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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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Solution Solution using graphical logic | Comment 3 of 12 |
1. The target square is 1u²
2. The distance of available travel is 0.5u in any direction. (circle with radius of 0.5u)

The maximum distance of the ending position is 0.5u. Thus we can represent this by an extended perimiter of 0.5u circling the 1u² box. This creates a square with round edges.

The probility of the target being inside the square is basically the area of the square / the area of the square with the round edges.

Area of the Target Square

Area of the Round edge Square
or Area of large square - area of 4 rounded corners
=l * w - 4[(w- Pi*r*²)/4]
=2*2 - 4[(1 - Pi*0.5²)/4]
=4 - (1-Pi*0.25)
=3 + 1/4 Pi (aprox 3.7854u²)

Probility of the ending position in Square
=1u² / (3 + 1/4 Pi u²)
=0.2642 (aprox)

The probility of the ending position is within the square is 26.42%
  Posted by Mag on 2003-10-19 16:48:19
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