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355/113 is also mentioned by Zu Chongzhi, which is much earlier |
Added section that explains Laplace's extension for the case of a grid |
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Dutch science journalist Hans van Maanen argues, however, that Lazzarini's article was never meant to be taken too seriously as it would have been pretty obvious for the readers of the magazine (aimed at school teachers) that the apparatus that Lazzarini said to have built cannot possibly work as described.<ref name="vMaanen2018">Hans van Maanen, [https://skepsis.nl/lazzarini/ 'Het stokje van Lazzarini' (Lazzarini's stick)], "Skepter" 31.3, 2018.</ref>
==Laplace's Extension (Short Needle Case)==
Now consider the case where the plane contains two sets of parallel lines orthogonal to one another, creating a standard perpendicular grid. We aim to find the probability that the needle intersects at least one line on the grid. Let <math>a, b</math> be the sides of the rectangle that contain the midpoint of the needle whose length is <math>\ell</math>. Since this is the short needle case, <math> \ell < a, \ell < b </math>. Let <math>(x,y)</math> mark the coordinates of the needle's midpoint and let <math>\varphi</math> mark the angle formed by the needle and the ''x''-axis. Similar to the examples described above, we consider <math>x, y, \varphi</math> to be independent uniform random variables over the ranges <math> 0\leq x \leq a, \quad 0\leq y \leq b, \quad \frac{-\pi}{2}\leq \varphi \leq \frac{\pi}{2} </math>.
To solve such a problem, we first compute the probability that the needle crosses no lines, and then we take its compliment. We compute this first probability by determining the volume of the domain where the needle crosses no lines and then divide that by the volume of all possibilities, <math>V</math>. We can easily see that <math> V = \pi a b</math>.
Now let <math>V^*</math> be the volume of possibilities where the needle does not intersect any line.
Developed by [[J.V. Uspensky]] <ref name="Uspensky">J.V. Uspensky, [https://ia601407.us.archive.org/17/items/in.ernet.dli.2015.263184/2015.263184.Introduction-To.pdf 'Introduction to Mathematical Probability'], 1937, 255.</ref>,
:<math>V^* = \int_{\frac{-\pi}{2}}^{\frac{\pi}{2}}
F(\varphi)d\varphi</math>
where <math>F(\varphi)</math> is the region where the needle does not intersect any line given an angle <math>\varphi</math>. To determine <math>F(\varphi)</math>, let's first look at the case for the horizontal edges of the bounding rectangle. The total side length is <math>a</math> and the midpoint must not be within <math>\frac{l}{2}\cos(\varphi)</math> of either endpoint of the edge. Thus, the total allowable length for no intersection is <math>a - 2(\frac{l}{2}\cos(\varphi))</math> or simply just <math>a - l\cos(\varphi)</math>. Equivalently, for the vertical edges with length <math>b</math>, we have <math>b \pm l\sin(\varphi)</math>. The <math>\pm</math> accounts for the cases where <math>\varphi</math> is positive or negative. Taking the positive case and then adding the absolute value signs in the final answer for generality,
:<math>F(\varphi) =(a - l\cos(\varphi))(b - l\sin(\varphi)) = ab - bl \cos ( \varphi) - al|\sin (\varphi)|
+ \frac{1}{2}l^2|\sin(2\varphi)|</math>.
Now we can compute the following integral:
:<math>V^* = \int_{\frac{-\pi}{2}}^{\frac{\pi}{2}}
F(\varphi)d\varphi = \pi ab-2bl-2al+l^2 </math>
Thus, the probability that the needle does not intersect any line is
:<math>\frac{V^*}{V}= \frac{\pi ab-2bl-2al+l^2}{\pi a b}
= 1 - \frac{2l(a+b)-l^2}{\pi a b}.
</math>
And finally, if we want to calculate the probability, <math>p</math>, that the needle does intersect at least one line, we need to subtract the above result from 1 to compute its compliment, yielding
:<math>p = \frac{2l(a+b)-l^2}{\pi a b}</math>.
== See also ==
*[[Bertrand paradox (probability)]]
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* {{cite journal| last = Dell|first = Zachary| author2 = Franklin, Scott V.|title = The Buffon-Laplace needle problem in three dimensions|journal = Journal of Statistical Mechanics: Theory and Experiment|date=September 2009| volume = 2009| issue = 9| pages = 010| doi = 10.1088/1742-5468/2009/09/P09010| bibcode = 2009JSMTE..09..010D| s2cid=32470555 }}
* Schroeder, L. (1974). "Buffon's needle problem: An exciting application of many mathematical concepts". ''Mathematics Teacher'', 67 (2), 183–6.
* Uspensky, James Victor. "Introduction to mathematical probability." (1937).
== External links ==
{{commons category|Buffon's needle}}
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