Showing posts with label Bertrand's Random Chord Paradox. Show all posts
Showing posts with label Bertrand's Random Chord Paradox. Show all posts

Wednesday, January 14, 2015

Bertrand's Random Chord Paradox Methods 2 and 3


   In my April 21, 2014 post, Bertrand’s Random Chord Paradox 1  (http://www.scratch-blog.com/2014/04/bertrands-random-chord-paradox-1.html) I presented the first of three methods for randomly selecting a chord in a circle with an inscribed equilateral triangle and then using the Monte Carlo method (coded in Scratch) to experimentally determine the probability that a randomly selected chord is longer than the side of the inscribed triangle. For the random points on the circumference method discussed in the post, the probability is one-third (1/3).
   The paradox arises from the fact that two other equally valid methods for picking the chord each give a probability different from one-third. 
   In my Scratch project Bertrand’s Random Chord Paradox 2 (http://scratch.mit.edu/projects/20392511/)
the Monte Carlo method correctly approximates the probability that a randomly selected chord, using the random radius method, is one-half.
   In my Scratch project Bertrand’s Random Chord Paradox 3 
(http://scratch.mit.edu/projects/21000750/)
using the random midpoint method, the probability is found to be one-fourth!
   The theoretical derivations for the three methods are presented in a three free, PDF files that can be obtained by sending an email request to grandadscience@gmail.com.