# NavList:

## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

**Re: Still on LOP's**

**From:**Trevor Kenchington

**Date:**2002 Apr 21, 21:43 -0300

Geoffrey, Thank you for posting your diagrams. You have examined the problem from the perspective of a known (by the Almighty) true position and three fixed points from which bearings are taken. Given the assumption of symmetrical errors (which seems reasonable, at least as a first approximation), I cannot find fault with your argument that there are eight possible, equally-likely classes of outcomes, of which only two place the cocked hat around the true position. The probability of the cocked hat thus enclosing the true position (equal to that of the true position being in the 'hat) is 0.25. That assumes, of course, that the errors in the three bearings are independent but I cannot see how they would not be if we were talking about bearings taken from landward by different observers. Nor would I suggest that those bearings would lose independence simply by having their reciprocals taken from seaward. I have been arguing from a model in which we have a cocked hat and seek to know the probability of the true position lying in various areas in and around it. That way of viewing the problem leads to there only being seven classes of outcomes (all equally likely, if the observations were independent), only one of which places the true position inside the cocked hat, and hence to a probability of the true position being in the 'hat of about 0.143. However, for the reasons I outlined in my last, I doubt that the observations are independent when considered in that model. Could it be that we are both right? Could the effect of the non-independence in my model be such as to raise the probability from 0.143 to 0.25? But that would still leave us with the problem of what happens when the cocked hat chances to be very small or very large. Can you explain how the probability remains at 0.25? As to George's questions: >How many list members still dispute that, or remain unconvinced? > >How many are prepared to say that they still believe that a cocked hat MUST >embrace the true position? > Count me as still unconvinced for the former (over the years, I've seen too many mistakes in non-statisticians' attempts to figure out such problems!) but I'd definitely agree that there is not now nor ever was any reason to think that the true position MUST lie within the cocked hat. That was always a misunderstanding of the consequences of statistical distributions. Trevor Kenchington