Uniformly distributed n-dimensional probability vectors over a simplex

#/Total[#,{2}]&@Log@RandomReal[{0,1},{m,n}] will give you a sample of m points from a uniform distribution over an n-1-dimensional regular simplex. (An equilateral triangle is a 2-dimensional regular simplex.) Here's what m = 2000, n = 3 should look like, where {x,y} = {p[[2]]-p[[1]], Sqrt@3*p[[3]]} are the barycentric coordinates of the 3-element probability vector p:

Here's what you get if you omit the Log@ and normalize Uniform(0,1) variables, which is what both of the OP's examples do:

Old question, but I didn't see this method. Generates $n$ points uniformly randomly distributed on a simplex embedded in $d$ dimensions.

    genSimplex[n_, d_] := 
       Table[Differences[Sort[Flatten[{0, RandomReal[1, d – 1], 1}]]], {n}];

The algorithm generates points that are randomly distributed on an outer face of a simplex. The way to generate them is, for a d-dimensional problem…

1. Generate d-1 uniformly distributed random values in the range [0,1]
2. Add a 0 and a 1 to the list
3. Sort the list
4. Extract a list of the differences between the elements

You now have a list of random values that sum to 1 (so they are on a plane that is defined by points that sum to one) and that are otherwise independent of each other, so their dispersion is uniform.

Updating the answer with a picture of example data with 1,000 points.

This topic is well-covered here... https://stackoverflow.com/questions/3010837/sample-uniformly-at-random-from-an-n-dimensional-unit-simplex

Starting in M10.2, you can just use RandomPoint:

pts=RandomPoint[Simplex[{{0,0,1},{0,1,0},{1,0,0}}], 1000];
Graphics3D[Point[pts]]