SOCR EduMaterials Activities BuffonNeedleExperiment - Revision history

IvoDinov at 05:33, 28 December 2007

2007-12-28T05:33:53Z

IvoDinov at 05:32, 28 December 2007

2007-12-28T05:32:34Z

IvoDinov at 19:13, 18 December 2007

2007-12-18T19:13:30Z

IvoDinov: /* See also */

2007-12-18T19:08:34Z

‎See also

IvoDinov: /* Buffon's needle experiment and estimation of the constant $\pi$ */

2007-12-18T19:02:25Z

‎Buffon's needle experiment and estimation of the constant <math>\pi</math>

IvoDinov at 17:09, 18 December 2007

2007-12-18T17:09:50Z

IvoDinov: /* Buffon's needle experiment and estimation of the constant $\pi$ */

2007-12-18T17:03:53Z

‎Buffon's needle experiment and estimation of the constant <math>\pi</math>

IvoDinov: /* Buffon's needle experiment and estimation of the constant $\pi$ */

2007-12-18T00:59:11Z

‎Buffon's needle experiment and estimation of the constant <math>\pi</math>

IvoDinov: /* Buffon's needle experiment and estimation of the constant $\pi$ */

2007-12-18T00:50:34Z

‎Buffon's needle experiment and estimation of the constant <math>\pi</math>

IvoDinov: /* Buffon's needle experiment and estimation of the constant $\pi$ */

2007-12-18T00:48:21Z

‎Buffon's needle experiment and estimation of the constant <math>\pi</math>

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 == Description ==
-The Buffon’s Needle Experiment involves dropping a needle on hardwood floor with floorboards of width 1. The experiment is shown graphically in the first graph box.
+The [http://www.socr.ucla.edu/htmls/SOCR_Experiments.html Buffon’s Needle Experiment] involves dropping a needle on hardwood floor with floorboards of width 1. The experiment is shown graphically in the first graph box (see image below).
 The angle X of the needle relative to the floorboard cracks and the distance Y from the center of the needle to the lower crack are recorded in the first table on each update. In the distribution table which is the distribution table, the values of the density of ''I'', empirical density of ''I'' are recorded, where variable ''I'' indicates the event that the needle crosses a crack. The values of ''pi'' (<math>\pi</math>) and Buffon’s estimate of ''pi'' are recorded in the last table.

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 The Buffon’s Needle Experiment involves dropping a needle on hardwood floor with floorboards of width 1. The experiment is shown graphically in the first graph box.
-The angle X of the needle relative to the floorboard cracks and the distance Y from the center of the needle to the lower crack are recorded in the first table on each update. In the distribution table which is the distribution table, the values of the density of ''I'', empirical density of ''I'' are recorded, where variable ''I'' indicates the event that the needle crosses a crack. The values of ''pi'' (<math>\pi</math> )and Buffon’s estimate of ''pi'' are recorded in the last table.
+The angle X of the needle relative to the floorboard cracks and the distance Y from the center of the needle to the lower crack are recorded in the first table on each update. In the distribution table which is the distribution table, the values of the density of ''I'', empirical density of ''I'' are recorded, where variable ''I'' indicates the event that the needle crosses a crack. The values of ''pi'' (<math>\pi</math>) and Buffon’s estimate of ''pi'' are recorded in the last table.
 Each point (X,Y) is shown as a red dot in the scatter plot. The density of ''I'' is shown in blue and the empirical density of ''I'' is shown in red in the distribution graph. In the last graph, ''pi'' is shown in blue and the Buffon’s estimate of ''pi'' is shown in red.

← Older revision		Revision as of 19:13, 18 December 2007
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−	== Buffon Needle Experiment (estimation of <math>\pi</math> ) ==	+	== [[SOCR_EduMaterials_ExperimentsActivities \| SOCR Experiments]] - Buffon Needle Experiment (estimation of <math>\pi</math> ) ==

	== Description ==		== Description ==

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 ==See also==
 * [[SOCR_EduMaterials_Activities_Uniform_E_EstimateExperiment | Uniform e-Estimation experiment (estimating ''e'')]]
 {{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_EduMaterials_Activities_BuffonNeedleExperiment}}

