# NavList:

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

**Re: Position lines, crossing.**

**From:**George Huxtable

**Date:**2006 Dec 9, 23:36 -0000

I had written- | > In fact, if any systematic errors have been properly corrected for, | > and only random scatter remains, the simple truth is this. Only on one | > time in 4 will the vessel be inside that triangle at all, and 3 times | > in 4 it will be somewhere outside it, though in the vicinity. This is | > a simple statistical truth, easily proved, but one that mariners are | > most reluctant to accept, because it is so contrary to what they have | > been taught. And Lu Abel responded. | ... Is the probability that I'm inside the cocked hat | 1 in 4 or 1 in 8?? | | I presume your "simple statistical truth" is that, absent any additional | information about the quality of the LOPs, there is a 50% chance that my | true position is on one side or the other of the LOP. | | To be inside the cocked hat, it seems to me that I have to be on one | particular side of each of the LOPs, which makes for a probability of | 1/2^3 or 1/8. | | What am I missing? =========================== Lu is nearly there. The supposition is indeed that " there is a 50% chance that my true position is on one side or the other of the LOP." To make the argument simpler, let's consider a particular, simple, geometry, though the rule is indeed true for any shape of triangle. Let's take three objects A, B, C, at azimuths 0 deg, 120 deg, and 240 deg. If there's no systematic error, then it's equally likely that each resulting position line, at right angles to those azimuths, will be towards the object (T) or away from it (A). There are then 8 possible combinations, TTT, TTA, TAT, TAA, ATT, ATA, AAT, AAA. All are equally likely. Of those combinations, both TTT and AAA put the true position within the triangle (that's the bit of the argument that Lu missed). The others put it outside a side or outside a corner of the resulting "cocked hat". So 2 combinations in 8 means that there's only one chance in 4 of being actually inside the cocked hat. Of course, there's an analogous situation when compass bearings are being taken of 3 landmarks. As long as deviation and variation have been properly allowed for, then each bearing has an equal chance of being in error clockwise, as anticlockwise. In the same way, there's only a 1 in 4 chance that the vessel will be contained within the triangle. George. contact George Huxtable at george@huxtable.u-net.com or at +44 1865 820222 (from UK, 01865 820222) or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK. --~--~---------~--~----~------------~-------~--~----~ To post to this group, send email to NavList@fer3.com To unsubscribe, send email to NavList-unsubscribe@fer3.com -~----------~----~----~----~------~----~------~--~---