# NavList:

## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

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Re: Still on LOP's
From: Martin Gardner
Date: 2002 Apr 18, 07:25 -0700

```George Huxtable wrote -

> Well, let's say we are determining our position by bearings on three
> distant landmarks, 1, 2, and 3.
>
> There is an equal chance that, due to errors in taking the bearing from
> landmark 1, that bearing will lie to the left of the true position as to
> the right of it. If we take the possibility that the bearing can be exactly
> on the line of the true position to be zero, the probability of it being on
> the left (L) is 0.5, the same as it being on the right (R). We can say the
> same about landmarks 2 and 3.
>
> There are eight possible combinations, if we list the three bearings, taken
> in the order 1, 2, 3, as follows-
>
> LLL, LLR, LRL, LRR, RLL, RLR, RRL, RRR.
>
> For each such combination, because it combines 3 terms each with a
> probability of 0.5, its probability is (0.5) cubed, or 0.125. There are 8
> such combinations, each with a probability if 0.125, so that looks right,
> doesn't it?

George, it sure does.  But when I draws it out, the plane is divided into
only seven areas: the inside of the triangle, the areas outside each side,
and the areas outside each vertex.  One of the combinations cannot occur on
a real plane (which one depends on how you draw and number your lines).
>
> However, of those 8 combinations, there are only two which place the true
> position inside the cocked hat. These are LLL and RRR. This can be seen if
> a drawing is made showing all the 8 options. The other combinations put the
> true position outside a side or outside a corner. So the probability of the
> true position being inside the cocked hat is exactly 0.125 x 2, or 0.25,
> which is what we set out to show.

When I drew this, RRR and LLL lay outside my triangle.  I was able to redraw
it so that RRR lay inside, but then there is no area LLL.

I think something went awry in your reasoning:  for any given three lines
intersecting to form a triangle, there is only one area  'inside'.  If two
different patterns of R and L both described some of the 'inside' then the
'inside' would have to be partitioned, which it is not.
>
> What seems at first so surprising is that this result is quite independent
> of the skill of the navigator. The reason for this is that the better
> navigator will produce, on average, a smaller cocked hat, But the
> probability of it embracing the true position will remain at 1 in 4.

I don't have an opinion on whether the probability is 1 in 4 (or 1 in 7, or
one in 8); I have encountered this line of reasoning before, and I think
there's something amiss with it, though I can't exactly quantify what.

My worry is that I don't think the three bearings are actually independent.
It's not as if we were dropping three long straws at random on a chart -
we're in some sense measuring the same thing all three times.

Martin Gardner
Venice CA

```
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