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    Re: Cocked hats, again.
    From: George Huxtable
    Date: 2007 Mar 15, 21:42 -0000

    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
    conclusions.
    
    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
    observations.
    
    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.
    
    George.
    
    contact George Huxtable at george---.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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