Bertrand's Pair 'o Ducks

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Bertrand's Pair 'o Ducks

• An exploration of choices

Ray McGovern
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Joseph Louis Francois Bertrand
1882 - 1900

• French mathematician naturalist
• Known for applications of differential equations
  to physics
• Worked on differential geometry and probability
  theory
• One prime between n and 2n-2 for every n gt
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• Calcul des Probabilities - 1888 - his book
  exploring continuous probabilities, became known
  as Bertrands Paradox

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Get a Job!
He sat around making observations
Watched his wife sewing,
she kept dropping the needles, he made her nervous
Computed probability of the needles landing on
the lines of the floorboards
She finally chased him out of the house
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Why a duck?
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Lets consider
First, assume a unit circle
Then, drawing random cords whose end points lie
on the circle
What is the probability the cord length exceeds
is the length of the inscribed side of an
equilateral triangle. A cord will have a length
gt if the midpoint has a distance d lt
1/2 from the origin
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Given a circle,
x
A
B
y
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How do we reference the cords?
The cord of a circle is perpendicular To the
radial line to the midpoint of the cord
Rectangular coordinates of the midpoint, M
Polar coordinates of the midpoint, M
Polar coordinates and of the
endpoints A and B of the cord
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Case 1 (x,y) of the midpoint, M
Choose random values for (x,y) in the range -1,1
Verify
Must be true or else M (x,y) is outside the
circle
Then, the length
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Case 2 Polar
of the midpoint, M
Circle rotation doesnt change length of
cord, Assume a horizontal line
Choose random values of r in the range -1,1
Length of cord with midpoint
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Case 3 Polar
of the endpoints A and B
Assume B, one of the end points, lies at (1,0),
then
Choose random values for
In the range
Then by the law of Cosines,
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Care to guess?
1/4
1/2
1/3
Youre probably right!
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And the answer, It depends
L gt v3 if(x, y) lies inside a circle of radius
1/2, which occurs with probability
L gt v3 if r lt 1/2, which occurs with
probability
L gt v3 if 2p/3 lt a lt 4p/3, which occurs with
probability
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So, in conclusion
1/4 1/2 1/3
Your answer will depend on your perspective.
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Sources Introduction to Probability, Grinstead,
pp. 47-50

• betrand's paradox demo