ON INVERSE PROBABILITIES. 55 



ball (chance -0, is \ x -y or -^ ; while our chance of 

 going to the second urn (which is -<), and thence draw- 

 ing a given white ball (chance J), is 1 X J or -J. But 

 since we do not alter the chance of producing a white 

 ball from either urn, if we double, or treble, &c. the 

 number of white balls, provided we at the sanle time 

 double, or treble, &c. the number of black balls, let us 

 put four times as many balls into the first, and seven 

 times as many into the second, as there are already. 

 Thus we have : 



(8 white, 20 black) (21 white, 7 black). 



There are now 28 balls in each : every individual ball 

 has the antecedent probability -J- x --% ', and since our 

 knowledge of the event (a white ball was drawn) ex- 

 cludes the black balls, the question is simply this : 

 Out of 29 possible, and equally probable cases, was 

 the event which did happen one out of a certain 8, or 

 one out of the remaining 21 ? The chances of these 

 are 7% and -g-J- ; consequently it is 21 to 8 that the 

 second lottery was that which was drawn, and not the 

 first. 



On looking at the resulting chance for the first urn, 

 namely, -j^, or the (21 -f- 8)th part of 8, we see that/ 

 8 and 21 are in proportion to the two chances for a 

 white ball being drawn, when we know that we are 

 drawing from the first urn, or from the second. For 

 these chances are ^ and ^, which, reduced to a common 

 denominator, are -^ and -|i, which are in the propor- 

 tion of 8 to 21. The same reasoning may be applied 

 to any other cases, and the result is as follows : 



Principle V. When an event has happened, and 

 the state of things under which it happened must have 

 been one out of the set A, B, C, D, &c., take the 

 different states for granted, one after the other, and 

 ascertain the probability that, such state existing, the 

 event which did happen would have happened. Divide 

 the probability thus deduced from A by the sum of the 

 probabilities deduced from all, and the result is the 

 E 4 



