PROBABILITY. 



PROBABILITY. 



773 



both arguments very powerful, it is to be assumed that the mind does 

 not see the ratio of the small quantities, and the question is evenly 

 balanced. 



2. No evidence; only argument for. The odds are 1 to 1 A; 

 taken by all the logicians who have considered the subject to be A to 

 1 A. The first result is always greater than the second; which 

 means that even if the argument be invalid, the conclusion may still 

 be true, and has some probability left. 



3. A witness of given credibility does not add so much to the force 

 of the conclusion as an argument of the same probability of validity. 

 We introduce this assertion as one of common sense : if the witness be 

 wrong the assertion is false : if the argument be invalid, the assertion 

 may still be true ; consequently the argument is better than the 

 witness. How do the formula! verify this ? As follows : if a be the 

 probability of both, the odds for the truth are altered by the witness 

 in the proportion of a to 1 a and by the argument in the greater 

 proportion of 1 to 1 a. 



4. Any argument, however slight its force, adds'something to the 

 force of its conclusion : for alteration in the ratio of 1 to 1 a, is 

 increase, let a be ever so small. But it must be remembered that this 

 is only true of the argument per se, and has no reference to the possi- 

 bility of the weakness of the argument being itself ah argument on the 

 other side. If a weak argument be brought forward by a person who 

 most probably would know the stronger ones, if there were any, the 

 argument on the other side just spoken of may arise. On this, how- 

 ever, and other points, our space obliges us to content ourselves with 

 the reference already made. 



The various problems of which the solutions have been given are 

 mathematical consequences of the definition of probability. Every 

 such problem is simply one of combinations, however much the length 

 of detail, and the number of mathematical abbreviations of the process 

 of combining, may tend to make us lose sight of the first principle. 

 At the same time it is found requisite to establish a few simple funda- 

 mental propositions, which we shall cite, with some consequences. 

 As we are not writing an elementary treatise, we shall not demonstrate 

 these propositions, referring the student to any of the modern works 

 hereinafter cited. The probabilities of the events A, B, c, &c., are 

 denoted by a, b, c, &o. 



1. By a, the probability of A, is meant the fraction which the number 

 of cases favourable to the happening of A is of the whole number of 

 CUM, that is, both of those in which A can happen, and those in which 

 it cannot : all cases being equally likely. And the probability that A 

 wjll not happen is Ia. 



2. When A and B are events independent of each other, so that the 

 happening of either in no way promotes nor retards the happening of 

 the other, the probability that both shall happen is ab ; that neither 

 shall happen, 1 a It + ab; that one only shall happen, a + 4 2a6; 

 that one or both shall happen, a + b a 6. 



3. When A and B are mutually exclusive, that is, when if one happen 

 the other cannot, the probability that one or other shall happen is a + 6 ; 

 and that neither shall happen, 1 a b. 



4. If either A or B must happen, o + 6 = l ; and if trials are to be 

 made, the several terms in the expansion of (a + *)" are the chances of 

 the arrivals denoted by their exponents. Thus 



a* is the chance that all are AS. 

 na'-'b, that n 1 are AS and one is B. 



n^ a"- 5 *, that 2 are AS and two are BS, fie. 



5. When an event has happened which may have sprung from any 

 one of the sets of circumstances which we may describe by A, B, c, D, 

 &.C., and it is desired to find what is the probability of the event which 

 did happen having arisen from one or another set, proceed as follows : 

 Had the circumstances denoted by A certainly existed, and the 

 event been a contingency, let a be the (probability that those circum- 

 stances would have produced the event ; or briefly, let a be the pro- 

 bability that A, when certain, brings about the event. Let 4 be the 

 same for B certain, e for c, &c. Then the probability that it was A 

 which brought about the event which did happen, is the first of the 

 following set : 



a+b + c+kc., 



a + b -r c + tc. 



, &c. 



the probability that it was B is the second, that it was c the third, and 

 so on. Or, when convenient, instead of a, b, c, &c., may be substituted 



attach to our conclusions. That person was Dr. Wollatton, who, upon a given 

 point, wa Induced to offer me a wager of two to one on the affirmative. I 

 rather impertinently quoted Butler's well-known lines 



Quoth she, I'ye heard old cunning stagers 

 Say fool* for arguments use wagers,' 



bout the kind of persons who use wagers for argument, and he gently explained 

 to me that he considered nuch a wager not as a thoughtless thing, but as the 

 expression of the amount of belief in the mind of the person offering it ; com- 



bimn,; tl.i* cm ion- :>[>|:lication of the waiter, as a meter, with the necessity that 

 ever rxisU-d of drawing conclusions, i:ot absolute, but proportionate to the 

 rrkknce." 



