a? 8 THE SCIENCE OF LOGIC 



pennies? The total number of possible combinations of head and tail in either hypo 

 thesis is 4 : h\h. a h^ /,^ 2 , /,/ 2 ; and out of these four alternatives, three yield a 

 head. The probability is therefore \. Now this is the sum of the two separate 

 simple probabilities. For, if we toss one coin twice the chance of getting a head 

 the first time is \ ; the chance of the second toss taking place at all is contingent 

 on the failure of the first, i.e. is \ ; and the chance of its securing a head if it do 

 take place is \. Therefore the total chance from the second toss is \. Con 

 sequently the total chance by both tosses is \ + * = . If we toss two coins, 

 A and B, simultaneously, the chance of A turning a head is i, and the event of 

 its failure being combined with the heading of B (/^ 2 ) is \ ; or that of heading 

 B I, and of combining its failure with the success of A \ ; in either case the 

 total probability of securing a head is 4 + \ = |. Or again : What is the prob 

 ability that a throw of two dice, A and B, will yield any given number be 

 tween 2 and 12, say 7 ? There are 36 alternatives in a throw. The desired 

 number, 7, can be obtained in six different ways, viz. when A and B are re 

 spectively I and 6, 2 and 5, 3 and 4, 4 and 3, 5 and 2, 6 and i. Now the prob 

 ability of the occurrence of any individual alternative of these she is & ; and 

 as any one of them will yield the desired result, 7, the probability of the occur 

 rence of the latter is the sum of these six probabilities, or 3 ff = i . 



267. INVERSE PROBABILITY : BERNOULLI S THEOREM : 

 ELIMINATION OF CHANCE. So far, the problems examined have 

 been all of the same general character, namely : Given the con 

 ditions capable of realizing indifferently any one of a known num 

 ber of possible alternative events, what is the probability that a 

 particular alternative, a, will occur ? This is called direct or a 

 priori probability. The inverse problem is the following : Given 

 that a certain event has happened, to which of a number of pos 

 sible alternative causes is it most probably due ? The event here 

 in question always yields some data for an indirect or a posteriori 

 probability, and is, of course, always an actual application or 

 realization of some a priori probability. The determination of 

 this a priori or antecedent probability will give a probable know 

 ledge of the actual causes from which the phenomenon or event 

 has followed. Thus, for example, an urn is known to contain 

 three balls, but their colour is unknown. We are asked to deter 

 mine their colour by repeatedly drawing one and returning it. 

 We draw a white : it gives us no reason to conclude anything 

 about the colour of the others. But if we next draw a black we 

 thereby know that the contents of the box may be either w w b 

 or w b b ; but that these two do not exhaust all the alternatives, 

 another possible one being w b x, where x may be another colour. 

 If, however, we continue to draw only blacks and whites, the 

 probability against x being some third colour (say red) rapidly 

 decreases. For if a red ball were there, the probability against 



