Geometric probability of intersection of a square and a circleIn the unitary square we choose a point (X,Y) with iid coordinates U[0,1] and a radius R, independent of (X,Y) and U[0,1], and we draw the circle of radius R with center (X,Y). Find the probability that this circle intersects the circumference of the unit square.

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thenurssoullu · Accepted Answer

Step 1In order to not intersect for given R, the center must be in a smaller square area   (  1  −  2  R      )    2   (which is course only possible when   R  &amp;lt;      1    2  ).Step 2Hence we have intersection with probability   1  −      ∫    0                  1        2              (  1  −  2  r      )    2          d    r  .

In the unitary square we choose a point (X,Y) with iid coordinates U[0,1] and a radius R, independent of (X,Y) and U[0,1], and we draw the circle of radius R with center (X,Y). Find the probability that this circle intersects the circumference of the unit square.

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