any whole numbers which are proportional to them. For example, let 

 there be three lotteries, containing balls as follows : 



all white, 



2 white, 7 black. I 



A drawing has been made three times from one of these, but from 

 which is not known, the ball being replaced after the drawing, and 

 every drawing has given a white ball. Required the chances in favour 

 of each of these lotteries having been the one drawn from. If it had 

 been the first, the chance of the event would have been 1 (or certainty) ; 

 if it had been the second, the chance of the event would have been 



444 64 



;r x -^ x g- , or, j-jj ; if it had been the third, the chance would have 



2 2 2 " 8 

 been ^ x jj- x 5> or >729' The numerators of these fractions, reduced 



to a common denominator, are 91125, 46656, and 1000; whence the 

 probability that the lottery drawn from was the first, ia 



91125 91125 



91125 + 46656 + 1000' or 138781 ' 



and the probabilities of the second and third lotteries are 

 46656 1000 



138781 



and 



138781' 



The preceding question is well calculated to show the meaning of 

 questions in this theory, which is tlras seen to be applied to events, 

 not because they are uncertain, but because they are unknown. So 

 soon as the lottery is chosen, it is certain which it is, but since it is 

 not known to the drawer, it is to him as much a contingency as it was 

 before it was chosen. 



6. When, in such a case as the preceding, it is required to know the 

 probabilities of the events which may happen at any further trials, the 

 probability of each lottery having been the one in question is to be 

 multiplied by the probability of the new event arising from that lottery- 

 taken as certain, and the results added together. Thus, suppose there 

 are two lotteries, one having all white balls, and the other equal numbers 

 of white and black balls ; two drawings have been made from one of 

 them (not known) and both drawings have given a white ball ; what 

 is the probability that a third drawing from the same lottery would 

 also give a white ball ? The chances for the two lotteries are found 



by the last to be -g and y, while the chances of a white ball from 

 one and the other are 1 and -jr. It follows that jxl+-px-;r,or 



~ V , V M 



9 



TQ, is the chance of the third draw ing giving a white ball. 



7. If A or B must happen at every trial, and if hi m trials nothing 

 but A has happened, and if we know nothing whatever about the 

 nature of preceding circumstances, then it is m + lto 1 that A shall 

 happen at the next trial, and m + 1 to k that A shall happen throughout 

 the next t trials. But if in m + n trials A have occurred m times and 

 B times, it is m + 1 to n + 1 that A shall occur at the next trial. 



8. Every event that ran happen must happen, if trials enough be 

 made ; and not only must happen, but must happen any number of 

 times in succession. For example, if there be only one white ball to 

 100 black ones in an urn, and if drawings enough be made, the ball 

 drawn being always returned and the urn properly shaken before a 

 new trial, a person who goes on must at last draw that single white 

 ball 1000 times running. This is a conclusion which the beginner 

 cannot receive, particularly when he is told that 1,000,000 might be 

 written for 1000, or any other number however high. Nevertheless, 

 it will, in the course of his studies, not only be made clear to him, but 

 it will also be shown that it is a conclusion which may be made obvious 

 to common sense without any very profound mathematics. 



9. If the odds against an event happening at any one trial be n to 1, 

 there is an even chance of its happening in '69 x n trials,* it is 10 to 1 

 that it happens in 2'40 x n trials, 100 to 1 that it happens in 4 '62 x n 

 trials, 1000 to 1 that it happens in 6'91 x n trials, and 10,000 to 1 that it 

 happens in 9'21 x n trials. Thus if it be 200 to 1 against success iu 

 any one trial, 9'21 x 200 being 1842, it is 10,000 to 1 that there shall 

 be one success at least in about 1842 trials. But if a person should make 

 1000 trials without one success, he would have no right to say that 

 it is 10,000 to 1 in favour of one or more successes in the next 842 

 trials. 



10. When any sum depends upon a contingent event, the present 

 value of that sum is such a fraction of it as the probability of the 

 event happening is of unity. Thus 201. to be gained if an event 

 happen against which it is 7 to 1, is now worth only the eighth part 

 of 20/. 



There are no questions in the whole range of applied mathematics 

 which require such close attention, and in which it is so difficult to 

 escape error, as those which occur in the theory of probabilities. This 

 makes it hard to lay down in a short space either maxims or examples 

 sufficient to be rually of use hi advancing the student's progress : and 



* The numbers 'CD, 2'JO, &c., arc all aMiroslmations. 



