A while back, my brother and I ended up in a discussion of the probability of three random points in a plane or plane figure always making a triangle. The method of producing this would be akin to shooting darts at a board, one at a time. This reduces then to any given line, and the likelihood of a randomly picked (the third) point being on the line. (I know, it’s backwards to the way I said it, but that’s how the discussion progressed.)
I claimed that if you’re talking about mathematical objects, the line is infinitesimally thin and the probability is zero. I seem to recall hearing that in a course while discussing continous distribution; the whole thing may have a density but a particular segment has zero prob. My brother (who earned a degree doing a lot of numerical analysis) wanted me to write a quick program to test this (I don’t have a copy of Fortran, unfortunately for him). I refused to do the program, saying it wouldn’t show anything. Once he corrected me on that matter, I refused to do it on the grounds that it would go to zero, as I could see (or thought I could see) already.
I never did write the program, but neither was I exceptionally satisfied with my answer. So, real-world situations kept out of it, is this probability zero? Or is it computable?
Yes, the probability is indeed zero. The probability that a given point lies in a give region is (area of given region) / (total area). Here, the area of given region is zero, so the probability is zero.
However, in a computer simulation, a number is represented with a finite number of digits. So in your program, if the third point lies sufficiently close to the line, the program thinks the distance is zero when in fact it is an extremely small non-zero distance. Therefore your computer program will come up with a meaningless non-zero number as a result. You can show that the result is meaningless by compiling it on different computers, or changing the data type (float or double). You should get different results.
Just thought I’d mention something about the numerical methods, since this was the very misconception that by brother cleared me up on.
When done properly (or ‘effectively’, if you will), the numerical analysis doesn’t just go straight for getting the minimum displayable number. Floating point arithmetic is often avoided to prevent machine error, and fixed point (using integers) is used instead. That’s not the crux, though. The important idea is convergence. The analysis detects the pattern; in this case, a pattern of heading for zero as the area of a patch of the grid is decreased. So you get some finite value, then wait and see where it goes as you approach the theoretical ideal.
Hope this doesn’t confuse anyone. Further questions, ask me.