9. Buffon's needle problem

• Buffon's needle problem is a classic question, see for details in Wikipedia or in WolframMathWorld. We will generalize this question. Take a sufficiently large sheet of paper that has parallel lines with unit distances from each other. Drop a needle of length \(L\) randomly onto the paper. Let \(X\) be the number of lines touched by the needle. Write out the values of the frequency function of \(X\) based on the \(N\) experiments at \(0, 1, 2, \dots,\lfloor L \rfloor + 1\).
• The command line input of the program is a positive integer number \(N\) and a positive real number \(L\).
• The output is
  • a sequence of \(\lfloor L\rfloor+2\) integer numbers,
  • the histogram belonging to this sequence, and
  • the value of \(2L/a\), where \(a\) is the relative frequency of those cases when exactly one line is crossed.
• We remark that if \(L<1\), then the probability of line-crossing is \(2L/\pi\), so the relative frequency of the line-crossings approximates this value.
• Simulation can be simplified because of symmetry reasons in several ways. For example, the angle between the needle and the lines can be considered a uniformly distributed variable on the interval \([0,\pi/2]\). Similarly, the distance of its center from the nearest line is uniformly distributed on \([0,0.5]\). One may calculate by supposing that the "left" end point of the needle is uniformly distributed on \([0,1]\), and so on.
• Those who love challenges may try to calculate the probabilities for a particular \(L\) (e.g. \(L = 2\) or \(L = 5\)) comparing the results with the experimental values. This isn't a compulsory part of this exercise!