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Please help <a href="http://en.wikipedia.org/w/index.php?title=Buffon%27s_needle&action=edit" class="external text" title="http://en.wikipedia.org/w/index.php?title=Buffon%27s_needle&action=edit" rel="nofollow">improve this article</a> by adding citations to <a href="/wiki/Wikipedia:Reliable_sources" title="Wikipedia:Reliable sources">reliable sources</a>. <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">Unverifiable</a> material may be challenged and removed. <small><i>(October 2007)</i></small></td></tr></table><p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, <b>Buffon's needle problem</b> is a question first posed in the 18th century by <a href="/wiki/Georges-Louis_Leclerc,_Comte_de_Buffon" title="Georges-Louis Leclerc, Comte de Buffon">Georges-Louis Leclerc, Comte de Buffon</a>:</p><dl><dd>Suppose we have a <a href="/wiki/Floor" title="Floor">floor</a> made of <a href="/wiki/Parallel_(geometry)" title="Parallel (geometry)">parallel</a> strips of <a href="/wiki/Wood" title="Wood">wood</a>, each the same width, and we drop a <a href="/wiki/Sewing_needle" title="Sewing needle">needle</a> onto the floor. What is the <a href="/wiki/Probability" title="Probability">probability</a> that the needle will lie across a line between two strips?</dd></dl><p>Using <a href="/wiki/Integral_geometry" title="Integral geometry">integral geometry</a>, the problem can be solved to get a <a href="/wiki/Monte_Carlo_method" title="Monte Carlo method">Monte Carlo method</a> to approximate <a href="/wiki/Pi" title="Pi">蟺</a>.</p><table id="toc" class="toc" summary="Contents"><tr><td><div id="toctitle"><h2>Contents</h2></div><ul><li class="toclevel-1"><a href="#Solution"><span class="tocnumber">1</span> <span class="toctext">Solution</span></a></li><li class="toclevel-1"><a href="#Lazzarini.27s_estimate"><span class="tocnumber">2</span> <span class="toctext">Lazzarini's estimate</span></a></li><li class="toclevel-1"><a href="#See_also"><span class="tocnumber">3</span> <span class="toctext">See also</span></a></li><li class="toclevel-1"><a href="#External_links_and_references"><span class="tocnumber">4</span> <span class="toctext">External links and references</span></a></li></ul></td></tr></table><script type="text/javascript">//<![CDATA[ if (window.showTocToggle) { var tocShowText = "show"; var tocHideText = "hide"; showTocToggle(); } //]]></script><p><a name="Solution" id="Solution"></a></p><h2><span class="editsection">[<a href="/w/index.php?title=Buffon%27s_needle&action=edit&section=1" title="Edit section: Solution">edit</a>]</span> <span class="mw-headline">Solution</span></h2><div class="thumb tright"><div class="thumbinner" style="width:222px;"><a href="/wiki/File:Buffon_needle.gif" class="image" title="The a needle lies across a line, while the b needle does not."><img alt="" src="http://upload.wikimedia.org/wikipedia/commons/f/f6/Buffon_needle.gif" width="220" height="132" border="0" class="thumbimage" /></a><div class="thumbcaption">The <i>a</i> needle lies across a line, while the <i>b</i> needle does not.</div></div></div><p>The problem in more mathematical terms is: Given a needle of length <span class="texhtml"><i>l</i></span> dropped on a plane ruled with parallel lines <i>t</i> units apart, what is the probability that the needle will cross a line?</p><p>Let <i>x</i> be the distance from the center of the needle to the closest line, let <i>胃</i> be the acute angle between the needle and the lines, and let <img class="tex" alt="t\ge l" src="http://upload.wikimedia.org/math/c/2/5/c259c5aee0011feee9aa7e5f94e13f46.png" />.