Sheet1Simulating theBuffon Needle ProblemThis DIGMath module
simulatesthe Buffon Needle Problemwith n needles of length L.How
many random needles (50-200)?84How long L is each needle
(1-12)?8How wide W is the strip (10-90)?61Out of 84 random needles
of length 8,122 crossed the strip line0According to probability
theory,0the expected number of 'hits' = 7.0Click each item below
for suggestions and investigationsItem 1Item 2Item 3Created by:
Sheldon P. Gordon & Florence S. GordonFarmingdale StateCollege
NYITDevelopment of this module was supported by theNSF's Division
of Undergraduate Educationunder grants DUE-0310123 and
DUE-0442160.
(1) The Buffon Needle Problem involves dropping a needle of
length L onto a floor made up of planks whose width is W. The
question is: What is the probability that the needle will land
entirely on a plank compared to the probability that it will land
crossing over the edge of a plank? This DIGMath spreadsheet
simulates the Buffon Problem by generating many random needles of
the indicated length and counting how many of them actually "land"
across the edge of the plank to estimate that probability.(2) Start
with a fairly small number of needles, so that you can see more
clearly what is happening. Select a value for the width of the
plank and then investigate what happens as you increase the length
of the needles. Is it more or less likely that longer needles will
land entirely in a single strip? Is it more or less likely that
longer needles will land across the edge of the strip? Increase the
number of random needles, even up the maximum number of 200 that is
allowed.
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Sheet2nx0y0qxyCount198.575582226130.48949156423.8198.575582226130.48949156420.010295.732600711259.25961721663.9392.311779015125.51307022620100.010393.387404482910.5450668175.07492.601316165351.64600062652.1295.732600711259.2596172166071583.821319845667.42328993422.5490.076776705453.601732911010071661.45525563525.52996448643.47754.191660221923.73515379544.3593.387404482910.54506681700864.270181264917.15118386740.1896.17484918393.04639013121100995.068998993227.92629548732.931069.784464547917.90605601233.5492.601316165351.6460006265Number
of
hits1221183.629657812867.76624078731.8388.429082224258.471867893101299.323177897123.67119662573.34Expected=7.01363.920802498617.21931248672.7683.821319845667.42328993421499.018363909748.16921110851.1277.219145303571.941175647111541.981079794370.58187127265.871648.746185236733.44864252313.8261.45525563525.52996448641779.278319250128.28996544290.9653.874383754322.974503367801890.976558111127.46204891322.661969.438554291460.87832196076.0554.191660221923.73515379542081.476733834766.6269754693.8651.32797737816.265259514402121.063739385553.58535906463.052260.796068961627.95080222554.7164.270181264917.1511838674239.392719646553.91083268685.2572.138646384218.595917983102411.609605664118.88284437162.482526.457488812931.85307079681.4395.068998993227.92629548732639.598141278559.09609642182.3087.250017002329.618489514402717.789628649962.14158486244.732867.280736289133.10628447766.1869.784464547917.90605601232914.134283331439.39342696931.3962.413232895914.797206718303017.362542271133.83694966295.013164.264631754327.98076418011.9183.629657812867.76624078733262.753703488223.82105678342.9981.592811702175.502600261913393.637025114124.55732021595.37348.222698969530.78248746175.8699.323177897123.67119662573568.347208129766.41622255490.8191.472499300622.132741044803614.509452001612.87899986292.823725.899180697157.21740823665.9863.920802498617.21931248673870.274740539864.81412753795.6756.489830603620.182529774603914.602248686862.41617130191.644044.068215158724.78104599845.7499.018363909748.16921110854199.573359620962.0923759283.23102.503310336555.37026300404258.434459132658.683687814.504366.302336210930.58340400514.8641.981079794370.58187127264444.179018641519.6196190191.6649.317164025367.390967301104533.820297312743.89719759295.044695.164228444535.40232878541.9348.746185236733.44864252314785.102907987869.55597793070.5