21 replies to this topic

[Prof. Templeton]

I agree with CaptainEd. Jaynes' solution to the Bertrand paradox shows that random radius (method 2) is the only robust theoretical means of choosing random chord.

So, then it is more random. I looked up Jaynes' proposal and it based on the fact that the size of the circle is unknown, here the OP has given us a size. Is throwing straws at a circle a better method than selecting a number at random between 0 and πr² , letting that number represent an area and drawing a chord to section off that area?

[bonanova]

Indeed, more than one way.

Bonanova, what is the goal of this puzzle?
a) show us that the notion of "randomly chosen" depends on how you define it?
b) have us show lots of ingenuity in defining truly different ways to define it?
c) elicit from us the really-really-best way to define it?
d) all of the above
e) none of the above
f) other

All of the above, probably.

Let's make it a competition to show the greatest and least among any reasonable definition of randomness.

[bonanova]

Well since I neglected to use the word "average" in the OP, another answer is anything between 0 and 2.
That was the "fourth" answer.

[joef1000]

Spoiler for

Well the longest chord possible is the diameter, and the shortest possible is a tangent (0 length). On average, you will get something between the two so the average is half the diameter, AKA, the radius!

Edited by joef1000, 22 November 2012 - 04:40 PM.

[joef1000]

My above solution assumes that you define random in a way that makes all chord lengths equally likely.
(I hate the edit system on this site)

Edited by joef1000, 22 November 2012 - 04:52 PM.

[bonanova]

Spoiler for

Well the longest chord possible is the diameter, and the shortest possible is a tangent (0 length). On average, you will get something between the two so the average is half the diameter, AKA, the radius!

This is an interesting approach. The radius is a middle measure of the extremes of zero and the diameter, but It is the median, not the mean. One must represent (find an expression for) all the uncountably infinite chords and then examine the distribution of their lengths. If the distribution were found to be uniform from 0 to 2r then r would be correct. All the examples given previously in this thread, however, show the lengths not to be uniformly distributed.

[bonanova]

My above solution assumes that you define random in a way that makes all chord lengths equally likely.
(I hate the edit system on this site)

I'm not sure you can define random in any sense other than an unbiased selection from all available instances. The challenge that arises is that several exhaustive, random [in some sense] and uncountable subsets each yield a distinct value of average length.

[Prof. Templeton]

After doing some research and none of the math, I've found

Spoiler for that

The shortest mean chord length was 1.00 and the longest 1.81 depending on the method used to make a random chord.

[bonanova]

After doing some research and none of the math, I've found

Spoiler for that

The shortest mean chord length was 1.00 and the longest 1.81 depending on the method used to make a random chord.

Inscribe a regular polygon of n sides where n > 1.
Take the chord to be one side of the polygon; (n = 2 gives a diameter.)
Let n be chosen at random.

What is that average chord length?

[Prof. Templeton]

Inscribe a regular polygon of n sides where n > 1.
Take the chord to be one side of the polygon; (n = 2 gives a diameter.)
Let n be chosen at random.

What is that average chord length?

Quite small. The larger n is the smaller the chord length. I would argue that this is not an acceptable way to chose "random chords", however.