NavList:

Anytime I want a" simulation based in a model WITHOUT NAVIGATIONAL DATA" or a navigation problem without reference to the manner in which the data is obtained, I'll be sure to use a random number generator with a Gaussian distribution.

Whenever I wish a thoughtful analysis of the issue, I will certainly begin with that of John Karl.  It starts with the manner in which data is obtained, assigns a distribution about each LOP based upon your proficiency of acquisition and then assigns a most likely location based upon the product of three distributions.  Its a probabilistic analysis of St. Hilaire.  Well reasoned and mathematically robust.

I most certainly agree with Gary, when he shows that the actual position is most likely to not agree with the determined position within the cocked hat. Indeed, there is very little importance to being precisely correct.  In the middle of an ocean of 3000 miles, what is +/- 3 miles anyway?  That would be 1/10 of 1%.  When coming to landfall, that tiny error vanishes when coastal pilotage begins.

For PRACTICAL NAVIGATION, the most cogent advice is:"When in coastal or pilotage waters, to pick the worst scenario/most dangerous position in the cocked hat. Otherwise when well offshore I generally picked the position closest to the centre"  of the cocked hat.  Thank you William Hawes.  That's sound advice by an actual practitioner of celestial navigation, performed without the precision of GPS.