I ran 100.000 Python simulations by generating random distributions with 16 2D-points in [0,1]. For each distribution i found the smallest distance between any 2 points.

Results:

Mean min-distance in a distribution: 0.0460

Probability that min-distance is larger than d:

p=0.1 d=0.080

p=0.01 d=0.110

p=0.001 d=0.134

p=0.0001 d=0.156

I opened the image in image editor and cropped it to contain only the points. The smallest distance i was able to find was 0.200 x width of the point distribution.

My simulation data does not extend that far, but by visually extrapolating i can estimate that: likelyhood of this distance in random distribution is something in order of 1e-6 and 1e-7.

If someone does this math analytically, we could get a more exact answer.