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?
(In reply to
Solution? by Prab)
I came up with exactly the same solution, i.e. given the distance to travel r (0<=r<=1) the probability of staying in the square is P = (r^24r)/pi+1
Another interesting problem is to calculate the probability of staying within the square when distance r is randomly chosen from an interval [r0,r1] where 0<=r0<=r1<=1 (Answer 1 at the bottom of the message)
Yet another extension of the problem is to calculate the probability of staying within the square for any r > 0. Obviously for r >= 2^(1/2) P=0 so we only have to consider 1<=r<=2^(1/2). (Answer 2 at the bottom)
The probability of staying within the square when r is randomly chosen from an interval [r0;r1] where 0<=r0<=r1 can be calculated for any specific case of r0 and r1 using the formulae from the above problems, but general formula is a bit messy.
Anyone to offer a solution for a rectangle with one unit side and the other side of s where s > 1?
Answer1:
P=((r1^2+r1r0+r0^2)/3  2(r1+r0))/pi+1
Answer2:
P=[4(r^21)^(1/2)r^224arctg(1/(r^21)^(1/2))]/pi1

Posted by Yevgen
on 20031112 08:39:23 