针长大于间距的蒲丰投针问题及应用

Buffon Needles Problem with Needle Length Greater Than Spacing and Application

本文聚焦于针长l大于平行线间距d的蒲丰投针问题.首先,探讨了d<l<2d时针与平行线恰好有两个交点的概率.接着,研究了l<2d时针与平行线至少有两个交点的概率.然后,对于边长分别为l_(1)、l_(2)、l_(3)的三角形,在l_(1)<l_(2)<l_(3)的条件下,应用得到的引理求解了该三角形与平行线相交的概率.进一步,通过数值模拟,得到该三角形与平行线相交概率的变化趋势.

This paper focuses on the Buffon needle problem with the needle length l greater than the spacing d.The probability that there are exactly two intersections between the needle and the parallel lines when d<l<2d and the probability of at least two intersections with parallel lines when l<2d are discussed.For a triangle with side lengths l_(1)、l_(2)、l_(3) and l_(1)<l_(2)<l_(3),the probability of the intersection between the triangle and the parallel lines is solved.By numerical simulation,the variation trend of the intersection probability between the triangle and the parallel lines is obtained.