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Interestingly, the method in which you determine your "random point" can greatly affect the end result, so swapping out the methods (e.g. "generate three coordi
by chias 8y ago
Interestingly, the method in which you determine your "random point" can greatly affect the end result, so swapping out the methods (e.g. "generate three coordinates and discard" method for the "generate a direction and a length" method) may produce wildly different results.
An interesting illustration of this is Bertrand's Paradox, which asks the question: "if you draw a random chord inside a circle, what is the probability that its length is greater than sqrt(3)?". The correct answer is 1/2. The correct answer is also 1/3. Oh, and the correct answer is also 1/4. It depends on three equally valid (and uniformly distributed) ways of defining your random chord.
See: http://web.mit.edu/tee/www/bertrand/ http://web.mit.edu/tee/www/bertrand/
- improbable22 8y agoParadox seems like a strong word for this. "Draw a random chord" is almost a pun, draw from what distribution? Until you tell me, there really is no one answer.
- chias 8y ago> "Draw a random chord" is almost a pun No more so than 'pick a random point in a circle'. Note in all cases we're talking the uniform distribution. The question of using Cartesian coordinates, parametric coordinates, etc. is what causes the differences.
- improbable22 8y agoI'm complaining that there isn't one uniform distribution. For example uniform in x on [1, 2] is different to uniform in 1/x on the same interval, and which one you draw from may matter. You have to be told. In fact every (one-variable) distribution is uniform in something! "pick a random point inside a circle" comes with an implicit "uniformly according to the ordinary euclidean metric on the paper". I guess it's a judgement call that this is obvious enough, but "a random chord" is clearly not... otherwise we wouldn't be here!
- deleted 8y ago[deleted]