On a regular two dimensional coordinate plane, you have a square with side length 1 unit.
Pick a point within the square at random, and from there travel a random but straight direction .5 units.
What is the probability that you end up still within the square?
I defined the square in the range  0.5 < x < 0.5 and 0.5
The points of type 1, haven two angle zones where the end point lies outside. The points of type 2 only have an angle zone where the end point lies outside.
To find the probability (already multiplied by four):
Type 1:
The integral from x=0 to x=0.5 of
the integral from y=0 to 0.5sqrt(0.5*0.5(x0.5)*(x0.5)) of
(
2*pi
 2*arcsin( sqrt(0.5*0.5(0.5x)(0.5x))/0.5)
 2*arcsin( sqrt(0.5*0.5(0.5y)(0.5y))/0.5)
) /2/pi * dx * dy /0.25
Type 2:
The integral of x=0 to x=0.5 of
the integral of y=0.5sqrt(0.5*0.5(x0.5)(x0.5)) of
(
2*pi
pi/2
 arcsin( sqrt(0.5*0.5(0.5y)(0.5y))/0.5)
 arcsin( sqrt(0.5*0.5(0.5x)(0.5x))/0.5))
) /2/pi * dx * dy /0.25
These two integrals has three parts each that give the result
(4pi)/4
(pi*pi12)/16/pi
(pi*pi12)/16/pi
3Pi/16
(pi*pi4)/32/pi
(pi*pi4)/32/pi
Adding these 6 parts gives the result of:
17/4/pi aproximately 0.4429576992
Pablo Meraz

Posted by Pablo
on 20040129 12:30:44 