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Chuck Taylor suggests a computer experiment, testing varied
erroneous bearings:

> 4. Repeat the following 1000 times:
>
> a. For each bearing line, generate a random number from a Normal distribution
> with mean 0 and standard deviation 1. (This is the distribution associated with
> the classic bell-shaped curve; about half the time this number will be negative
> and about half the time positive.) Treat these numbers as the bearing errors in
> degrees. Add these numbers to the bearings.  These will give the simulated
> bearings for one iteration. With this distribution most of the simulated
> bearings will be within about +/- 3 degrees of the actual bearing, which seems
> fairly reasonable for a good hand-bearing compass.
>
> b.  Using the simulated bearings, compute the equations of simulated bearing
> lines from each landmark.
>
> c.  Compute the pairwise intesections of the three simulated bearing lines.
> Thise will be the vertices of the triangle (the "cocked hat").
>
> d.  Determine analytically whether or not the true position lies within the
> triangle.  If yes, tally one for the "yes" column. If no, tally one for the "no"
> column.

You don't have to be so careful about the distribution, or about computing
equations of lines.  The only thing that determines whether the triangle
encloses the true position is the direction of the error of each bearing.
The magnitude of the error affects the triangle's size, and how far it is
from the true position, but doesn't affect which region of the figure contains
the true position.  The triangle encloses the true position if all three
bearings are increased a small amount, or if all three bearings are reduced
a small amount, but in no other case.  This is the heart of George Huxtable's
argument for the 25% probability that the triangle encloses the true position.
Once the argument has been simplified to this degree, it's easy to solve
analytically...

   

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