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.

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