http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_EduMaterials_Activities_LawOfLargeNumbers&feed=atom&action=historySOCR EduMaterials Activities LawOfLargeNumbers - Revision history2024-03-29T09:44:45ZRevision history for this page on the wikiMediaWiki 1.15.1http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_EduMaterials_Activities_LawOfLargeNumbers&diff=8792&oldid=prevIvoDinov at 22:33, 16 January 20092009-01-16T22:33:36Z<p></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>* '''Misconception 1''': If we observe a streak of 10 consecutive heads (when p=0.5, say) the odds of the <math>11^{th}</math> trial being a Head is > p! This is of course, incorrect, as the coin tosses are independent trials (an example of a ''memoryless'' process).</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>* '''Misconception 1''': If we observe a streak of 10 consecutive heads (when p=0.5, say) the odds of the <math>11^{th}</math> trial being a Head is > p! This is of course, incorrect, as the coin tosses are independent trials (an example of a ''memoryless'' process).</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>* '''Misconception 2''': If run large number of coin tosses, the '''number of heads''' and '''number of tails''' become more and more equal. This is incorrect, as the LLN only guarantees that the sample proportion of heads will converge to the true population proportion (the p parameter that we selected). In fact, the difference |Heads - Tails| diverges!</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>* '''Misconception 2''': If run large number of coin tosses, the '''number of heads''' and '''number of tails''' become more and more equal. This is incorrect, as the LLN only guarantees that the sample proportion of heads will converge to the true population proportion (the p parameter that we selected). In fact, the difference |Heads - Tails| diverges!</div></td></tr>
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<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">==References==</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">* Dinov, ID., Christou, N., Gould, R [http://www.amstat.org/publications/jse/v17n1/dinov.html Law of Large Numbers: the Theory, Applications and Technology-based Education]. [http://www.amstat.org/publications/jse JSE], [http://www.amstat.org/publications/jse/ Vol. 17, No. 1, 1-15, 2009].</ins></div></td></tr>
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</table>IvoDinovhttp://wiki.stat.ucla.edu/socr/index.php?title=SOCR_EduMaterials_Activities_LawOfLargeNumbers&diff=8791&oldid=prevIvoDinov: /* Overview */2009-01-16T18:23:35Z<p><span class="autocomment">Overview</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>== Overview==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>== Overview==</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>This is part I of a heterogeneous activity that demonstrates the theory and applications of the Law of Large Numbers (LLN). [[SOCR_EduMaterials_Activities_LawOfLargeNumbers2 | Part II]] and [[SOCR_EduMaterials_Activities_LawOfLargeNumbersExperiment | Part III]] of this activity contain more examples and diverse experiments.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>This is part I of a heterogeneous activity that demonstrates the theory and applications of the Law of Large Numbers (LLN). [[SOCR_EduMaterials_Activities_LawOfLargeNumbers2 | Part II]] and [[SOCR_EduMaterials_Activities_LawOfLargeNumbersExperiment | Part III]] of this activity contain more examples and diverse experiments<ins class="diffchange diffchange-inline">. The [http://socr.ucla.edu/htmls/exp/LLN_Simple_Experiment.html SOCR LLN applet is available here]</ins>.</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>===Goals of the SOCR LLN activity===</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>===Goals of the SOCR LLN activity===</div></td></tr>
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</table>IvoDinovhttp://wiki.stat.ucla.edu/socr/index.php?title=SOCR_EduMaterials_Activities_LawOfLargeNumbers&diff=7271&oldid=prevIvoDinov: /* Experiment 3= */2008-04-28T04:20:07Z<p><span class="autocomment">Experiment 3=</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Similarly, one may approximate the transcendental number <math>\pi</math>, using the [[SOCR_EduMaterials_Activities_BuffonNeedleExperiment#Buffon.27s_needle_experiment_and_estimation_of_the_constant_.CF.80 | SOCR Buffon’s Needle Experiment]]. Here, the LLN again provides the foundation for a better approximation of <math>\pi</math> by virtually dropping needles (many times) on a tiled surface and observing if the needle crosses a tile grid-line. For a tile grid of size 1, the odds of a needle-line intersection are <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.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Similarly, one may approximate the transcendental number <math>\pi</math>, using the [[SOCR_EduMaterials_Activities_BuffonNeedleExperiment#Buffon.27s_needle_experiment_and_estimation_of_the_constant_.CF.80 | SOCR Buffon’s Needle Experiment]]. Here, the LLN again provides the foundation for a better approximation of <math>\pi</math> by virtually dropping needles (many times) on a tiled surface and observing if the needle crosses a tile grid-line. For a tile grid of size 1, the odds of a needle-line intersection are <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.</div></td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>==Experiment 3<del class="diffchange diffchange-inline">=</del>==</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>==Experiment 3==</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Suppose we row 10 loaded hexagonal (6-face) dice 8 times and we are interested in the probability of observing the event A={3 ones, 3 twos, 2 threes, and 2 fours}. Assume the dice are loaded to the small outcomes according to the following probabilities of the 6 outcomes (''one'' is the most likely and ''six'' is the least likely outcome).</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Suppose we row 10 loaded hexagonal (6-face) dice 8 times and we are interested in the probability of observing the event A={3 ones, 3 twos, 2 threes, and 2 fours}. Assume the dice are loaded to the small outcomes according to the following probabilities of the 6 outcomes (''one'' is the most likely and ''six'' is the least likely outcome).</div></td></tr>
