276 THE SCIENCE OF LOGIC 



mathematical principles laid down for the estimation of prob 

 ability, considering those processes of estimation as modes of 

 reasoning from various combinations of disjunctive judgments. 



266. RULES FOR ESTIMATING PROBABILITY. (i) Simple 

 events. The probability of a simple event is expressed by a 

 fraction whose numerator is the number of favourable alternatives, 

 and denominator the total number of alternatives. 



Thus, if A can occur, or has occurred, in any of n different ways, their 

 sum, A, will represent certainty (for we know A will occur, or has occurred), 

 and is best expressed by unity. The probability of any particular case out of 

 the n is, therefore, i/. Out of , the total number of alternatives, (n- i) are 

 against, and i for, the occurrence of any given alternative. As there are six 

 faces to a die, one only of which is marked 4, and the others differently, the 

 chance of turning some of the others, besides the 4, is (n i) or five times 

 greater than the chance of turning the 4 ; i.e. the probability for the 4 is i/, 

 or i, the probability against it (- i)/, or $, Again, as there are fifty-two 

 cards in a pack, but each particular value or number is present in four 

 different ways, the probability in favour of drawing some particular value will 

 be /j, and the probability against it $f, as there are four chances in favour of 

 any particular value and forty-eight against. If there be a question of draw 

 ing some particular card, e.g. the ace of hearts, the probability is only ^. 



(2) Compound events. If an event is compound, i.e. made up 

 of a number of simple events connected together, these latter will 

 be (a) either independent, or () dependent on one another ; hence, 

 two cases arise. 



(a) Compound of independent events. The probability of an 

 event composed of a number of independent simple events, whether 

 simultaneous or successive, is the product of the probabilities of 

 the latter taken separately. 



Suppose, for example, A and B to be two urns, A containing two white 

 balls and one black, say w^ -a&amp;gt;, and b v and B likewise containing two white 

 balls and one black, say a/,, / 4 and a : what is the probability of drawing 

 (simultaneously or successively, it matters not) a black ball from each ? The 

 draw from A may be represented by the proposition, &quot; If S is A it is either 

 o/i or a/2 or b\ &quot; ; that from B by the proposition, &quot; If S is B it is either vu 3 or 

 o/ 4 or &amp;lt;V . Now, if we combine both experiments, we find that the total 

 number of possible ways of drawing two balls is 3 x 3, as any one of the three 

 possible draws from A may combine with any one of the three possible 

 draws from B. The double draw will be expressed by the proposition : &quot;If 5&quot; 

 is AB it is either a/ t 0/3, or TV. a/ 4 , or w 1 b^ ; or w\ w s , or w % a/ 4 , or o&amp;gt; 2 , ; or 

 b l it/-.,, or bi o/ 4 , or & l b* &quot; : from which we see that the probability of a double 

 black is , i.e. the product of the two simple probabilities i x i. We see, 

 likewise, that the probability of drawing two whites is |, or J x $, a further 

 verification of the principle. &quot; Lastly, as the probability of white is in each 

 case !, and that of black is I, so [| x J = ] f is the probability both that the 



