So for the 4x4 triangle, we find the probabilities are:
1/32, 5/32, 10/32, 10/32, 5/32, 1/32
For example, there is one path to the far left pin, and its probability is 1/2^N. The probability of exiting when at that pin is (1/2), so the total probability of exiting at the far left pin is 1/2^(N+1).
To reach the next lower pin, there are (N+1 Choose 1) = N+1 paths, each with a probability of 1/2^(N+1). To exit from there is a probability of 1/2, for a total probability of (N+1)/2^(N+2).
If we number the exits from 0 to 2N+1, the probability of exiting at each location A (0<=A<=N) is
For the 4x4 square, there are 10 exits, and their respective probabilities
are:
1/32, 5/64, 15/128, 35/256, 70/512, 70/512, 35/256, 15/128, 5/64, 1/32
With the same denominator of 256, those numerators are:
8, 20, 30, 35, 35, 35, 35, 30, 20, 8
An interesting note is that the four central probabilities will always be the same for any size square.