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.

Greyson Landry 2022-07-20 Answered
Geometric probability of intersection of a square and a circle
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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Answers (1)

thenurssoullu
Answered 2022-07-21 Author has 13 answers
Step 1
In 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 < 1 2 ).
Step 2
Hence we have intersection with probability 1 0 1 2 ( 1 2 r ) 2 d r .
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