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    Probabilities 75/25: a "realistic" scenario
    From: Frank Reed
    Date: 2010 Dec 18, 23:27 -0800

    I think part of the problem that people have wrapping their brains around this issue of the probability of being inside the "cocked hat" has to do with the more general issue of the meanings of probabilistic statements. Here's a classic that I am sure many of you have puzzled over: "Tomorrow, there is a 25% chance of snow." What does that mean exactly?? Will it snow definitely but only at 25% of locations in the forecast area? Will the weather system flip coins at dawn and only snow if they come up "heads-heads"?? I won't answer this one specifically, since this isn't a weather group, but suggest that you google "weather probability" if you're wondering. Back to navigation...

    What does it mean to say that there is a 75% chance of being outside the triangle formed by a three-body fix? Let's imagine an ocean race where the participants will be rewarded, in part, based on the accuracy of their celestial navigation fixes. I send two vessels sailing back and forth across the Pacific. One is a well-equipped sailing yacht, call it Alpha, and the team has chosen to use a plastic Davis sextant and other "low accuracy" celestial tools (they get a handicap of some sort for this). The other vessel, call it Beta, is a large commercial ship, and the navigator is using a fine old Plath sextant. The navigators on both vessels are competent. They sail about for three years straight and each collect a grand total of 1000 three-body fixes. Each records the "cocked hat" and the plotted fix. To judge this event, each vessel also carries a sealed GPS unit without a display. This device is kept near the navigator and records the actual position from which the observations are made to within a few meters. When the game is ready to be judged, we collect the plots and the fixes, open up the GPS recording units and compare. From this we calculate the mean error in position in nautical miles. We ALSO count the number of times that the recorded GPS position falls with the "cocked hat" on the plots.

    Here are some typical results we could expect:
    Alpha: Mean error in position: 2.2 n.m. GPS position in hat: 256.
    Beta: Mean error in position: 0.7 n.m. GPS position in hat: 239.

    Beta has done better navigationally. The fixes are closer to the vessel's actual location at the time, on average. But Alpha's GPS position was inside the cocked hat more often. What does that mean? Did they somehow do better? No. Of course not. If we repeat this game, again and again, with different instruments and different navigators, the number of fixes within the triangle would always be around 250 for every 1000 fixes, and there would be no connection with the accuracy or quality of the navigation if the number is a little higher or lower. The number of times that the true position of the vessel, as determined by GPS or some other very high accuracy position-finding system, falls within the three-body fix triangle of a celestial fix will always be about 25%. The number in an actual run of a thousand trials may be a little higher or a little lower than 250 only because "that's the luck of the draw".

    I don't know if this "realistic" scenario is actually helpful to anyone, but i figure it can't hurt. I guess I would estimate that it won't hurt 95% of the time. :)

    -FER


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