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 don't know why the previous message was crash)
I defined the square in the range
 0.5 < x < 0.5 and
 0.5 < y < 0.5
I can divide the square in four sub squares,
one in each cuadrant.
I will find the probability that the end point
lies on the original square given that the start
point lies on the first quadrant
finally i must multiply this probability by four
since there are four quadrants.
On the first quadrant two types of points are
identified:
 The one that lies under the curve
0.5sqrt(0.5*0.5(x0.5)(x0.5))
 The one that lies over the same curve.
the diference between these two types of point is:
Type 1: If we draw a circle with center in a
point of this type, the circle intercept the
original square 4 times.
Type 2: The circle intercept the original Square
only 2 times
The importance of knowing how many times the
square is intercepted is that one can know
exactly from what angle to what angle a end
point is outside of the original Square.
In Type 1 there are two angle zones where the
end point is outside in Type 2 there is only one
zone
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 or aproximately 0.4429576992
Pablo Meraz

Posted by Pablo
on 20040129 15:25:48 