Title: Bertrand's Pair 'o Ducks
1Bertrand's Pair 'o Ducks
- An exploration of choices
Ray McGovern
2 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
3 - Calcul des Probabilities - 1888 - his book
exploring continuous probabilities, became known
as Bertrands Paradox
3Get 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
4Why a duck?
5Lets 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
6Given a circle,
x
A
B
y
7How 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
8Case 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
9Case 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
10Case 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,
11Care to guess?
1/4
1/2
1/3
Youre probably right!
12And 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
13So, in conclusion
1/4 1/2 1/3
Your answer will depend on your perspective.
14Sources Introduction to Probability, Grinstead,
pp. 47-50