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The long debate over the figure 0.25 probability of being in a Cocked
hat continues to spill out of the pub after navigation classes and
into the scholarly and mathematical literature of navigation.

I think we noticed this one in pre print form but it has now appeared
in the Journal of Navigation


https://www.cambridge.org/core/journals/journal-of-navigation/article/abs/cocked-hat-formal-statements-and-proofs-of-the-theorems/5264C1491A61116CAF7B890161CFD091

The cocked hat: formal statements and proofs of the theorems
Imre Bárány , William Steiger  and Sivan Toled

Abstract

Navigators have been taught for centuries to estimate the location of
their craft on a map from three lines of position, for redundancy. The
three lines typically form a triangle, called a cocked hat. How is the
location of the craft related to the triangle? For more than 80 years
navigators have also been taught that, if each line of position is
equally likely to pass to the right and to the left of the true
location, then the likelihood that the craft is in the triangle is
exactly 1/4. This is stated in numerous reputable sources, but was
never stated or proved in a mathematically formal and rigorous
fashion. In this paper we prove that the likelihood is indeed 1/4 if
we assume that the lines of position always intersect pairwise. We
also show that the result does not hold under weaker (and more
reasonable) assumptions, and we prove a generalisation to n lines.

   

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