 



~F 



The- probability of the happening, of m. .2 of the 

 events^ will be 



And, in general, the probability of the happening 

 of m n of the events, will be 



m(m-^l}(m. 2) . . . (m -f 1 ) a m -"b n 

 ~~ 



1 9 ,, 



j. . ^ . c> . ... // 



the factors in. the numerator being continued until 



their number be , and the same in the denominator. 



2d, The probability of .the-happ-uingof m, ml, 



m 2, on, at fewest, of but (> s), such events will 



\ + ^^^^^^^ b y^ &. 



the numerator of the fraction being supposed to 

 consist of -{- 1 terms. 



And the probability of the happening of at .least 

 one such event, will be 



>*+H 



--T- 



3d, The probability of the happening of 'neither, 

 or but one, or two, or at most, of but n, such events 

 will be, 



m(m-l). m(m-l)(m2) . 



if-'a ,}-.->- '&' *a*-j- - \. b m *a*+. &c. 



1 . t 1 . j . O 



the numerator of which fraction consists of w-j-1 

 terms. 



PROBLEM XXVI I. 



37. Suppose a lottery, in which the number of 

 prizes is to the number of blanks as 1 to 39 ; how 

 many tickets must be purchased, that the buyer may 

 have aa equal chance for one or more, prizes ? 



SOLUTION. The probability of having one- prize 

 or more in m tickets, in a lottery wherein the pri/.es 

 are to the blanks. as 1 to 39* is. the same with that o^ 

 throwing one ace or more in m throws, with a die 

 that has 1 -j- 39r 1O faces. Therefore, puttiiig4:0 mrt) 



n m (n 1 V 



thisprobability will (by Prob. X.)be, * - '- t 



which, by putting 1 =: a, and 39 = b, is also 

 b" 



But 



And the probability of missing, is 



by hypothesis, there is to be an equal chance of the 

 having or missing one or more prizes. Now, if the 

 certainty of the having or missing one or more prized 

 be denoted by unity, then the probabilities of an equal 

 chance, for- having or missing one or more of them> 

 will each of them be denoted by 4> 



rp, f (a + 6)-^ 6 

 Therefore ^~ , = - -- rr = 4- : 

 (a + 6)"' (a-f-6)" 1 



whence 2b" l = ,-f-6)" J ; 

 which in logarithms will be, 

 log. 2-J-7M log. 4 m log. ( 



it lu. 2. 



an cl hence, m= 



lo 



Now, log. =. 30103, and log; (fl-f-i)=log. 40 

 1.6U206, aiid log. b=\og. 39=1.5ylOG ; therefore 



m=-'~ -y/;;, = 27.36. By which it appears^ that tire 

 n imber of tickets must be greater than 27. 



PROBLEM XX VII I. 



38. In a pack of 26 cards^ 13 of which are black, 

 and 13 red, if m cards be dealt, how many is there an 

 equal chance of being red ? 



SOLUTION". if the number of chances for the hap- 

 f the event be denoted by , and those for 



its failing by b, then (by Art. 2G.) -.;- -r^ is the 



probability of its not happening once iu-w -trials. In 



b' n -i r mb'"~ l a 



like manner, ~r r will express the probabi- 

 lity of its not happening twice in m trials ; and 

 b m -jr. m b'"~' ! a -j V . -b m ~ 1 a 1 



(/,-!_) 



expresses the probability of its not happening thrice ; 

 and so on. And because the question requires how 

 many times the event will happen in in trials upon an 

 equality of chance, it wiiljbilow, that when tiie event 

 b" 



is to happen once, then - -7^- 4. ; (See Art. 37.) 

 And the probability of the happening of at most , (a-J-6/" 



btit tttr-1) such events will be, . . . . . 



and when it is to happen twice, then- 



and when it is to happen thrice ; then 



Chances. 



