In the solar neighbourhood, the stellar density is around 0.002 stars per cubic lightyear - or to put it another way, 1 star per 500 cubic lightyears. That sounds really piddly, but it isn't - 500 cubic lightyears would be a cube that is about 7.93x7.93x7.93 lightyears. Or would be a spherical volume with a radius of 4.92 lightyears.
In an stellar cluster the stellar density is higher. In a globular cluster (outside the galaxy) the density is much higher.
e.g. The Pleiades star cluster apparently has a radius of about 8 lightyears, and contains 1000 stars (considerably more than the brightest seven or eight that we can see with the naked eye). That sounds crowded, but consider that the volume of a sphere that is 8 lightyears in radius is about 2150 cubic lightyears. So the stellar density in the Pleiades is about 0.47 stars per cubic lightyear.
If you assume that's a cubical volume (i.e. a 16x16x16 ly cube) rather than a sphere that is 16 ly in diameter then that volume goes up to 4096 lyÂ³, and the density is more like 0.25 stars/lyÂ³. Apparently the way to calculate the average separation (assuming a cubical lattice of stars at least) is to take the cube root of the (total volume/number of stars). So that would mean that the average separation of the stars in the Pleiades is 1.6 lightyears.
If we took that same cubical volume of 4096 lyÂ³ and filled it with stars equivalent to stellar neighbourhood density (0.002 stars/lyÂ³) then we'd expect to find only about 8 stars, and their average separation would be about 7.93 lightyears apart.
Put another way: The probability of having a star in a given cubic lightyear is 1/500 (0.002), so the probability of NOT having a star in a given cubic lightyear is 499/500 (0.998)
If you check two cubic lightyears, that probability of NOT finding a star is 499/500 * 499/500 (0.998Â²), which is about 0.996. In three cubic lightyears, it's (0.998Â³ =) 0.994, etc.
So, if the probability of finding a star in a cubic lightyear is p, and the probability of not finding a star in a cubic lightyear is q (which is the same as 1-p):
The probability P of finding a star in n cubic lightyears is given by:
P = 1 - (q)[super]n[/super]
This means that in a 500 cubic lightyear volume (corresponding to a spherical volume that is 3 pc in diameter), the probability of finding a star is only about 63%.
In an 18 cubic lightyear volume (corresponding to a spherical volume that is 1 pc in diameter - i.e. a Traveller hex), the probability of finding a star is only about 3%.