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Square thinking (Posted on 2003-10-19) Difficulty: 3 of 5
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?

No Solution Yet Submitted by Cory Taylor    
Rating: 4.1250 (8 votes)

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re: Solution? (& extensions of the problem) | Comment 7 of 12 |
(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^2-4r)/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^2-1)^(1/2)-r^2-2-4arctg(-1/(r^2-1)^(1/2))]/pi-1
  Posted by Yevgen on 2003-11-12 08:39:23

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