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?
Should we randomly chose the definition of random? I know our friend laid out three ways to draw a random chord while leaving out, arguably, the most intuitive way. I thought I remember reading that there were more ways than that even. Math is just a hobby for me so I am not immersed in it.