Suppose I throw k-sided dice n times and want to know the probability $p$ of observing a set of counts with individual probability higher than $x$.

Example, let k=2,n=2, fair dice. Possible sets of counts are (0,2),(1,1),(2,0). Individual probabilities of those counts are 1/4,1/2 and 1/4 respectively. Probability of getting outcome with individual probability above 0 is 1, above 1/4 is 1/2, above 1/2 is 0.

What is the relationship between $p$ and $x$? For k=3, line gives surprisingly good fit

This is a generalization of a related unanswered question

Douglas Zare suggests to think of counts as lattice sites of a random walk and use Central Limit theorem. This suggests that relationship is going to be quadratic for k=5, and indeed, parabola seems to give a decent upper bound in that case

n = 21; types = Flatten[ Permutations /@ (IntegerPartitions[n, {3}, Range[0, n]]/n), 1]; prob[p_, q_] := n! Times @@ MapThread[(#1)^(n #2)/(n #2)! &, {p, q}]; cum[p_, cutoff_] := Total[Select[prob2[p, #] & /@ types, # >= cutoff &]]; p0 = RandomChoice[Select[types, FreeQ[#, 0] &]]; pvals = prob[p0, #] & /@ Union[types]; cvals = cum[p0, #] & /@ pvals; data = Transpose[{pvals, cvals}]; Show[ListPlot[data, PlotRange -> All], Plot @@ {Fit[data, {1, x}, x], {x, 0, Max[pvals]}, PlotStyle -> Red}]