# Therefore the conclusion of proposition 474 reduces

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Unformatted text preview: ing variation of the Buﬀon’s needle problem (Example 4.6.1). Suppose a needle of unit length is thrown at random onto a plane with both a vertical grid and a horizontal grid, each with unit spacing. Find the probability the needle, after it comes to rest, does NOT intersect any grid line. Solution: Let Mh be the event that the needle misses the horizontal grid (i.e. does not intersect a horizontal grid line) and let Mv denote the event that the needle misses the vertical grid. We 2 seek to ﬁnd P (Mh Mv ). By the solution to Buﬀon’s needle problem, P (Mh ) = P (Mv ) = 1 − π . If 22 2 ≈ 0.132. But Mh and Mv were independent, we would have that P (Mh Mv ) = (1 − π ) ≈ (0.363) these events are not independent. Let Θ be deﬁned relative to the horizontal grid as in the solution of Buﬀon’s needle problem. Then the vertical displacement of the needle is sin(Θ) and the horizontal displacement is | cos(Θ)|. Assume that the position of the needle relative to the horizontal grid is i...
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## This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Institute of Technology.

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