I wrote a program that repeatedly generates a random grid and counts the number of adjacent pairs, and finally outputs the distribution of this number. Here are some results:
10*10 grid, 50 pairs, 1000000 games
0 1 2 3 4 5 6 7 8 9 10 11 12 158437 295726 272342 164827 73472 25585 7400 1784 350 67 8 2 0 100*100 grid, 5000 pairs, 10000 games
0 1 2 3 4 5 6 7 8 9 10 1361 2794 2729 1725 891 335 117 37 9 2 0 1000*1000 grid, 500000 pairs, 100 games
0 1 2 3 4 5 6 7 8 9 10 13 25 23 25 10 3 1 0 0 0 0