OPINION AND PROBABILITY 277 



drawings will give first white and then black, and that they will yield first 

 black and then white ; this is shown to be true by the fact that the final pro 

 position yields two cases in which white is followed by black, and two cases 

 in which black is followed by white.&quot; 1 



Similarly, if the first urn contains 3 white and 4 black, and the second 

 4 white and 5 black balls, the number of possible ways of drawing two balls 

 is 7 x 9 = 63, the number of ways of drawing two whites 3x4=12, and of 

 drawing two blacks 4 x 5 = 20 ; consequently the probability of drawing 

 two whites is H and of drawing two blacks f#. 



It will be seen that there is no difference in principle or theory between 

 this sort of compound, and the simple event itself, the disjunctive proposition 

 expressing the former being a combination of the alternatives of each inde 

 pendent simple event. 



() Compound of dependent events. If two simple events are 

 so connected that the occurrence of the first affects the probability 

 of the occurrence of the second, the probability of the compound 

 event will be the product of the probability of the first with the 

 probability of the second as affected by the first, 



For example, what is the probability of drawing two white balls in suc 

 cession, without replacing the first drawn, from an urn containing two white 

 balls and one black one? If a black is drawn first the estimate is rendered 

 impossible. The probability of drawing a white first is f . If it is drawn, the 

 constitution of the urn for the second draw is modified by the first draw. 

 There are now only two balls, one white and one black, in the urn ; and the 

 probability of drawing the white is $. The two draws may be represented by 

 the two propositions : 



A is either w } or /, or b\ 

 B is either TV* or b^ 



Where b.^ is the same ball as b v and o/ 2 the white ball that remains after vt^ or 

 o/ a has been extracted. Those two propositions combined will give six alter 

 natives, of which two are pairs of whites : &quot; AB is either a/, a/ 3 , or a/i 2 , or 

 TV?, -a/ ;J , or wa bi, or ^-0/3, or 6\ b^ &quot; ; from which we see the probability of 

 drawing two whites is |, or the first simple probability multiplied by the second 

 as affected by the occurrence of the first, i.e., \ x \. 



The same result may be stated in this way : The probability of drawing a 

 white first, and consequently of the second draw taking place at all, is \. In 

 the hypothesis that it does take place, the probability of drawing the white 

 again is \. Therefore the probability of drawing two whites is \ x \ = \. 



(3) Total probability of events which may happen in a plurality 

 of ways. When the same simple event may occur in many inde 

 pendent groups of conditions, its total probability is the sum of 

 the probabilities that the various conditions will be verified. 



For example, what is the probability of turning a &quot; head &quot; in one or other 

 of two consecutive throws of a penny, or (which is the same) in one throw of two 



1 W ELTON, op. cit., p. 173, whose treatment is largely followed in the present 

 section. 



