Q. A needle is dropped on a paper, keeping it parallel to the plane of paper. Find the probability that the needle cuts one of the parallel lines on the ruled sheet of paper, given that needle is of length 1 unit and distance between 2 parallel lines is one unit. The method I adopted- I assumed the needle to cut a length x units of a parallel line, then, I rotated it in a circle of radius 1/2, and I calculated the angle when the needle just touches the line. I found the arcsin of this value, dividing it by pi, integrated it from x=0 to x=1/2, and multiplied my answer by 2. I hope this is correct. If yes, please tell me a shorter method, if possible. Thanks.
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