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    Prob 1/4 of being in a cocked hat
    From: Bill Lionheart
    Date: 2021 Feb 20, 22:14 +0000

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