I'm not sure how much this will help you but I hope it gives some pointers.

Here is a Mathematica function which computes the probability under the distributions for a circle of radius separation/2  for two ships with a normal distribution of position with variance 0.2. A variance of 0.2 is close to the 95% certainly level.

In brief it defines a mixture distribution in 2 dimensions with covariance matrix {{0.2,0},{0,0.2}} (* other covariance matrices would account for elliptical distributions *). Forms the probability distribution function for that mixture and then integrates it over the required range.

(* Uses absolute separation distance rotated to the x axis *)

    probProximity[reportedSeparationMiles_, probabiityRangeMiles_] := 
     With[{dist = MixtureDistribution[{1, 1},
     {MultinormalDistribution[{-(reportedSeparationMiles/2),0}, {{0.2, 0}, {0,0.2}}], 
      MultinormalDistribution[{  reportedSeparationMiles/2, 0}, {{0.2, 0}, {0,0.2}}]}]}, 
      NIntegrate[
       PDF[dist][{x, y}] 
       Boole[Abs[\[Sqrt]((0 - x)^2 + (0 - y)^2)] <= probabiityRangeMiles/2], 
       {x, -(probabiityRangeMiles/2), probabiityRangeMiles/2}, 
       {y, -(probabiityRangeMiles/2), probabiityRangeMiles/2}]]

The probability distribution of position for two ships 5 miles apart with a 95% confidence of being within one mile of reported position.

![Separation Distribution][1]


For a range of 5 miles, the calculated value is 

    probProximity[5, 5]

>0.464173

Here is the probability of proximity over a range of distances:
![Probability of separation][2]




  [1]: https://i.sstatic.net/J8H1J.png
  [2]: https://i.sstatic.net/IPaFX.png