The distribution of chord lengths in a circle, Bertrand paradox - matlab

Question

I want to write a program which will calculate the statistic in Bertrand paradox.
In my way, I want select two dot in circle , and pass line using them (two dots), it is my chord. Then I want to calculate how many of these chords are longer than sqrt(3); but when I run this script some of the chords are bigger than 2 ! ( radius of my circle is 1 )
I don't know what is wrong with it, can anybody help me?
See this link please for the formula used.
r1 = rand(1,1000000);
teta1 = 2*pi * rand(1,1000000);
x1 = r1 .* (cos(teta1));
y1 = r1 .* (sin(teta1));
r2 = rand(1,1000000);
teta2 = 2*pi * rand(1,1000000);
x2 = r2 .* (cos(teta2));
y2 = r2 .* (sin(teta2));
%solve this equation : solve('(t*x2 +(1-t)*x1)^2 +(t*y2 +(1-t)*y1)^2 =1', 't');
t1= ((- x1.^2.*y2.^2 + x1.^2 + 2*x1.*x2.*y1.*y2 - 2*x1.*x2 - x2.^2.*y1.^2 + x2.^2 + y1.^2 - 2*y1.*y2 + y2.^2).^(1/2) - x1.*x2 - y1.*y2 + x1.^2 + y1.^2)/(x1.^2 - 2*x1.*x2 + x2.^2 + y1.^2 - 2*y1.*y2 + y2.^2);
t2= -((- x1.^2.*y2.^2 + x1.^2 + 2*x1.*x2.*y1.*y2 - 2*x1.*x2 - x2.^2.*y1.^2 + x2.^2 + y1.^2 - 2*y1.*y2 + y2.^2).^(1/2) + x1.*x2 + y1.*y2 - x1.^2 - y1.^2)/(x1.^2 - 2*x1.*x2 + x2.^2 + y1.^2 - 2*y1.*y2 + y2.^2);
length = abs (t1-t2) * sqrt (( x2-x1).^2 + (y2-y1).^2);
hist(length)
flag = 0;
for check = length
if( check > sqrt(3) )
flag = flag + 1;
end
end
prob = (flag/1000000)^2;

Answer 1

Your formula for length is probably to blame for the nonsensical results, and given its length, it is easier to replace it than to debug. Here is another way to find the length of chord passing through two points (x1,y1) and (x2,y2):
Find the distance of the chord from the center
Use the Pythagorean theorem to find its length
In Matlab code, this is done by
distance = abs(x1.*y2-x2.*y1)./sqrt((x2-x1).^2+(y2-y1).^2);
length = 2*sqrt(1-distance.^2);
The formula for distance involves abs(x1.*y2-x2.*y1), which is twice the area of the triangle with vertices (0,0), (x1,y1), and (x2,y2). Dividing this quantity by the base of triangle, sqrt((x2-x1).^2+(y2-y1).^2), yields its height.
Also, putting 1000000 samples into mere 10 bins is a waste of information: you get a crude histogram for all that effort. Better to use hist(length,100).
Finally, your method of selecting two points through which to pass a line does not take them from the uniform distribution on the disk. If you want uniform distribution over the disk, use
r1 = sqrt(rand(1,1000000));
r2 = sqrt(rand(1,1000000));
because for a uniformly distributed point, the square of the distance to the center is uniformly distributed in [0,1].
Finally, I've no idea why you square in prob = (flag/1000000)^2.
Here is your code with aforementioned modifications.
r1 = sqrt(rand(1,1000000));
teta1 = 2*pi * rand(1,1000000);
x1 = r1 .* (cos(teta1));
y1 = r1 .* (sin(teta1));
r2 = sqrt(rand(1,1000000));
teta2 = 2*pi * rand(1,1000000);
x2 = r2 .* (cos(teta2));
y2 = r2 .* (sin(teta2));
distance = abs(x1.*y2-x2.*y1)./sqrt((x2-x1).^2+(y2-y1).^2);
length = 2*sqrt(1-distance.^2);
hist(length,100)
flag = 0;
for check = length
if( check > sqrt(3) )
flag = flag + 1;
end
end
prob = flag/1000000;