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</table>IvoDinovhttp://wiki.stat.ucla.edu/socr/index.php?title=SOCR_EduMaterials_Activities_LawOfLargeNumbers&diff=7270&oldid=prevIvoDinov: added the multi-nomial experiment demonstrating LLN2008-04-28T04:19:28Z<p>added the multi-nomial experiment demonstrating LLN</p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Similarly, one may approximate the transcendental number <math>\pi</math>, using the [[SOCR_EduMaterials_Activities_BuffonNeedleExperiment#Buffon.27s_needle_experiment_and_estimation_of_the_constant_.CF.80 | SOCR Buffon’s Needle Experiment]]. Here, the LLN again provides the foundation for a better approximation of <math>\pi</math> by virtually dropping needles (many times) on a tiled surface and observing if the needle crosses a tile grid-line. For a tile grid of size 1, the odds of a needle-line intersection are <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.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Similarly, one may approximate the transcendental number <math>\pi</math>, using the [[SOCR_EduMaterials_Activities_BuffonNeedleExperiment#Buffon.27s_needle_experiment_and_estimation_of_the_constant_.CF.80 | SOCR Buffon’s Needle Experiment]]. Here, the LLN again provides the foundation for a better approximation of <math>\pi</math> by virtually dropping needles (many times) on a tiled surface and observing if the needle crosses a tile grid-line. For a tile grid of size 1, the odds of a needle-line intersection are <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.</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">==Experiment 3===</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">Suppose we row 10 loaded hexagonal (6-face) dice 8 times and we are interested in the probability of observing the event A={3 ones, 3 twos, 2 threes, and 2 fours}. Assume the dice are loaded to the small outcomes according to the following probabilities of the 6 outcomes (''one'' is the most likely and ''six'' is the least likely outcome).</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"><center></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">{| class="wikitable" style="text-align:center; width:75%" border="1"</ins></div></td></tr>
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<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">| ''x'' || 1 || 2 || 3 || 4 || 5 || 6 </ins></div></td></tr>
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<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">| ''P(X=x)'' || 0.286 || 0.238 || 0.19 || 0.143 || 0.095 || 0.048</ins></div></td></tr>
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<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">: ''P(A)=?''</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">Of course, we can compute this number exactly as:</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">: <math>P(A) = {10! \over 3!\times 3! \times 2! \times 2! } \times 0.286^3 \times 0.238^3\times 0.19^2 \times 0.143^2 = 0.00586690138260962656816896.</math> </ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">However, we can also find a pretty close empirically-driven estimate using the [[SOCR_EduMaterials_Activities_DiceExperiment | SOCR Dice Experiment]].</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">For instance, running the [http://socr.ucla.edu/htmls/SOCR_Experiments.html SOCR Dice Experiment] 1,000 times with number of dice n=10, and the loading probabilities listed above, we get an output like the one shown below.</ins></div></td></tr>
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<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"><center>[[Image:SOCR_EBook_Dinov_Multinimial_030508_Fig1.jpg|500px]]</center></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">Now, we can actually count how many of these 1,000 trials generated the event ''A'' as an outcome. In one such experiment of 1,000 trials, there were 8 outcomes of the type {3 ones, 3 twos, 2 threes and 2 fours}. Therefore, the relative proportion of these outcomes to 1,000 will give us a fairly accurate estimate of the exact probability we computed above</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">: <math>P(A) \approx {8 \over 1,000}=0.008</math>. </ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">Note that that this approximation is close to the exact answer above. By the Law of Large Numbers, we know that this SOCR empirical approximation to the exact multinomial probability of interest will significantly improve as we increase the number of trials in this experiment to 10,000.</ins></div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>== Hands-on activities==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>== Hands-on activities==</div></td></tr>
