290 ELEMENTARY ILLUSTRATIONS. [CHAP. XVIII. 



and, proceeding with this according to the rule, we have, finally, 

 Prob. x = p(l - q) + cpq. (4) 



where c is the probability that if the event st happen, x will 

 happen. Now if we form the developed expression of st by mul- 

 tiplying the expressions for s and t together, we find 



c = Prob. that if x and y happen together, or x and z happen 

 together, and y fail, or y and z happen together, and x fail, the 

 event a will happen. 



The limits of Prob. x are evidently p (1 - q} and p. 



This solution is more definite than the former one, inasmuch 

 as it contains a term unaffected by an arbitrary constant. 



SOLUTION OF THE THIRD CASE. 



Here the data are 



Prob. {x + (1 - x)yz] = p, 

 Prob. [y + (1 - y) xz\ = q, 

 Prob. (z+ (I- z)xy] = r. 



Let us, as before, write x for 1 - #, &c., and assume 



x + xyz = s, 



z + ~zxy = u. 

 On reduction by (VIII. 8) we obtain the equation 



(x + xyzjs + sx (yz + y z + yz) 



+ (y + yxz) t + ty (zx + xz + xz) 



-f (z + zxy) u+ uz (xy + Icy + xy) = 0. (5) 



Now instead of directly eliminating y and z from the above 

 equation, let us, in accordance with (IX. Prop, in.), assume the 

 result of that elimination to be 



Ex + E(\ -tf) = 0, 



then E will be found by making in the given equation x = 1, 

 and eliminating y and z from the resulting equation, and E' will 

 be found by making in the given equation x = 0, and eliminating 

 y and z from the result. First, then, making x = 1, we have 



