773 



PROBABILITY. 



PROBABILITY. 



774 



whole theory of probabilities, for all the rest is mathematical deduction. 

 Let x be the proper numerical measure of the probability of drawing 

 1, or of drawing 2, &c. ; these probabilities being equal, since there is, 

 by the hypothesis, nothing to render one more likely than the other. 

 Then, if the preceding be admitted, 2x is the probability that either 1 

 or 2 is drawn ; 3x, that either 1, 2, or 3 is drawn ; and so on up to lOx, 

 which is the probability that one or other of the set 1, 2, 3, .... 10, 

 shall be drawn. But since that a drawing shall take place is an 

 absolute condition, one of the ten numbers must be drawn ; hence x 

 must be so taken that lOx shall be the numerical measure of certainty. 

 It is indifferent what number is taken to stand for the exponent of 

 certainty, so far aa principles are concerned ; but in a mathematical 

 point of view, unity is more convenient than anything else. Let 



unity be adopted, then 10.r=l, or x= . Hence the chance of draw- 



Q 



ing, say one of the three, 1, 2, 3, is ; and that of drawing one of the 

 remaining 7 is - . If then the first three should be white balls, and 



q 



the last seven black, the chance of a white ball is , that of a black 

 one is ; and a block ball is more likely than a white one in 



the proportion of 7 to 3, which being inequality' in the proportion 

 of 2 1 to 1, the (xldi are said to be 2| to one in favour of a black ball, or 

 against a white one. By this we do not mean that every man doet, 

 in such a case, look for a black ball with an expectation 2} times aa 

 great as his expectation of a white ball, but that, if he muld measure 

 the strength of hU own feelings and adjust them with mathematical 

 precision, he would proportion the strength of the two expectations in 

 the preceding manner. And if money were to be spent upon the 

 expectations, he may as reasonably give 2>/. for a black ball, before it 

 appears, as 11. for a white one. We do not naturally reckon probabi- 

 lities by numbers ; but nevertheless, we have some kind of estimating 

 apparatus in our brains : Kant called it a weiyhiny machine with 

 wutamptd weight!. 



It thus appears that the theory of probabilities is simply the re- 

 duction to a numerical estimation, in cases which are perfectly known 

 as to the number of events which may happen, of our comparative 

 right to expect one or another event in preference to the rest. In the 

 events of common life, we make estimations of this comparative right, 

 but not numerical///, because we are not in sufficient possession of the 

 event* which might have happened instead of the one which does 

 happen. In such terms as barely possible, very unlikely, improbable, 

 not improbable, as likely as not, rather likely, highly probable, almost 

 certain, &c., we see a gradation which amounts to a rough attempt 

 to make those comparisons which might be made numerically if 

 the proper data could be obtained. The truth is, that every one 

 naturally admit* and practises the fundamental principles of this 

 theory, though often, it may be, unskilfully. But it i.i not to be 

 imagined that persons in general make an investigation like the pre- 

 ceding, nor indeed any investigation at all : how is it then that not 

 only are these principle! acquired, however imperfectly, by the 

 majority of mankind, but the mathematical results which are obtained 

 by those who professedly study the subject, are received almost univer- 

 sally ; not only are they felt to be agreeable to common sense in a 

 great majority of caws, but they are soon admitted to be sufficient 

 indications that common sense is wrong, in the few cases in which they 

 at first appear repugnant to it? The answer to this question leads to 

 another view of the subject. 



We all find, by every -day observation, that whenever an event of one 

 kind happens permanently more often than one of another kind, there 

 exi.iU some reason for such frequency of occurrence, which, had it been 

 inquired into before any event happened, would have enabled us to 

 predict the frequency in question. So much is this the case, that if 

 we were to take an observer to an urn in which were black and white 

 balls, but how many of each he is not told, and were to make 1000 draw- 

 ings, replacing the ball drawn after each drawing, and shaking the urn 

 before every trial ; if of the thousand drawings 822 were white and 

 178 black, he would be irresistibly led to conclude that there must be 

 more white balls than black ones in the urn. Not that this is abso- 

 lutely necessary ; for it is barely possible that there may be only one 

 white ball and 999 black ones, and that by u remarkable coincidence 

 the sole white ball may have come up 822 times out of the 1000 trials. 

 More than this, a person used to observation would conclude, not only 

 that there are more white balls in the urn, but that the proportion of 

 white and black balls does not differ very greatly from that of 822 to 

 178. There is a disposition, <l.'rivc-.l from experience, to think that 

 events happen in the long run in some sort of manner connected with 

 the facilities afforded for their happening beforehand : and hence 

 follows a disposition to judge what those facilities were from observed 

 events, as well as to predict events from observation of those facilities ; 

 and it may happen that one person would draw his notion of likelihood 

 from the first, and another from the second. Thus, if we were to put 

 the question, " What do you mean by saying that it is 5 to 3 for A 

 against B ? " one person might reply, " I mean that in the long run 



A will happen 5 times for every 3 times which u happens ; " while 

 another might say, "I mean that precedent circumstances are so 

 arranged that for every three contingencies under which B may happen, 

 there are 5 under which A may happen." If these two persons were to 

 dispute which was the true mode of viewing the subject, they would 

 be fighting about nothing ; for both are true, and each of them foUows 

 from the other : but if they were to differ as to which is the common 

 mode, we should feel rather disposed to side with the one who was for 

 the first answer. 



