I would like to play with the simulation(s) for Betrand's paradox in order to find the flaw in my reasoning. :)
I am a software engineer and a very casual armchair mathematician, but the following thought experiment leads me to the conclusion that one of the three selection methods is invalid. However it also stands to reason that there must be a flaw in the following since this problem has stood for over 100 years and no one has declared the following observation a fault.
Consider the "random radial point" method. Described on Wikipedia as “Choose a radius of the circle, choose a point on the radius and construct the chord through this point and perpendicular to the radius” If we assume for a moment that the circle has its center at the origin, these directions could be rendered simply as choose an angle theta and a distance r’ < r. Or more simply choose a point in the interior of the circle using polar notation. If we imagine that r’ is the radius of an inscribed circle we can observe that all chords drawn tangent to the inscribed circle have equal length. This again tells us that the angle theta at which the chosen radius is drawn has no bearing on the length of the resulting chord, or the probability of its length being longer or shorter than s. Since the angle is not important, it may be instructive to choose a convenient angle like 0 (or the positive X axis in the Cartesian plane). Doing this we observe that the probability of our chord being longer than s is 0.5. But we can also make a couple of other observations. First we note that the numerical representation of our chosen point is the same for the polar coordinate (r,theta) and the Cartesian coordinate (X,Y). Somehow it seems we doing something equivalent to choosing a random interior point with its Y coordinate fixed to 0. Second we should observe that the midpoint of each chord lies on the X axis. So this method is simply the “random midpoint” method in disguise. This method appears to be a fraud and a cheat, as it makes a show of choosing two random numbers to select a chord even though only one of them has any bearing on the length of the chord.
On the one hand the method seems valid in that it can select any chord of the circle. But on the other hand seems invalid since it discards half the randomness (entropy?) generated in the selection of the chord’s midpoint.
I would like to play with the simulation(s) for Betrand's paradox in order to find the flaw in my reasoning. :)