Numerators        1     2      4     8     <<<---=== Denominators

    1             1    1/2    1/4   1/8

    2             2     1     1/2   1/4

    4             4     2      1    1/2

    8             8     4      2     1


So, the basic space is the space of 4*4 = 16  ratios in the cells of this matrix.


We randomly select two ratios from this table: so, there are 16*16 = 256 elements in the space of events.


We compare the ratios and count in how many cases they have identical values.


They are identical in  4*4 + (3*3 + 3*3) + (2*2 + 2*2) + 2 = 44 cases:


     4*4   for 4 pairs of ratios along the major diagonal;
     then  (3*3 + 3*3)  pairs of ratios along two next "diagonals";
     then  (2*2 + 2*2)  pairs of ratios along two next-next "diagonals";
     and, finally, 1+1 = 2  pairs of ratios along the next-next-next "diagonals".


In all, among 256 possible (potential) pairs of ratios, there are 44 cases when they have identical values.


So, the answer to the problem's question is   = .