CHANCES. 



probability required will be = - --- - 



2. Let the probability be required of 

 its happening- once in three times. Pro- 

 vided it has happened once at least in the 

 first two trials, the event will have equally 

 succeeded, whether it happens or fails in 



the third trial, and therefore i --- ^ will 

 a-f-6' 



represent the probability in this case. But 

 it may have failed in the first two and hap- 

 pened in the third trial, the probability of 



which is - ^ ; adding this to the preced- 

 3 



found. Let it be supposed to have happen- 

 ed only once in three times, the probability 

 of which, by the preceding problem, in 



; then will the probability of its hap- 



pening the fourth, after having happened 



once in the three preceding, be *, 



q + 6) 

 and therefore the whole probability will be 



q3+3 a 1 6 3 a 1 b\ _ 



By proceeding in the same manner, it may 

 be found that the probability of an event's 

 happening twice at least in five trials, will 



ingfractionwehave 



- - 



a+6r 



babm 



required. In like manner the proba- 

 bility of its happening once at least in 



be 



X 



4 a 63 



1 -f-10 3 b- 



And if 



four trials will be = 



a 63 q*+6q36+6 



the probability of the event's happening 

 thrice in 4, 5, 6, &c. trials be required, 

 they may, by pursuing the same steps, be 



the probability of its happening once at 



a+6" b n 

 least in n times will be = - - In 



other words, since the event must happen 

 once at least, unless it fails every time, the 

 probabilityrequired(by Def. l)will always 

 be expressed by the difference between 



re. 



spectively. Hence it follows, that if the 

 binomial a -\- \ be raised to rath power, 

 the probability of an event's happening at 

 least d times in n trials will be = 



ni. _ 



F" a " 2 62 ( n ^_ i </) 



3. Let the probability be required of an 

 event's happening twice at least in three 

 trials. In this case it will succeed, if it hap- 

 pens the first and second, and fails the third 

 time, if it happens the first and third, and 

 fails the second time, if it happens the 

 second and third, and fails the first time, 

 or if it happens each time successively. 



The first three probabilities are - r and 



the fourth is 





q + 6', 

 bility required will be 



; therefore the proba- 

 Ifthe 



event is to happen twice at least in four 

 times, the probability of its happening dur- 

 ing the first three times has been already 



that is, the series in the numerator must 

 be continued till the index of a becomes 

 e qualtoJ. 



Cor. From this solution it appears that 

 the series. 



ni 

 bn _}- n 6n-lq-f ^."r 6n-2 a 2 to r/ terms, 



will express the probability of the event's 

 not happening so often as d times in n 

 trials. 



Ex. Supposing a person with six dice 

 undertakes to throw two aces or more 

 in the first trial, what is the probability 

 of his succeeding ? In this case q, 6, n, 

 and d, being respectively equal to 1, 5, 

 6, and 2, the above expression will be- 

 come =. 



1 + 30+15x25+20x125+15x625 

 6 6 