</p><p>The <a href="/wiki/Probability_density_function" title="Probability density function">probability density function</a> of <i>x</i> between 0 and <i>t</i> /2 is</p><dl><dd><img class="tex" alt=" \frac{2}{t}\,dx. " src="http://upload.wikimedia.org/math/a/6/f/a6f3e403a54b07792ba75bc874976049.png" /></dd></dl><p>The probability density function of 胃 between 0 and 蟺/2 is</p><dl><dd><img class="tex" alt=" \frac{2}{\pi}\,d\theta. " src="http://upload.wikimedia.org/math/2/b/5/2b5d21744ef929eb6d804cc1acd97010.png" /></dd></dl><p>The two <a href="/wiki/Random_variables" title="Random variables" class="mw-redirect">random variables</a>, <i>x</i> and <i>胃</i>, are independent, so the joint probability density function is the product</p><dl><dd><img class="tex" alt=" \frac{4}{t\pi}\,dx\,d\theta. " src="http://upload.wikimedia.org/math/9/0/1/90101d78c6983c89091e5a6d5b9857bc.png" /></dd></dl><p>The needle crosses a line if</p><dl><dd><img class="tex" alt="x \le \frac{l}{2}\sin\theta." src="http://upload.wikimedia.org/math/a/9/3/a934fef4c3a41c031758a5d258391b53.png" /></dd></dl><p>Integrating the joint probability density function gives the probability that the needle will cross a line:</p><dl><dd><img class="tex" alt="\int_{\theta=0}^{\frac{\pi}{2}} \int_{x=0}^{(l/2)\sin\theta} \frac{4}{t\pi}\,dx\,d\theta = \frac{2 l}{t\pi}." src="http://upload.wikimedia.org/math/3/2/8/328f809d2393703b1ecfb0b5e0d8f5a8.png" /></dd></dl><p>For <i>n</i> needles dropped with <i>h</i> of the needles crossing lines, the probability is</p><dl><dd><img class="tex" alt="\frac{h}{n} = \frac{2 l}{t\pi}," src="http://upload.wikimedia.org/math/a/9/4/a942027a048b34e66155c1504d233c48.png" /></dd></dl><p>which can be solved for <i>蟺</i> to get</p><dl><dd><img class="tex" alt="\pi = \frac{2{l}n}{th}." src="http://upload.wikimedia.org/math/9/6/f/96fb81c935eeb22dae83d220b8de98b9.png" /></dd></dl><p>Now suppose <span class="texhtml"><i>t</i> < <i>l</i></span>. In this case, integrating the joint probability density function, we obtain:</p><dl><dd><img class="tex" alt="\int_{\theta=0}^{\frac{\pi}{2}} \int_{x=0}^{m(\theta)} \frac{4}{t\pi}\,dx\,d\theta ," src="http://upload.wikimedia.org/math/6/3/4/63470536a5d3d74df4e56823f07c9f95.png" /></dd></dl><p>where <span class="texhtml"><i>m</i>(胃)</span> is the minimum between <span class="texhtml">(<i>l</i> / 2)sin胃</span> and <span class="texhtml"><i>t</i> / 2</span>.</p><p>Thus, performing the above integration, we see that, when <span class="texhtml"><i>t</i> < <i>l</i></span>, the probability that the needle will cross a line is</p><dl><dd><img class="tex" alt="\frac{h}{n} = \frac{2 l}{t\pi} - \frac{2}{t\pi}\left\{\sqrt{l^2 - t^2} + t\sin^{-1}\left(\frac{t}{l}\right)\right\}+1." src="http://upload.wikimedia.org/math/c/2/a/c2a531dc129046097f34c30bd92472c3.png" /></dd></dl><p><a name="Lazzarini.27s_estimate" id="Lazzarini.27s_estimate"></a></p><h2><span class="editsection">[<a href="/w/index.php?title=Buffon%27s_needle&action=edit&section=2" title="Edit section: Lazzarini's estimate">edit</a>]</span> <span class="mw-headline">Lazzarini's estimate</span></h2><p><a href="/w/index.php?title=Mario_Lazzarini&action=edit&redlink=1" class="new" title="Mario Lazzarini (page does not exist)">Mario Lazzarini</a>, an <a href="/wiki/Italy" title="Italy">Italian</a> <a href="/wiki/Mathematician" title="Mathematician">mathematician</a>, performed the Buffon's needle experiment in 1901. Tossing a needle 3408 times, he attained the well-known estimate 355/113 for 蟺, which is a very accurate value, differing from 蟺 by no more than 3脳10<sup>鈭⌨️ 快捷键说明
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