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    Re: Cocked hats, again.
    From: Gary LaPook
    Date: 2007 Mar 15, 15:34 -0700

    Gary LaPook wrote:
    Sorry about all the paper but I wanted to take it one step at a time so
    it could be easily followed.
    I agree that most cases would be clustered near the intersection. I
    could have drawn the diagrams with a bunch of little boxes making a
    finer mesh of the probabilities but I didn't  want to make it even more
    complex than it already was and I just wanted to illustrate the concept.
    Regarding your comment about diagram 11, you will agree that it is
    correct in stating that some of the boxes will have somewhat more and
    others somewhat less that one quarter of the cases. I didn't attempt to
    quantify the size of the differences and they may be quite small as you
    point out, but a true statement, nonetheless.
    I don't know why it comes out this way after considering your statement
    of the situation.  I am still hung up on the fact that your formulation
    doesn't take into account the size of the triangle. As I  mentioned in
    another post today, that if you draw circles of probability around the
    fix, that sometimes the circle will be larger and sometimes smaller than
    the triangle depending on the size of the triangle. This is because the
    radii of the circles are fixed by sigma. This then causes the proportion
    of the locations inside versus outside the triangle to also vary with
    the size of the triangle.
    Looking at the logic you use, basically 2^3 possibilities, I wonder if
    it doesn't break down as the third line is moved further from the
    intersection of the other two lines. At some point the line is so far
    away that it is no longer true that you have an equal chance to be on
    either side of that third line. You must be on the side of the line that
    allows you, at the same time, to be within the possible positions as
    limited by the first two lines as shown in figure 6. These situations
    were illustrated in figures 14, 16 and 18.
    I agree with you that as the third line is moved further from the first
    intersection that you find yourself in rare territory. The very extreme
    example shown by figure 18 would occur, at most,  only once in a billion
    times but I put it in to illustrate my argument that the size of the
    triangle changes the ratio of positions within and without the triangle.
    So, I am still confused. Your explanation makes sense to me too and I
    don't claim to be a mathematician, I only have a plotter and a pencil.
    But, on the upside, we do agree on the main point of this discussion,
    that a navigator needs to be aware, that in spite of what he might have
    been taught, his true position may be significantly outside the cocked
    hat and he better allow for that possibility.
    Regarding your other post today, I too learned that your position was
    always inside the triangle with the most likely point being the center.
    We now know that was bad information.
    George Huxtable wrote:
    >Gary appears to be still finding it difficult to accept the 1 in 4
    >argument for the probability of a fix-triangle embracing the true
    >position of an observer.
    >First, I ask him if he has seriously considered the statistical
    >argument that has been put forward in support of that argument, and
    >whether he has found any holes or flaws that cause him to reject its
    >He has put forward a counter example, which I have done my best to
    >follow, ignoring the mock-biblical language. To me, it doesn't address
    >the point in question. It applies only to the restricted situation
    >where the intercepts are several sigmas from the true position, and
    >that applies to a tiny fraction only of possible observations. So even
    >if Gary's conclusion, in those rare situations, is correct, it has
    >little relevance, on average, to the 1 in 4 question. That applies to
    >the whole gamut of possible triangle, large ones and small ones,
    >hardly any of which will correspond to Gary's rare example.
    >Going to his figure 6, that shows a probability of 1/4 of the true
    >position being in each of those 4 quadrants, which is fair enough. But
    >what will be the actual distribution of possible positions? It will
    >correspond to a crowd of dots, like a swarm of bees, centred on where
    >the two position lines cross. Nearly all the observations will lie
    >near the centre of that shaded area, and hardly any near the edges,
    >because Gary has chosen to put its boundary several sigmas away from
    >the crossing point. So when he chooses, in figure 11, to restrict the
    >area further, by imposing a limit on its possible distance from the
    >new diagonal position lines, that may cut off two corners of the
    >shaded area, but will exclude none (or hardly any) of the possible
    >So when he writes- "Diagram 11 shows that the northwest and southeast
    >quadrants now have somewhat more than a one in four chance and the
    >southwest and northeast each have somewhat less than a one in four
    >probability", that isn't so. Although the areas may differ, the
    >populations of those four areas will differ only insignificantly.
    >Any argument based on such rare events has no bearing on the
    >statistical distribution of the vast majority of observations, which
    >will lie near the centre of his picture.
    >Finally, a request. If Garry has more sketches to send, I ask him to
    >scan several such diagrams to each sheet, as I'm now getting submerged
    >in paper.
    >contact George Huxtable at george@huxtable.u-net.com
    >or at +44 1865 820222 (from UK, 01865 820222)
    >or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
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