How to find the number of cells that one point can be in?

Question

I draw the attached plot drawn with the following code. Each color represents a cell. How can I find the average number of cells that one user (any point within the square) can be in? By average I mean "the user can be anywhere within the whole area". For example, averaging over 10000 random locations of a point within the square area.

ell = t [Function] t^-20.75];500, 2}]; 
 H = RandomVariate[ExponentialDistribution[1], 50]; 
 r = Table[H[[i]] ell[Norm[{x, y} - X[[i]]]], {i, 1, 20}]; 
 s = Table[r[[i]]/( Total[Delete[r, i]] + 30.99), {i, 1, 20}];

Show[Table[
 RegionPlot[s[[i]] >= 0.1, {x, -0.7, 0.7}, {y, -0.7, 0.7}, 
 MaxRecursion -> 2, PlotPoints -> 40, 
 PlotStyle -> {FaceForm[{Opacity[0.5], ColorData[97][i]}]}, 
 BoundaryStyle -> ColorData[97][i]], {i, 1, 20}], Graphics[Point[X]]]

Henrik Schumacher · Answer

Well, a Monte-Carlo approach could look like this. There is a considerable amount of number crunching involved, so we better compile the working horse function:
cf = Compile[{{H, _Real, 1}, {X1, _Real, 1}, {X2, _Real, 1}, {Z, _Real, 1}},
   Block[{n, R, r, x1, x2},
    x1 = Compile`GetElement[Z, 1];
    x2 = Compile`GetElement[Z, 2];
    r = H ((X1 - x1)^2 + (X2 - x2)^2)^-1.875;
    R = Total[r] + 3.99;
    Total[UnitStep[r - (0.1/1.1) R]]
    ]
   ,
   CompilationTarget -> "C",
   RuntimeAttributes -> {Listable},
   Parallelization -> True,
   RuntimeOptions -> "Speed"
   ];

Now:
pts = RandomReal[{-0.7, 0.7}, {10000000, 2}];
{X1, X2} = Transpose[X];
N@Mean@cf[H, X1, X2, pts]

2.05869

The function cf counts how many entries of the vector s are above 0.1 for a given {x,y} = Z. Applying cf in a listable way to all elements of pts and taking the mean yields the empirical average number of satisfied inequalities.

How to find the number of cells that one point can be in?

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