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Geoffrey,

You wrote:

>Coming to your question above where you are concerned that even when the
>cocked hat is very small, how can it be that the probability remains at 0.25?
>
>If I may say so, I think that this question is the result of trying to put
>too much statistical significance on one cocked hat. If you did three more
>measurements on the same landmarks and drew another cocked hat, the size of
>the cocked hat would not be the same. When plotted on the chart the cocked
>hat might not even be in the same place, as I have shown.
>

Indeed. But, unless I am mistaking something (which is by no means
impossible!), the probability that the cocked hat lies over the true
position is numerically equal to the probability that that true position
lies within the cocked hat.

The former you have shown to be 0.25 and I cannot find a mistake in your
logic. Thus, if we actually knew our true position but took a whole lot
of bearings anyway (each set independent of the others), we would end up
with lots of cocked hats, 75% of which would not enclose our position.
However, in practical applications, we do not know our true position and
we only construct a single cocked hat. From your proof that the number
is 0.25 and my postulate that the two probabilities are numerically
equal, I am led (painfully!) to conclude that George was right and the
probability that a navigator's true position lies inside his cocked hat
is only 0.25.

The fact that other possible cocked hats would have lain in other places
is a necessary part of your proof but does not, so far as I can see,
affect one particular navigator's conclusions about the particular 'hat
that he has just drawn.

Yet that 'hat can be very small or very large. Those variations in size
can be due to different levels of uncertainty in the measurement of each
bearing, in which case the probability of being in the 'hat need not
change. (Worse bearings give bigger area in which you might be located.)
But the varied sizes can also be due to chance effects: Three "bad"
bearings cam all pass through (very nearly) the same intersection point.
I could understand that the best estimate of the MPP was at the
intersection and the best estimate of a confidence circle around that
point would be a very small circle. But those are only estimates. Your
logic leads to the conclusion that, even when a cocked hat is very small
because of chance effects, there is exactly a 0.25 probability of the
true position lying inside it. And that doesn't make any sense to me at all.


Maybe I need to go and stand on a well-charted point of land with three
lighthouses in sight and take 100 bearings on each one to see how the
'hats fall out.  Then again, Canadian Hydrographic Service charts are
wrong in local detail so often that I might not believe the result anyway!


Trevor Kenchington

   

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