CHANCES. 



fail in the first, third, and fourth trials ; 

 or happen the third, and fail in the first, 

 second, and fourth trials ; or happen the 

 fourth, and fail in the first, second, and 

 third trials. The probability of each of 



these being ^L,, the required proba- 



n n ^ n 2 



tion, are =s 'X 3 X -- - continued 



-i- -o O 



to J terms ; the general rule therefore 

 a d foil d 



- n multiplied into n X 



will be 



;^ 4; and for the same rea- 



son, the probability of its happening 1 three 

 times and failing onlv once in four trials 



will be i|* 

 a-f ft) 



4. Let the probability be required of 

 its happening twice and failing twice in 

 four trials : here the event may be deter- 

 mined in either of six different ways : 1st, 

 by its happening the first and second, and 

 failing in the third and fourth trials; 2dly. 

 by its happening the first and third, and 

 failing the second and fourth trials; odly, 

 by its happening the first and fourth, and 

 failing the second and third trials ; 4thly, 

 by its happening the second and third, 

 and failing the first and fourth trials; 

 5thly, by its happening the second and 

 fourth, and failing the first and third 

 trials; or, 6thly, by its happening the 

 third and fourth, and failing the first and 

 second trials. Each of these probabili- 



ties being expressed by - 4 , it follows 

 a + 6' 



that the sum of them, or _" 



press the probability required. 



By proceeding in the same manner, 

 the probability in any other case may be 

 determined. "But if the number of trials 

 be very great, these operations will be- 

 come exceedingly complicated, and there- 

 fore recourse must be had to a more ge- 

 neral method of solution. 



Supposing n to be the whole number of 

 trials, and dthe number of times in which 

 the event is to take place, the probability 

 of the event's happening d times succes- 

 sively, and 'failing the remaining nd times, 



. . a d nd _ 



will be^ -.-d _. nd - _ 



I> 



But as there is the same probability of its 

 happening any other d assigned trials aod 

 failing in the rest, it is evident that this 

 probability ought to be repeated as often 

 as d things can be combined in n things, 

 which, by the known rules of combina- 



^ ^ . continued to .d 



234 



terms. 



Er. Supposing a person with six dice 

 undertakes to throw two aces, and no 

 more ; or, which is the same thing, that 

 he undertakes with one die to throw an 

 ace twice, and no more, in six trials ; it is 

 required to determine the probability of 

 his succeeding', a being in this case = 1, 

 6= 5, n 6, and d = 2, the above expres- 



sions will become 



, multiplied into 



6 x-; 



5 625 X 15 



- very nearly. 



2 46656 

 Hence, since there is only one chance for 

 his succeeding, while there are four for 

 his failing, the odds against him will be 

 as four to one. 



Prob. 2. To determine the probability 

 that an event happens a given number of 

 times in a given number of trials, sup- 

 posing, as in the former problem, the 

 probability of its happening each time to 

 that of its failing to be in the ratio of a 

 to b. 



Sol. It will be observed that this pro- 

 blem materially differs from the preced- 

 ing, in as much as the event in that pro- 

 blem was restrained, so that it should 

 happen neither more or less often than a 

 given number of times, while in this pro- 

 blem the event is determined equally fa- 

 vourable by its happening either as often 

 or oftener than a given number of times, 

 so that in the present case there is no 

 further restriction than that it should not 

 fall short of that number. 



1. Let the probability be required of an 

 event happening once at least in two trials. 

 If it happens the first and fails the second 

 time, or fails the first and happens the 

 second time, or happens both times, the 

 event will have equally succeeded. The 



probability in the first case is - 1} the 



probability in the second is ^- ; ; ,andthe 

 a-{-b\ 



probability in the third is^=- ; hence the 



