What is the probability that a randomly drawn chord will be longer than the radius of the circle?
(In reply to re(2): Different Approach (Continued)
DJ. I admit that restricting my sample to discrete degrees resulted in a slight understatement, but I take some solace that my response was approx. 99.5% of yours. I'm not certain I understand Aaron's statement. It is true that the sine function changes as the degree of the angle changes, but that is not what we have here. We are asking if the number of chords within one segment of a circle is idential to the number withing an identical segment within a different portion of the circle. The answer is yes, there is no difference. The chords that can be drawn from 0 degrees to 5 degrees are precisely the same as those that can be drawn in any other 5 degree sector of the circle. The proof I think is straight-forward. Incidentally, are you the same person who proposed the Origami question (computing the length of a fold in a rectangle?). That was challenging! I spent a significant amount of time on it (admittedly struggling as to how to explain my "solution" clearly). Assuming you are the same person, please let me know, when convenient, if my proposed answer is correct. I posted it a few days ago. I believe the length of the fold is equal to the square root of the diagonal of the rectangle multiplied by the width over the length, or W/L x square root of (W squared + L squared).
Thanks for providing the mental challenges.