MATHEMATICAL SECTION. 87 
The value of the discriminate association ratio is not affected by 
interchange of a and 6. It is assumed that association and recip- 
rocal association are equal. 
The concurrence residual is significant in proportion to its square; 
and this is in accordance with the general comparative importance 
of residuals in the theory of least squares. 
When s is infinite, with a, 6, and ¢ each finite, the discriminate 
2 
association ratio reduces to the indiscriminate =. 
chances are as infinity to one against fortuitous concurrence; and the 
extent of association must be regarded as based wholly on special 
relations. 
In this case the 
This paper was briefly discussed by Messrs. GILBERT, H. Far- 
QUHAR, BAKER, and the author. 
25TH MEETING. , Marcu 2, 1887. 
The Chairman presided. 
Present, twenty-one members and two guests. 
A symposium was held on the following problem in probabilities: 
ONE-THIRD OF THE APPLES ON A TREE ARE ROTTEN AND ONE- 
FOURTH ARE WORMY, WHAT IS THE PROBABILITY THAT AN APPLE 
TAKEN AT RANDOM FROM THE TREE WILL BE (1) ROTTEN ONLY; 
(2) WoRMY ONLY; (8) BOTH ROTTEN AND WORMY; AND (4) 
SOUND? 
The discussion was begun by Mr. M. Baker who gave an analyt- 
ical statement of the conditions of the problem and concluded that 
the problem was an indeterminate one until some assumptions were 
introduced. Without introducing anything additional to that stated 
he concluded that the four probabilities were respectively 
oo. 2 d dO d 
rae na PHN ig? 1 Nema i a Paras 
where d = the number of sound apples and s the total number of 
apples. 