742.524470389628.419699854804898.176471152853.63065662753.704941.009437305216.30982625660.8579.278319250128.28996544295022.514951385138.72875155040.5583.857498780734.849776021405146.600779268750.62789487743.235288.444700219462.19705647852.9290.976558111127.46204891325380.011026734353.23524752953.8383.896971150231.187562094205413.732875551134.66554798125.915598.733219749741.46789157113.7069.438554291460.87832196075622.676343806359.25582257173.9777.227329375759.052128464105735.222084946717.88892359134.465897.767418898342.99867210554.3381.476733834766.6269754695940.908478032554.82169601022.5575.477849037361.334208594606035.677338018442.06637203153.54615.385548087216.10446387265.3821.063739385553.58535906466294.142647315930.44235495791.3713.097090236254.315085952506373.280464577957.40952570244.316482.31350109269.04555847780.7660.796068961627.95080222556547.777142595367.11835479925.2160.750298029219.950933162806651.904537701762.91872372242.676736.191218870432.13192639513.389.392719646553.91083268686870.317764451432.63000424781.3413.472471159547.029290156806988.458551604345.37156163120.847098.473799086232.28336189670.2211.609605664118.8828443716717.885311991369.14712959161.405.275055227123.768892955207212.713923147948.8060051493.167325.603776864749.77555451543.5726.457488812931.85307079687483.909504361310.07969325850.3527.616377758539.768686781107571.888757243927.22375712414.077644.80177379215.3377949454.3839.598141278559.09609642187760.235803705526.29694115544.4834.262864686765.057205688407867.501855568763.88894584380.387977.824827354451.49794674370.3917.789628649962.14158486248071.1422141529.9907091281.8817.900399655254.142351787708176.20347634616.76357251565.378254.281554992655.42266795275.6567.280736289133.10628447768383.459843625119.69688079483.7275.241899951132.318964245808440.83508630732.68278539663.5285000.0014.134283331439.393426969386000.0015.557664887147.2657827125087000.0088000.0017.362542271133.836949662989000.0019.724746043226.193652381090000.0091000.0064.264631754327.980764180192000.0061.630990778435.5348316414093000.0094000.0062.753703488223.821056783495000.0054.841871510925.0055038118096000.0097000.0093.637025114124.557320215998000.0098.514447629218.2161255643099000.00100000.008.222698969530.7824874617101000.0015.508889430927.47924230570102000.00103000.0068.347208129766.4162225549104000.0073.863242985972.21047483331105000.00106000.0014.509452001612.8789998629107000.006.923384007515.41899442750108000.00109000.0025.899180697157.2174082366110000.0033.525253317554.80017502740111000.00112000.0070.274740539864.8141275379113000.0076.804998776360.19290179530114000.00115000.0014.602248686862.4161713019116000.0014.045796298370.39679538390117000.00118000.0044.068215158724.7810459984119000.0050.902855302920.62319870680120000.00121000.0099.573359620962.092375928122000.0091.606157032761.36871653340123000.00124000.0058.434459132658.68368781125000.0056.72911412850.8675635520126000.00127000.0066.302336210930.5834040051128000.0067.493621617922.67259879460129000.00130000.0044.179018641519.619619019131000.0043.451704420527.58648879470132000.00133000.0033.820297312743.8971975929134000.0036.409653305736.32783595920135000.001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(3) Now repeat the previous investigations by varying the width
of the strip. What happens as you increase the length of the
needles. Is it more or less likely that the needles will land
entirely in a single strip if the strip is wider or narrower? Is it
more or less likely that needles will land across the edge of the
strip if the plank is wider? Increase the number of random needles,
even up the maximum number of 200 that is allowed. Mathematicians
have developed a pair of formulas to predict the probability P that
an individual needle will land across the boundary of the strip,
depending on the length L of the needle and the width W of the
strip. When L < W, it turns out that P = 2L/pD; a much more
complicated formula gives the probability when L > W.
Sheet3
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