Geometric Probability Brittany Crawford-Purcell

Bertrand’s Paradox “Given a circle. Find the probability that a chord chosen at random be longer than the side of an inscribed equilateral triangle.”

Solution 1 We need to randomly choose 2 points on the circle. First point doesn’t matter, only the second point does. Make the first point fixed. Focus on the chords that extend from the fixed point

Solution 2 Chords are determined by midpoints. So, let’s focus on the midpoints. Circle inscribed into an equilateral triangle that is inscribed in a circle.

Area of small circle Area of large circle

Solution 3 Focus on the distance of the chord to the center of the circle The chord is greater than √3 (length of the side of the equilateral triangle) if the distance to the center of the circle is smaller than 1/2

Which is correct? Look at the distribution