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Bill (in his video) assumes that errors attached to the three LOPs have a symmetrical Gaussian distribution.  This implies that there is a 50% chance that the true position lies on one side or the other of the plotted line.  Crudely, this implies that the probability of the true position lying within the 'cocked hat' is 0.5 x 0.5 x 0.5 - ie, a 7 to 1 chance that it lies outside.  I recall a discussion some years ago which refined the calculation of this probability, but still resulting in a greater chance of the true position lying outside the triangle.  Rather than concentrating on the niceties of finding the point of highest probability within the triangle, how would one approach the more useful specification of a boundary (circle?) which defines the inclusion of the true position with some specified probability?

Geoff Butt