[VA] SRC #012a - Then and Now: Probability

[Albert Chan]

(11-02-2022 10:56 PM)Valentin Albillo Wrote:  Running the program for 5 rows and a sufficiently large number of steps (S=100), we get in SCI 6:

>RUN
       Rows,Steps=5,100 -> 2.666667e-1


            1/30
         1/15 1/15
      1/15  1/10  1/15
   1/15  1/10  1/10  1/15
1/30  1/15  1/15  1/15  1/30

and we see that P(corners) = 1/30, P(edges) = 1/15 = 2/30 and P(inner) = 1/10 = 3/30, so we conjecture that the ratios are

P(corners) : P(edges) : P(inner) = 1 : 2 : 3

which A. Chan also discovered and posted here.

(10-27-2022 02:07 PM)Werner Wrote:  @ P-> Q, adjust corners and edges
...
@ Q->P
@ P(X) := Q(X-1)+Q(X+1)+Q(X-I-1)+Q(X-I)+Q(X+I+1)+Q(X+I+2)
...

Asymptotic P ratios can be easily derived from the code.
If Q entries are all the same, nextP = P

Say, all entries in Q = k (for above example, k = 1/60), Q -> P:

P(corners) : P(edges) : P(inner) = 2k : 4k : 6k = 1 : 2 : 3