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 Suppose the length of the needle is <math>0\le L\le 1</math> and the distance between the lines is set to 1. When the needle is dropped, there are two important measurements we make to determine if the needle crosses any of the the grid-lines - these are the ''angle at which the needle falls'' (<math>\alpha</math> ) and the ''distance between the center of the needle to the closest line'' (''d''). Note that <math>0\le \alpha \le \pi</math> and <math>0\le d \le {1 \over 2}</math>. The needle will cross a grid-line if <math>d \le {L \over 2} \sin(\alpha)</math>. What are the odds of this happening?
-The distribution of the pairs of (<math>\alpha, d)</math> for which this inequality is satisfied is presented in the graph below. The shaded region on the right represents the situation when the needle crosses a grid-line (i.e., <math>d \le {L \over 2} \sin(\alpha)</math>). Therefore, the probability of an needle-line intersection is determined by the ratio of the shaded area to the area of the enclosing rectangle. The area of the shaded area is computed by <math>\int_{0}^{\pi} {{L \over 2} \sin(\alpha) d \alpha} = L </math> and the area of the rectangle is <math>{1 \over 2}\pi</math>. Then, the '''odds of a needle-line intersection''' is <math>{ L \over {\pi\over 2}}={2 L \over \pi}</math> (if <math>L=1</math>, then <math>{2 \over \pi}\approx 0.63662</math>). In practice, to estimate <math>\pi</math> from a number of needle drops (''N''), we take the reciprocal of the sample odds-of-intersection. This allows us to empirically estimate <math>\pi \approx {1  \over odds-of-intersection} = {2 L \over { Number-Of-Intersections \over N}}.</math>
+The distribution of the pairs of (<math>\alpha, d)</math> for which this inequality is satisfied is presented in the graph below. The shaded region on the right represents the situation when the needle crosses a grid-line (i.e., <math>d \le {L \over 2} \sin(\alpha)</math>). Therefore, the probability of an needle-line intersection is determined by the ratio of the shaded area to the area of the enclosing rectangle. The area of the shaded area is computed by <math>\int_{0}^{\pi} {{L \over 2} \sin(\alpha) d \alpha} = L </math> and the area of the rectangle is <math>{1 \over 2}\pi</math>. Then, the '''odds of a needle-line intersection''' are <math>{ L \over {\pi\over 2}}={2 L \over \pi}</math> (if <math>L=1</math>, then <math>{2 \over \pi}\approx 0.63662</math>). In practice, to estimate <math>\pi</math> from a number of needle drops (''N''), we take the reciprocal of the sample odds-of-intersection. This allows us to empirically estimate <math>\pi \approx {1  \over odds-of-intersection} = {2 L \over { Number-Of-Intersections \over N}}.</math>
 <center>[[Image:SOCR_Activities_BuffonNeedleExperiment_Dinov_121707_Fig4.jpg|600px]]</center>

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-== Buffon Needle Experiment ==
+== Buffon Needle Experiment (estimation of <math>\pi</math> ) ==
 == Description ==
@@ Line 5: / Line 5: @@
 The Buffon’s Needle Experiment involves dropping a needle on hardwood floor with floorboards of width 1. The experiment is shown graphically in the first graph box.
-The angle X of the needle relative to the floorboard cracks and the distance Y from the center of the needle to the lower crack are recorded in the first table on each update. In the distribution table which is the distribution table, the values of the density of ''I'', empirical density of ''I'' are recorded, where variable ''I'' indicates the event that the needle crosses a crack. The values of pi and Buffon’s estimate of pi are recorded in the last table.
+The angle X of the needle relative to the floorboard cracks and the distance Y from the center of the needle to the lower crack are recorded in the first table on each update. In the distribution table which is the distribution table, the values of the density of ''I'', empirical density of ''I'' are recorded, where variable ''I'' indicates the event that the needle crosses a crack. The values of ''pi'' (<math>\pi</math> )and Buffon’s estimate of ''pi'' are recorded in the last table.
-Each point (X,Y) is shown as a red dot in the scatter plot. The density of ''I'' is shown in blue and the empirical density of ''I'' is shown in red in the distribution graph. In the last graph,'' pi'' is shown in blue and the Buffon’s estimate of ''pi'' is shown in red.
+Each point (X,Y) is shown as a red dot in the scatter plot. The density of ''I'' is shown in blue and the empirical density of ''I'' is shown in red in the distribution graph. In the last graph, ''pi'' is shown in blue and the Buffon’s estimate of ''pi'' is shown in red.
 The parameter is the needle length L which can be modified with the scroll bar.
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 One example may be for throwing darts onto a dart board. When using a dart made of a fixed material, a shorter length dart will have lighter weight so it may be easier to control the dart, thus there will be a higher chance that the dart will land within the target. But a longer dart may experience more air resistance and have a heavier weight, causing the probability to land within the target to decrease.
 {{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_EduMaterials_Activities_BuffonNeedleExperiment}}

