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## Behavior of Learning Formula

To understand the behavior of the learning formula , we continue our previous numerical example . Table below show again the user responses together with the failure probability and learning probability at = 0.50.

Initially, the learning probability distribution is set as uniform distribution with equal probability of for each character. The learning distribution was then updated using learning formula. Based on the learning distribution, we can design the program response so that the characters that have large learning distribution value will be asked more frequently. In the very long trials, if the user has known all the syllables, the learning distribution should approach uniform distribution again.

## User response

 trial no 0 1 2 3 4 5 6 7 8 9 10 a Wrong Right Wrong Right b Right c Wrong Right Right d Wrong Right

## Probability distribution of failure

 F(a) 0% 30% 31% 20% 21% 24% 26% 26% 27% 25% F(b) 33% 3% 6% 15% 16% 20% 23% 24% 23% 24% F(c) 33% 33% 31% 32% 32% 32% 26% 23% 23% 24% F(d) 33% 33% 33% 33% 32% 24% 26% 26% 26% 27%

## Learning Probability Distribution (= 0.50)

 F(a) 25% 13% 21% 26% 23% 22% 23% 24% 25% 26% 26% F(b) 25% 29% 16% 11% 13% 15% 17% 20% 22% 23% 23% F(c) 25% 29% 31% 31% 31% 31% 32% 29% 26% 25% 24% F(d) 25% 29% 31% 32% 33% 32% 28% 27% 27% 26% 27%

Observe in the table above that the initial probability is very important to determine the behavior of the learning probability distribution. The probability distribution of failure is similar to the learning distribution at zero learning rates. Higher the value of learning rate makes the fluctuation smaller and longer to reach the equilibrium. In this example, the equilibrium values are the initial probabilities (i.e. for each character).