meanRunDensity[p_, len_, trials_] :=
 Mean[Length[Cases[Split@#, {1, ___}]] & /@
    Unitize[Chop[RandomReal[1, {trials, len}], 1 - p]]]/len

Column@Table[
  Grid[Join[{{p, n, K, diff}},
    Table[{q, n, x = meanRunDensity[q, n, 100] // N,
      q (1 - q) - x}, {n, {100, 1000, 10000, 100000}}], {}],
   Alignment -> Left], {q, {.1, .3, .5, .7, .9}}]