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 The distribution of the pairs of (<math>\alpha, d)</math> for which this inequality is satisfied is presented in the graph below. The shaded region on the right represents the situation when the needle crosses a grid-line (i.e., <math>d \le {L \over 2} \sin(\alpha)</math>). Therefore, the probability of an needle-line intersection is determined by the ratio of the shaded area to the area of the enclosing rectangle. The area of the shaded area is computed by <math>\int_{0}^{\pi} {{L \over 2} \sin(\alpha) d \alpha} = L </math> and the area of the rectangle is <math>{1 \over 2}\pi</math>. Then, the odds of a needle-line intersection is <math>{ L \over {\pi\over 2}}={2 L \over \pi}</math> (if <math>L=1</math>, then <math>{2 \over \pi}\approx 0.63662</math>). In practice, to estimate <math>\pi</math> from a number of needle drops (''N''), we divide 2 by the sample odds-of-intersection. This allows us to empirically estimate <math>\pi \approx {1  \over odds-of-intersection} = {2 L \over { Number-Of-Intersections \over N}}.</math>
-<center>[[Image:SOCR_Activities_BuffonNeedleExperiment_Dinov_121607_Fig4.jpg|600px]]</center>
+<center>[[Image:SOCR_Activities_BuffonNeedleExperiment_Dinov_121707_Fig4.jpg|600px]]</center>
 For example, suppose <math>N=10</math>, <math>L=0.2</math> and we observe 1 needle-line crossing in the 10 trials. Then our estimate of <math>\pi \approx { 2 \over {1 \over 10\times 0.2}} = 4</math>, as shown on the image below.

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 <center>[[Image:SOCR_Activities_BuffonNeedleExperiment_Dinov_121607_Fig5.jpg|400px]]</center>
-In general, when the grid-line tiles are of size ''S'', the formulas slightly change as follows: area of the rectangle is <math>{S \over 2}\pi</math> and the odds of a needle-line intersection is <math>{ L \over {S \times \pi\over 2}}={2 L \over S \times \pi}</math>. Estimation of <math>\pi</math> from a number of needle drops (''N'') is then obtained by <math>\pi \approx {1  \over odds-of-intersection} = {2 L \over S} \times {1 \over Number-Of-Intersections \over N}}.</math>
+In general, when the grid-line tiles are of size ''S'', the formulas slightly change as follows: area of the rectangle is <math>{S \over 2}\pi</math> and the odds of a needle-line intersection is <math>{ L \over {S \times \pi\over 2}}={2 L \over S \times \pi}</math>. Estimation of <math>\pi</math> from a number of needle drops (''N'') is then obtained by <math>\pi \approx {1  \over odds-of-intersection} = {2 L \over S} \times {1 \over {Number-Of-Intersections \over N}}.</math>
 == Applications ==

SOCR EduMaterials Activities BuffonNeedleExperiment - Revision history

IvoDinov at 05:33, 28 December 2007

IvoDinov at 05:32, 28 December 2007

IvoDinov at 19:13, 18 December 2007

IvoDinov: /* See also */

IvoDinov: /* Buffon's needle experiment and estimation of the constant \pi */

IvoDinov at 17:09, 18 December 2007

IvoDinov: /* Buffon's needle experiment and estimation of the constant \pi */

IvoDinov: /* Buffon's needle experiment and estimation of the constant \pi */

IvoDinov: /* Buffon's needle experiment and estimation of the constant \pi */

IvoDinov: /* Buffon's needle experiment and estimation of the constant \pi */

IvoDinov: /* Buffon's needle experiment and estimation of the constant $\pi$ */

IvoDinov: /* Buffon's needle experiment and estimation of the constant $\pi$ */

IvoDinov: /* Buffon's needle experiment and estimation of the constant $\pi$ */

IvoDinov: /* Buffon's needle experiment and estimation of the constant $\pi$ */

IvoDinov: /* Buffon's needle experiment and estimation of the constant $\pi$ */