布冯投针变体问题

图老师3

This problem is a variation on the Buffon Needle Problem.A needle of length 2 is bent at its midpoint forming a right angle. It is then dropped onto a floor on which a family of parallel lines spaced Sqrt[2] units apart have been drawn. What is the probability that the needle lands on one of the lines? Assume that where the midpoint of the needle lands and with what orientation are both uniformly distributed. What if the lines are spaced 2 units apart? 1 unit?
布冯投针变体问题

长度为2的针在其中点弯曲成直角。然后将其放到地板上，在该地板上绘制了一系列间隔为Sqrt [2]单位的平行线。针落在其中一条线上的概率是多少？假设针的中点落在任意处以及方向都均匀分布。如果针长度为2不变，线间隔2个单位概率是多少？1个单位概率是多少？
参考答案：
width of a board        probability the bent needle hits a crack
sqrt[2]                      (2+sqrt[2])/(pi*sqrt[2]) or about .7685
2                             (2+sqrt[2])/(2 pi) or about .5434
1                             (pi/2+sqrt[2])/pi or about .9502)