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
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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
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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
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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?
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Look at the distribution