The following problems, illustrative of the connection between tho 

 two modes, are the results of mathematical investigation. The table 

 used is that given in MEAN : but two decimal places must be made in 

 column A, and four in column B ; thus 451 opposite to 4 must be read 

 as -0451 opposite to -04, similarly '8931 comes opposite to 114 and 

 9825 opposite to 1'6S. 



PROBLEM 1. The probabilities of p and Q, at any one trial, are aa 

 ate b, that is, it is a to 6 for p against Q, and 6 to o for Q against p. A 

 large number of trials n(a + b) is made; what are the chances that the 

 number of PS which happen shall lie between n a + 1 and n a I, and (of 

 course) the number of QS between nbl and nb + l (I being small 

 compared with n a orn 6) 1 



ROLE. Calculate (2 ? + !)-=- 



- 

 0+6 



let this be the A of the 



table (altered as above) ; then the corresponding B, altered as above, 

 is the probability required. 



EXAMPLE. It is 2 to 1 that a throw with a die shall give 3, 4, 5, or 

 6, and not 1 or 2 ; what is the chance that, in 12,000 throws, the 

 number which gives 3, 4, 5, or 6 shall lie between 8000 + 100 and 

 8000-100. Here a = 2, 6 = 1, n=4000, l=WO, and the complete 

 calculation by logarithms * (which we insert merely as a guide to the 

 readiest mode of treating more complicated cases) is as follows : 



11=4000 3-6021 22 + 1 = 201 2-3032 



0= 2 '3010 2-1646 



4= 1 -0000 



8 -9031 1-376 '1386 



4-S062 

 a+6=3 -4771 



2)4-3291 



2-1046 



In the table, A being 1-38, B is -9490, which is more than near 

 enough for the purpose, say '95 : the result then is that 'B5 is tho 

 chance in favour of tho throws which give 3, 4, 5, or 6 out of 12,000 

 throws lying between 8000 + 100 and 8000-100, whence '05 is the 

 chance against it; that is, it is about 95 to 5, or 19 to 1, that tho 

 throws shall so lie. A few instances of this kind will show how com- 

 pletely, in the long run, events may be expected to happeu in numbers 

 nearly proportional to what are called, when the precedent circum- 

 stances are fully known, their probabilities of happening. 



PROBLEM 2. In the preceding problem, to find I so that there may 

 be a given probability that the PS shall lie between na + l and nal. 



ROLE. Reduce the fraction which represents the given probability 

 to a decimal fraction, and take the A corresponding to the B which is 

 nearest to this fraction in the table ; multiply by the square root above 

 described, subtract 1, and divide by 2 ; the whole number nearest to 

 the result is the answer required for I. 



EXAMPLE. In the preceding example, find I so that it is an even 

 chance that the number of throws out of 12,000 which give 3, 4, 5, or 

 6 shall lie between 8000 I and 8000 + 1. Here 4 or '5 is the given 

 probability, the nearest to it in column B is '5027 to which A is '48, 

 which multiplied by the square root (or the number whose logarithm 

 is 2-1646) is 66'9 : subtract 1 and divide by 2, which gives 33 very 

 nearly : hence it is an even chance (very nearly) that the number of 

 throws out of 12,000, which give 3, 4, 5, or 6 lie between 800033 

 and 8000 + 33. 



PROBLEM 3. In a + b trials, a and i not being small numbers, and 

 nothing whatever being known except what these trials tell, it is found 

 that P happens a times, and p. happens b times ; from which it is ex- 

 ceedingly likely that the probabilities of p and q happening at a' single 

 trial would have been nearly as a to 6, if we had known enough to 

 form an d priori opinion. This we cannot do, by hypothesis : but, 

 judging from the preceding events, with what degree of probability 

 may we infer that if we had been able to form an d priori opinion, tho 

 odds for p against q would have appeared to us to lie between a k to 

 6 + k and a + k to 4 it ? 



RULE. Calculate i-=-V(^)> let this be A (in the table): then 



the corresponding B is the probability required. 



IPLE. In 600 drawings from an urn, the ball being replaced 

 after each drawing, there were 418 white balls and 182 black ones. 

 What is the probability that the proportion of white and bhrck balls in 



* In this, and many other coses, the four-figure logarithms, published by 

 Taylor and Walton on a card, will be found by,far tho most convenient. 



