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    Re: Position lines, crossing.
    From: George Huxtable
    Date: 2006 Dec 9, 23:36 -0000

    I had written-
    
    | > In fact, if any systematic errors have been properly corrected
    for,
    | > and only random scatter remains, the simple truth is this. Only on
    one
    | > time in 4 will the vessel be inside that triangle at all, and 3
    times
    | > in 4 it will be somewhere outside it, though in the vicinity. This
    is
    | > a simple statistical truth, easily proved, but one that mariners
    are
    | > most reluctant to accept, because it is so contrary to what they
    have
    | > been taught.
    
    And Lu Abel responded.
    
    | ... Is the probability that I'm inside the cocked hat
    | 1 in 4 or 1 in 8??
    |
    | I presume your "simple statistical truth" is that, absent any
    additional
    | information about the quality of the LOPs, there is a 50% chance
    that my
    | true position is on one side or the other of the LOP.
    |
    | To be inside the cocked hat, it seems to me that I have to be on one
    | particular side of each of the LOPs, which makes for a probability
    of
    | 1/2^3 or 1/8.
    |
    | What am I missing?
    
    ===========================
    
    Lu is nearly there. The supposition is indeed that " there is a 50%
    chance that my
    true position is on one side or the other of the LOP."
    
    To make the argument simpler, let's consider a particular, simple,
    geometry, though the rule is indeed true for any shape of triangle.
    
    Let's take three objects A, B, C, at azimuths 0 deg, 120 deg, and 240
    deg. If there's no systematic error, then it's equally likely that
    each resulting position line, at right angles to those azimuths, will
    be towards the object (T) or away from it (A).
    
    There are then 8 possible combinations, TTT, TTA, TAT, TAA, ATT, ATA,
    AAT, AAA. All are equally likely. Of those combinations, both TTT and
    AAA put the true position within the triangle (that's the bit of the
    argument that Lu missed). The others put it outside a side or outside
    a corner of the resulting "cocked hat". So 2 combinations in 8 means
    that there's only one chance in 4 of being actually inside the cocked
    hat.
    
    Of course, there's an analogous situation when compass bearings are
    being taken of 3 landmarks. As long as deviation and variation have
    been properly allowed for, then each bearing has an equal chance of
    being in error clockwise, as anticlockwise. In the same way, there's
    only a 1 in 4 chance that the vessel will be contained within the
    triangle.
    
    George.
    
    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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