Geometric Probability

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Geometric Probability

• Brittany Crawford-Purcell

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Bertrands Paradox

• Given a circle. Find the probability that a
  chord chosen at random be longer than the side of
  an inscribed equilateral triangle.

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Solution 1

• We need to randomly choose 2 points on the
  circle.
• First point doesnt matter, only the second point
  does.
• Make the first point fixed.
• Focus on the chords that extend from the fixed
  point

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Solution 2

• Chords are determined by midpoints.
• So, lets focus on the midpoints.
• Circle inscribed into an equilateral triangle
  that is inscribed in a circle.

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Area of small circle Area of large circle
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Solution 3

• Focus on the distance of the chord to the center
  of the circle
• The chord is greater than v3 (length of the side
  of the equilateral triangle) if the distance to
  the center of the circle is smaller than 1/2

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Which is correct?

• Look at the distribution