ON INVERSE PROBABILITIES. 59 



the problem exists. If we number the urns A 1? A , 

 A 3 , B, the chances which they severally give to the 

 observed event are , -y> T> an ^ "f> ^ e numerators of 

 which, reduced to a common denominator, are 27? 27 5 

 27? and 14. Consequently, the probability that A 1 was 

 chosen, is ^i; and the same for A 2 and A 3 . There- 

 fore, the chance that one or other of the three, A 1? A 2 _, 

 and A,,, was chosen, is -|J-; which is the probability of 

 the ball having been drawn from the urn (3 white, 

 4 black,) in the first statement of the problem. 



The rule to which the preceding reasoning conducts 

 us is as follows : When the different states under which 

 an event may have happened are not equally likely to 

 have existed, then having found the probability which 

 each state would give to the observed event, multiply 

 each by the probability of the state itself before using 

 the rule in page 55. The following is another example. 



An event has happened, the possible preceding states 

 of which are represented by A, B, and C. The chances 

 of the existence of these different states (independently 

 of all knowledge of the observed event) are, say, -|, -J, 

 and -J : the probabilities that the observed event would 

 have happened are -f 19 -fa, and T 2 T , if A, B, or C were 

 certainly existing. Form the three products 



T> i U-TX f A f Probability that the event would! 

 Probability of A x | happeni / A were known to exist, j &c - 



there are 



T * 7T> T X TP and T * TT; 



the numerators of which (the denominators being com- 

 mon) are 20, 12, and 4 Then the probability that 

 A was the state under which the event happened, is 

 20 divided by 20 + 12 -f 4, or -||-; those of B and C are 



7T and IT" 



Let us now suppose, that having only a first event 

 by which to judge of the preceding state of things, we 

 ask what is the probability of a second event yet to 

 come. For instance, an urn contains two balls, but 

 whether white or black is not known ; the first draw- 



