In a famous 18th-century problem, known as Buffon's needle problem, a needle of length h is dropped

Question:

In a famous 18th-century problem, known as Buffon's needle problem, a needle of length h is dropped onto a flat surface (for example, a table) on which parallel lines L units apart, L ‰¥ h have been drawn. The problem is to determine the probability that the needle will come to rest intersecting one of the lines. Assume that the lines run east-west, parallel to the x-axis in a rectangular coordinate system (as in the figure). Let y be the distance from the "southern" end of the needle to the nearest line to the north. (If the needle's southern end lies on a line, let y = 0. If the needle happens to lie east-west, let the "western" end be the "southern" end.) Let Î¸ be the angle that the needle makes with a ray extending eastward from the "southern" end. Then 0 ‰¤ y ‰¤ L and 0 ‰¤ Î¸ ‰¤ Ï€. Note that the needle intersects one of the lines only when y h sin Î¸. The total set of possibilities for the needle can be identified with the rectangular region 0 ‰¤ y ‰¤ L, 0 ‰¤ Î¸ ‰¤ Ï€, , and the proportion of times that the needle intersects a line is the ratio
(area under y = h sin) / (area of rectangle)
This ratio is the probability that the needle intersects a line. Find the probability that the needle will intersect a line if h = L. What if h = 1/2L?
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