CHANCES. 



probabilities of those events separately 

 considered," and therefore, if a always 

 denote the probability of its happening, 

 and b the probability of its happening and 



the 



fraction will express the 

 b n 



failing 1 , 



probability of its happening n times suc- 

 cessively," and (by Def. 1) the fraction 



~~ " will express the probability of its 



b n 

 failing n times successively. 



Rern. It should be observed, that in 

 some instances the probability of each 

 subsequent event necessarily differs from 

 that which preceded it, while in others it 

 continues invariably the same through 

 any munb'er of trials. In the one case 

 the probabilities are expressed, as in the 

 theorem, by fractions, whose numerators 

 and denominators continually vary; in 

 the other they are expressed, as in the 

 corollary, by one and the same inva- 

 riable fraction. But this perhaps will 

 be better understood by the following 

 examples. 



1. Suppose that out of a heap of coun- 

 ters, of which one part of them are white 

 and the other red, a person were twice 

 successively to take out one of them, and 

 that it were required to determine the 

 probability that these should be red coun- 

 ters. If the number of the white be 6, 

 and the number of the red be four, it is evi- 

 dent, from what has already been shown, 

 that the probability of taking out a red 

 one the first time will be _*. : but the 

 probability of taking it out the 2d time 

 will be different ; for since one counter 

 has been taken out, the-e are now only 

 nine remaining; and since, .in order to 

 the 2d trial, it is necessary that the coun- 

 ter taken out should have been a red 

 one, the number of those red ones must 

 have been reduced to 3. Consequently, 

 the chance of drawing out a red coun- 

 ter the 2d time will be 3, and the pro- 

 bability of drawing it out the first and 



4x3 

 2d time will (by this theorem) be 1(J 



2. Suppose next, that with a single die 

 a person undertook to throw an ace 

 twice successively : in this case the pro- 

 bability of throwing it the first does not 

 in the least alter his chance of throwing it 

 the second time, as the number of faces 

 on the die is the same at both trials. The 

 probability, therefore, in each will be ex- 

 pressed by the same fraction, so that the 

 probability, before any trial is made, will, 



by the preceding corollary, be Jx \ -g. 

 On these conclusions depend all the com- 

 putations, however complicated and labo- 

 rious, in the doctrine of chances. But this, 

 perhaps, will be more clearly exemplifi- 

 ed in the two following problems, which 

 will serve to explain the principles on 

 which every other investigation is found- 

 ed on this subject. 



Prob. 1. To determine the probability 

 that an event happens a given number of 

 times, and no more, in a given number of 

 trials. 



Sol 1. Let the probability be required 

 of its happening only once in two trials, 

 and let the ratio of its happening to that 

 of its failing be as a to b. Then, since the 

 event can take place only by it happen- 

 ing the first, and failing the second time, 



the probability of which is jrfc X 



b ab 



or by its failing the 



first and happening the second time, the 



probability of which isr^=^ , the sum of 

 a -f- b\ 



these two fractions, or will be 



the probability required. 



2. Let the probability be required of 

 its happening only twice in three trials. 

 In this case, the event, if it happens, must 

 take place in either of three different 

 ways : 1st, by its happening the first two, 

 and failing the third time, the probabili- 

 ty of which is ; 2dly, by its 



failing the first, and happening the other 

 two times, the probability of which is 



- a(l : or, 3dly, by its happening the 



a -r li\ } 



first and third, and failing the second 



time, the probability of which is -==s , 

 The sum of these fractions, therefore, or 



y will be the required probabili- 

 a -f- &Y 



ty. By the same method of reasoning, 

 the probability of its happening only 

 once in three trials, or, which is the 

 same thing, of its failing twice in three 



* i i r i i 3 b da 

 trials, may be found equal to - >. 



3. Let the probability of the event's 

 happening only once in four trials be re- 

 quired. In this case it must either hap- 

 pen the first and fail in the three suc- 

 ceeding trials ; or happen the second and 



