Generation of internally restricted random-dot patterns

The probability of a successful draw of a dot position within a random dot pattern is considered for the special case where a minimal city block distance is imposed separating any two dots within the random pattern. An analytical solution is derived and is tested against Monte Carlo simulations. The agreement is considered to be good, except at extremely low probabilities of a successful draw. An alternative method, based upon the utilization of successively coarser grids, is also given. Formulas are also provided for circular and for diamond-shaped separation regions.