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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

   

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