Hi mpresic:
mpresic3 said:
We should indeed be careful about using the word "random".
I agree that some care is needed when discussing randomness, but I think that the real issue that creates confusion is insufficient care in defining what "equally likely" means.
mpresic3 said:
I heard a story about astronomers, who for some reason needed a random star. One astronomer chose random right ascension and declination and found the star closest to this polar coordinate. Another astronomer listed the stars in the stellar catalog from 1 to N N very large. Then generated a random number from 1 to N and looked at the star corresponding to this N. Clearly, the definition of "random" leads to different results.
Here the search for a random selection involves that the selection is from a
finite number of choices.
If one wants each of the finite possibilities to be equally likely to be chosen, then selecting from the stellar catalog gives the desired result, while basing the choice on closeness to random orientation clearly fails to achieve equal likelihood of choice.
mpresic3 said:
One can see from Bertrand's paradox for choosing a chord at random on a circle you can get three different probabilities for this seemingly well posed problems.
These problems involves a transcendental cardinality. I think my interpretations are correct, but I am not strongly confident that I have avoided mistakes.
If I were given the task to choose a method, it would be that the
length of the chord would be randomly chosen with equal likelihood, and the angle of a radius bisecting the chord would be randomly chosen with equal likelihood.
Argument 1:
The "random endpoints" method: Choose two random points on the circumference of the circle and draw the chord joining them.
This involves the choices of chords whose corresponding arcs have lengths which are equally likely to be chosen.
Argument 2:
The "random radial point" method: Choose a radius of the circle, choose a point on the radius and construct the chord through this point and
perpendicular to the radius.
This involves the choices of chords whose corresponding pair (
r,
a) of values are equally likely to be chosen. The value of
r is chosen as a distance from center of the circle to a point on a line of length equal to the radius. The value of
a is is a length along the circumference from an arbitrary reference point on the circumference.
Argument 3:
The "random midpoint" method: Choose a point anywhere within the circle and construct a chord with the chosen point as its midpoint
This involves the choices of chords whose corresponding centers are equally likely to be chosen from all the points inside the circle.
Regards,
Buzz