NavList:

 Frank Reed wrote:

Imagine this:
Suppose we take an area on the ocean one degree on a side and we divide up it into little patches one tenth of a minute of arc on a side (one tenth of a nautical mile). That gives us a grid of 600 by 600 small patches, for a total of 360,000 squares in all. Now suppose I do some sights near the center of this one degree wide zone and I find three LOPs that cross in a nice equilateral triangle where the side length is one nautical mile --a fairly typical celestial navigation triangle. Note that there are ten of those little patches along one side of the triangle so the total area of the triangle is 43 patches. Now we can ask, what is the probability of being in that square patch right at the center of the triangle where I mark my fix? It's pretty low. I'm not gonna work out exact numbers, mostly because I don't want to waste more time on it, but it's a low number. Let's say 0.65%. So if I did this set of sights a hundred times, the probability of being inside that SPECIFIC patch right at the center where the fix is marked is less than even money.

Now let's examine other square patches in the triangle. As we move away from the center, the probability of being in any small square patch falls off. It's lower and lower as we move away from the center. But the probability of being somewhere in the triangle must total to 25% so, on average, it's about 0.58% in each square (0.58% multiplied by 43 patches adds up to 25%). When we reach the patches right at the corners of the triangle, it would be lower than that, perhaps 0.53% in each square. You can see that the probability DENSITY is falling away rather gradually, so there's not much difference. We're less likely to be in any one of the corners than in the center, but not by a whole lot. So we keep on going... The probability density actually falls off radially with distance away from the center patch so that some points just outside the triangle actually have higher probabilities than the corners of the triangle (they're closer to the triangle's center), but eventually we get to patches where the probability of being inside a particular patch is 0.10%, but there are lot of those little squares all at the same distance from the triangle's center. They add up to a greater total NET probability. Even further out, there are square patches where the probability is only 0.01% --the odds are 10,000-to-one that you are in one of those squares-- but there are lots and lots of those low probability patches. When you add up ALL of those squares outside the triangle, you find that the odds of being outside are THREE TIMES greater than being inside the triangle. But the patches with the highest probability within them are all closest to the center of the triangle and the highest one, by just a bit, is right at the center.

This is a very good explanation, Frank.  You have written an eloquent word-picture that is possibly the clearest post yet on this subject, which has been bounced about here for years.