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    Re: That darned old cocked hat
    From: John Karl
    Date: 2010 Dec 9, 09:53 -0800

    Yes, George once had me convinced that the probability of being outside the hat is 75% for all cases. But after giving it more thought, I don’t believe it. My logic goes like this:

    We’re considering a normal distribution of random errors (no other type of errors) that is known a priori from many tests. The probability of the true fix being at any given point is the product of these distributions (see the attached PDF file). This Probability P(x,y) is the probability per unit area. And it can never exceed 1.0. The probability of the correct fix being inside any given area is the integral of P(x,y) over that area. As the area of the cocked hat gets smaller, and tends to zero, P(x,y) does not get arbitrarily large. In fact it can never exceed 1.0. Therefore as the hat’s area get smaller, the probability that the true fix is inside the hat goes to zero. If you get a tiny cocked hat, in all likelihood you’re not there.

    The probabilities quoted in the second attachment below are approximate because, for the integration, I approximated the hat triangle with the closest contour (inside and outside the triangle to get a good estimate). Whatever the accuracy of these numerical integrations are, they don’t change the above argument.

    I think where George’s argument goes wrong is not thinking about integrating a variable probability/area over some specified area (like inside the hat).

    The attached file also shows that all the best estimates that I have seen (Fermat's point, the center of gravity, the bisecting angle point, and the Steiner point) can not be the most probable location of the true fix. (An observation that's of no practical use.)

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