CHANCES. 



tion -; if, therefore, any other person 



should be willing to purchase his chance, 

 he must give for it the half of 51. or "21. 

 10s. This is one of the most simple ca- 

 ses : before, however, we proceed, it 

 may be proper to give some definitions 

 introductory to the doctrine. 



Def. 1. The probability of an event is 

 the ratio of the chance for its happening 

 to all the chances for its happening or 

 failing: thus, if out of six chances for its 

 happening or failing, there were only two 

 chances for its happening, the probabili- 

 ty in favour of such an event would be in 

 the ratio of two to six; that is, it would be 

 a fourth proportional to 6, 2, and 1, or JL. 

 For the same reason, as there are four 

 chances for its failing, the probability that 

 the event will not happen will be in the 

 ratio of 4 to 6, or, in other words, it will 

 be a fourth proportional to 6, 4, and 1, 

 or I. Hence, if the fractions expressing 

 the prbabilities of an event's both hap- 

 pening or failing be added together, they 

 will always be found equal to unity. For 

 let a be the number of chances for the 

 event's happening, and b the number of 

 chances for its failing, the prob'ability in 



the first case being- and in the se- 

 cond case ; ' their sum will be = 

 a-\-b 



7 = 1. Having therefore determin- 

 ed the probability of any event's either 

 happening or failing, the probability of 

 the contrary will always be obtained by 

 subtracting the fraction expressing such 

 probability from unity. 



Def. '2. The expectation of an event 

 is the present value of any sum or thing, 

 which depends either on the happening 

 or on the failing of such an event. Thus, 

 if the receipt of one guinea were to de- 

 pend on the throwing of any particular 

 face on a die, the expectation of the per- 

 son entitled to receive it would be worth 

 3s. 6</.; for since there are six faces on a 

 die, and only one of them can be thrown 

 to entitle the person to receive his mo- 

 ney, the probability that such a face will 

 be thrown being |. (according to Def. 1.) 

 it follows, that the value of his interest 

 before the trial is made, or, which is the 

 same thing, that his expectation is equal 

 to one-sixth of a guinea, or 3s. 6d. Were 

 his receiving the money to depend on his 

 throwing either of two faces, his expecta- 

 tion would be equal to two-sixths of a 

 guinea, or 7s. And, in general, supposing 



the present value of the money or thing 

 to be received to be A, the probability of 

 the event's happening to be denoted by 

 a, and of its failing by b, the expectation 



will be either expressed by- ' or by 



, according as it depends either on 



the event's happening, or on its failing. 



Def. 3. Events are independent, when 

 the happening of any one of them does 

 neither increase nor lessen the probabi- 

 lity of the rest. Thus, if a person un- 

 dertook with a single die to throw an ace 

 at two successive trials, it is obvious 

 (however his expectation may be effect- 

 ed) that the probability of his' throwing 

 an ace in the one is neither increased nor 

 lessened by the result of the other trial. 



Tlieor. The probability that two subse- 

 quent events will both happen, is equal to 

 the product of the probabilities of the 

 happening of those events considered se- 

 parately. 



Suppose the chances for the happening 

 and failing of the first event to be denot- 

 ed by b, and those for its happening only 

 to be denoted by a. Suppose, in like 

 manner, the chances for the second 

 event's happening and failing to be de- 

 noted by d, and those for its happening 

 only by c ; then will the probability of the 

 happening of each of those events, sepa- 

 rately considered, be (according to Def. 



1) and- respectively. Since it is ne- 



b a 



cessary that the first event should happen 

 before any thing can be determined in 

 regard to the second, it is evident that 

 the expectation on the latter must be 

 lessened in proportion to the improbabi- 

 lity of the former. Were it certain that 

 the first event would happen, in other 



words, were a = b, or - = 1, the expec- 

 tation on the second event would be = 



-. But if a is less than b, and the ex- 

 d 



pectation on the second event- is restrain- 

 ed to the contingency of its having hap- 

 pened the first time, that expectation will 

 be so much less than it was on the former 



supposition as - is less than unity. 



Hence we have 1 : :: C - : , for the 

 c d bd 



true expectation in this case. 



Cor. By the same method of reasoning 

 it will appear, that the probability of the 

 happening of any number of subsequent 

 events is equal to the "product of the 



