CH A 



Chance*, bility that he draws one white counter and no more ? 



**~Y*~~' SOLUTION. The probability of his drawing a 

 black counter out of th; first heap, and a white coun- 

 ter out of the second, is (by Art. l!>. ) 

 a \ a a b 



( a 

 ~^p 



And the probability of his drawing a white counter 

 out of the first heap, and a black counter out of the 



second, is, in like manner, : TT\* ^ ow l ^ e P ro " 

 bability required, is manifestly the sum of these two : 

 Therefore its measure is 



Cor. If a and b represent the number of chances 

 for the happening and failing of an event at one trial, 

 then every chance relating to two such events that 

 can possibly happen in two trials, may be expressed 

 by a fraction, whose numerator consists of one or 

 more terms of the second power of the binomial a-\-b t 

 and whose denominator is that power itself. 



For instance, the probability in two trials of the 

 happening of 



Two events, 

 Only one of them, 

 Neither of them, 



NC E S. 



SOLUTION. If the person draws a black counter 







the first time, the probability of which is 7, 

 then he must draw two white counters out of the re* 

 maining heaps, the probability of which %7 TTffl 



and therefore the probability of succeeding by draw- 



I, o a*6 



ing in that manner, is -T-T ; i /\ = / a 4.i\i 



Again, if he draws a white counter the first time, 

 the probability of which is -T-T then he must draw 

 only one white counter out of the two remaining 

 heaps, the probability of which is . ,(Art.32.) 



therefore the probability of succeeding in this way, is 



fl + b- 



Now the whole proba- 



Prob. 6. 



Prob. 23. 

 Prob. 7. 



The sum of these three probabilities is manifestly 

 unity, as it ought to be. 



bility required, is manifestly the ura of thete two ; 



o'A 2/i'A 



therefore it is 7 



(a 



Cor. As the probability of drawing two white 

 counters or more, is the sum of the probabili- 

 ties investigated iu the two last problems, it will be 

 0*6 



PROBLEM XXVI. 



35. The same things being supposed as in the two 



Again, the probability of both the events happen- last questions, what is the probability that one white 



ng, s 



and the probability that both will not happen, n 



counter and no more shall be drawn 



SOLUTION. If he draws a white counter the first 

 time, then at the other two trials he must draw two 

 black ones, the probability of doing both of which if 



a 6* aft* 



PHOBLEM XXIV. 



33. Suppose three heaps, each containing n white 

 and 6 black counters, and that a person draws one 

 out of each, what probability is there that they shall 

 be all white ? 



SOLUTION. If he draws a white counter out of 



the first heap, the probability of which is ^-j, 



rt-j-O 



then he must draw two white counters also out of 

 the remaining heaps, the probability oF which is 



-, ,fl\> Dllt neither of these events will be effectu- 

 al w ithout the happening of the other, and therefore 



If he draws a black counter the first time, then at 

 the other two trials he must draw one black and 

 one white counter, the probability of doing which is 



ft 



required. 



Cor. In like manner, the probability of drawing 

 three black counters, or of failing to draw a white 



one at each attempt, is - T-T-. 



PROBLEM XXV. 



34-. The same things being supposed as in the last 

 problem, what is the probability that two of the 

 drawn and no more shall be white ? 



Therefore the probability required will be, 

 ab" 2a& 306* 



Cor. 1. The probability of drawing either one 

 two white counters, will be 



Sa*b Sab* Sab 



Cor. 2. The probability of drawing either one, 

 two, or three white counters, is 



Cor. 3. The probability of drawing none* or at 



. 

 most but one white counter, is 



Cor. 4-. The probability of drawing none, 

 or at most but two white counters, is 



Cor. 5. Hence it appears, that one or more of the 

 terms of the binomial a -f- 6, raised to the third power, 

 will be the numerators of fractions which express the 

 probabilities of all the varieties that can possibly hap- 

 pen in three trials concerning events, the number of 

 chances for the happening or failing of which are rr 



