This preview shows page 1. Sign up to view the full content.
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
seek to ﬁnd P (Mh Mv ). By the solution to Buﬀon’s needle problem, P (Mh ) = P (Mv ) = 1 − π . If
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...
View Full Document
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.
- Spring '08
- The Land