{
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  "body": "\n## Problem\n\nA floor has parallel lines at distance $t \u003e 0$ apart. A needle of length $l$ with $0 \u003c l \\le t$ falls at random on the floor.\n\"At random\" means this.\nThe distance $x$ of the centre of the needle from the nearest line is uniform on $[0, t/2]$.\nThe acute angle $\\theta$ between the needle and the lines is uniform on $[0, \\pi/2]$.\nThe variables $x$ and $\\theta$ are independent.\nFind the probability that the needle crosses a line.\n\n## Remarks\n\nThe answer is $\\dfrac{2l}{\\pi t}$. The needle crosses a line exactly when $x \\le \\frac{l}{2} \\sin \\theta$, and the integral of this region gives the answer.\nFor a needle with $l = t$, the probability is $2/\\pi \\approx 0.637$.\nThe answer contains $\\pi$, so repeated throws give a random estimate of $\\pi$.\nFor a long needle with $l \u003e t$, the answer is a different formula.\n\n## History\n\nGeorges-Louis Leclerc, Comte de Buffon, posed the problem in the 18th century. It was one of the first problems of geometric probability.\n\n## Explanations\n\n### L1 human\n\nThe needle crosses a line when it lands close to a line and at a steep angle.\nThe chance is the area of the \"good\" pairs (distance, angle) divided by the area of all pairs.\nA sine function appears in this area. That is how $\\pi$ comes into the answer.\n",
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