Does Information Theory shed any light on Bertrand's Paradox?

Whether a "random cord" is less or more than the inscribed equilateral triangle is a binary attribute. Information theory asserts that a binary choice has the least infomation (thus is the most random) when the binary choices are 50% each. For example the result of a coin flip contains less infomation then whether or not a 6 was rolled with a die.

Thus it would seem to be that the solution where the midpoints are distributed uniformly over the radius and the probability becomes 1/2 is the solution with the "most random" choice for the cord.

>Does Information Theory shed any light on Bertrand's
>Paradox?

Indeed, looks like it does.

>Thus it would seem to be that the solution where the
>midpoints are distributed uniformly over the radius and the
>probability becomes 1/2 is the solution with the "most
>random" choice for the cord.

The reason the situation is dubbed a paradox is that there are more than one plausible solution. Yes, Jaynes' and your arguments show that in one sense or another one of the solutions is more plausible or natural than the rest. They do not negate the presense of three solutions.

I do not know, perhaps these arguments should apply to the definition of probability so that Bertrand's chord problem would have a single solution.

>The reason the situation is dubbed a paradox is that there
>are more than one plausible solution.

Agreed. You could of course restate the problem to better define the probability distribution for a random cord and thus remove the ambiguity. The real point I think is to point out how easy it is to make statements in natural language that are subtly ambiguous. Natural language is fuzzy compared to say predicate logic.

I just find these sorts of mind benders fascinating. On one hand we are trying to make computers “think,” but on the other hand we don’t want computers to make mistakes. This kind of mind bender seems to point out that we can’t have it both ways…

>The real point I think is to
>point out how easy it is to make statements in natural
>language that are subtly ambiguous.

Right. There is another current example at

http://www.cut-the-knot.org/htdocs/dcforum/DCForumID11/3.shtml

>Natural language is
>fuzzy compared to say predicate logic.

I am sure this is why we humans use it in the first place. There is an interesting and quite relevant book Words and Rules by S. Pinker. Following rules all the time would be too hard on the brain. But remembering (as opposed to following rules) depends on other memories which are not universally shared; hence frequent ambiguities.

>I just find these sorts of mind benders fascinating. On one
>hand we are trying to make computers “think,” but on the
>other hand we don’t want computers to make mistakes.

You allow here the kind of ambiguity which is the subject of the discussion. We would not of course envisage a computer that returns 5 to a 2×2 = ? querry. On the other hand, it is easy to program a computer to be in a frivolous mode wherein it knowably returns 5 as an answer.

>This
>kind of mind bender seems to point out that we can’t have it
>both ways…

Or at least very difficult. But there is say fuzzy logic which would allow some leeway in terming a mistake mistake.