The Buffon's Needle

Calculus Level 3

While messing with probability, I came across this problem:

Suppose we have a floor made of parallel strips of wood, each the same width, and we drop a needle onto the floor. What is the probability that the needle will lie across a line between two strips?

Details and Assumptions:

  • Assume that you drop a needle on your rough copy and you have to find the probability that it crosses one horizontal line .

  • Length of needle is equal to separation between two lines.

  • Plane of needle is parallel to plane of floor or you rough copy.

Note: This is famous Buffon's needle problem. If you can,then evaluate the probability 1.) by assuming length of needle is less than separation between two lines and 2.) Length of needle is more than separation between two lines.
Please post a solution assuming all three cases. I'm stuck in the case where length of needle is greater than separation between two lines.
Image Credit: Wikimedia McZusatz

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