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    Re: Still on LOP's
    From: Trevor Kenchington
    Date: 2002 Apr 21, 21:43 -0300

    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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