NavList:
A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
Prob 1/4 of being in a cocked hat
From: Bill Lionheart
Date: 2021 Feb 20, 22:14 +0000
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 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.
