In a famous 18th-century problem, known as Buffon's needle problem, a needle of length h is dropped
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?
Transcribed Image Text:
hsin θ 2 yLh
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