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.