FACTORS IN LEARNING BY WHITE RATS 343 



the forward direction x/2 entrances to the blind alley in question 

 will be made; #/4 returns will be made from this point; and 

 3#/4 cases of keeping the general forward direction will occur. 

 But on any return trip past the same blind alley these proba- 

 bilities are reversed; there is now in x such cases a probability 

 of x/2 that the blind alley will be entered, of #/4 that the for- 

 ward direction in the maze will be resumed, and of 3x/4 that 

 the return direction will be maintained. Adding these fractions 

 to those for corresponding directions above, we get a prob- 

 ability of x for entrance to the blind alley, of x for the forward 

 direction, and of x for returns. That is, for any equal number 

 of forward runs in the maze and returns to the place of starting — 

 with the forward and backward intermediary movements that 

 according to chance would result — the probability of entrances 

 to blind alleys, of return directions of movement from such 

 points, and of forward directions of movement would be 1:1:1, 

 respectively. In such a case there could be no learning at all 

 so far as frequency effects go. But since every trial in the maze 

 must end with a forward run reaching the food box, as well 

 as begin with a forward run, these respective probabilities would 

 become x -f- 1/2, x + 1/4, and x + 3/4. In our totals, then, 

 the expectations would be 40 (x + 1/2), 40 (x -+- 1/4), and 

 40 (x + 3/4) ; or 40.x + 20, 40* + 10, and 40* 4- 30. We 

 actually got 88, 71, and 96, respectively, a very close approx- 

 imation. The coefficients, 40 in each case, represent 4 trials 

 times 10 blind alleys to pass in each trial. 



If Xi, Xi, x», . . . . Xio represent, respectively, the number 

 of times that an animal gets successfully past blind alley 1, 2, 

 8, .... 10, and if for simplification we posit a condition such 

 that whenever at any point the animal begins a return move- 

 ment on emergence from a blind alley, it must continue the 

 return to the starting place whatever other blind alleys may be 

 entered on the return run, then according to pure probability 

 laws Xi> x 2 > x 3 > . . . . x l0 . Just how much greater in each 

 case? Evidently, according to the foregoing, x 2 = 3/4 of x u 

 x 3 = 3/4 of x 2 , Xi = 3/4 of x,, . . . . x 10 = 3/4 of x t . But x 10 

 equals the total number of trials 3 that the animal has made up 

 to the point of consideration in the experiment, or the total 

 number of times that it has reached the food box, since each trial 



3 Trial is here used in the sense of a continuous effort ending only when the 

 food box is entered. 



