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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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