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</table>IvoDinovhttp://wiki.stat.ucla.edu/socr/index.php?title=SOCR_EduMaterials_Activities_LawOfLargeNumbers&diff=5679&oldid=prevIvoDinov: Added hands-on activities2008-01-11T03:16:12Z<p>Added hands-on activities</p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>== Overview==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>== Overview==</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>This is part I of a heterogeneous activity that demonstrates the theory and applications of the Law of Large Numbers (LLN). [[SOCR_EduMaterials_Activities_LawOfLargeNumbers2 | Part II]] and [[SOCR_EduMaterials_Activities_LawOfLargeNumbersExperiment | Part III]] of this activity contain more examples and diverse experiments.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>This is part I of a heterogeneous activity that demonstrates the theory and applications of the Law of Large Numbers (LLN). [[SOCR_EduMaterials_Activities_LawOfLargeNumbers2 | Part II]] and [[SOCR_EduMaterials_Activities_LawOfLargeNumbersExperiment | Part III]] of this activity contain more examples and diverse experiments.</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">===Goals of the SOCR LLN activity===</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">The goals of this activity are to:</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">* illustrate the theoretical meaning and practical implications of the LLN;</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">* present the LLN in varieties of situations;</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">* provide empirical evidence in support of the LLN-convergence and dispel the common LLN misconceptions.</ins></div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>===Example===</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>===Example===</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>== Exercise 1==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>== Exercise 1==</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>This exercise illustrates the statement and validity of the LLN in the situation of tossing (biased or fair) coins repeatedly. Suppose we let H and T denote Heads and Tails, the probabilities of observing a Head or a Tail at each trial are <math>0<p<1</math> and <math>0<1-p<1</math>, respectfully. The sample space of this experiment <del class="diffchange diffchange-inline">consist </del>of sequences of H's and Ts. For example, an outcome may be <math>\{H, H, T, H, H, T, T, T, ....\}</math>. If we toss a coin n times, the size of the sample-space is <math>2^n</math>, as the coin tosses are independent. [[About_pages_for_SOCR_Distributions | Binomial Distribution]] governs the probability of observing <math>0\le k\le n</math> Heads in <math>n</math> experiments, which is evaluated by the binomial density at <math>k</math>. </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>This exercise illustrates the statement and validity of the LLN in the situation of tossing (biased or fair) coins repeatedly. Suppose we let H and T denote Heads and Tails, the probabilities of observing a Head or a Tail at each trial are <math>0<p<1</math> and <math>0<1-p<1</math>, respectfully. The sample space of this experiment <ins class="diffchange diffchange-inline">consists </ins>of sequences of H's and Ts. For example, an outcome may be <math>\{H, H, T, H, H, T, T, T, ....\}</math>. If we toss a coin n times, the size of the sample-space is <math>2^n</math>, as the coin tosses are independent. [[About_pages_for_SOCR_Distributions | Binomial Distribution]] governs the probability of observing <math>0\le k\le n</math> Heads in <math>n</math> experiments, which is evaluated by the binomial density at <math>k</math>. </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>In this case we will be interested in two random variables associated with this process. The first variable will be the ''proportion of Heads'' and the second will be the ''differences of the number of Heads and Tails''. This will empirically demonstrate the LLN and its most common misconceptions (presented below). Point your browser to the [http://socr.ucla.edu/htmls/SOCR_Experiments.html SOCR Experiments] and select the '''Coin Toss LLN Experiment''' from the drop-down list of experiments in the top-left panel. This applet consists of a control toolbar on the top followed by a graph panel in the middle and a results table at the bottom. Use the toolbar to flip coins one at a time, 10, 100, 1,000 at a time or continuously! The toolbar also allows you to stop or reset an experiment and select the probability of Heads ('''p''') using the slider. The graph panel in the middle will dynamically plot the values of the two variables of interest (''proportion of heads'' and ''difference of Heads and Tails''). The outcome table at the bottom presents the summaries of all trials of this experiment. From this table, you can copy and paste the summary for further processing using other computational resources (e.g., [http://socr.ucla.edu/htmls/SOCR_Modeler.html SOCR Modeler] or [http://office.microsoft.com/excel MS Excel]). </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>In this case we will be interested in two random variables associated with this process. The first variable will be the ''proportion of Heads'' and the second will be the ''differences of the number of Heads and Tails''. This will empirically demonstrate the LLN and its most common misconceptions (presented below). Point your browser to the [http://socr.ucla.edu/htmls/SOCR_Experiments.html SOCR Experiments] and select the '''Coin Toss LLN Experiment''' from the drop-down list of experiments in the top-left panel. This applet consists of a control toolbar on the top followed by a graph panel in the middle and a results table at the bottom. Use the toolbar to flip coins one at a time, 10, 100, 1,000 at a time or continuously! The toolbar also allows you to stop or reset an experiment and select the probability of Heads ('''p''') using the slider. The graph panel in the middle will dynamically plot the values of the two variables of interest (''proportion of heads'' and ''difference of Heads and Tails''). The outcome table at the bottom presents the summaries of all trials of this experiment. From this table, you can copy and paste the summary for further processing using other computational resources (e.g., [http://socr.ucla.edu/htmls/SOCR_Modeler.html SOCR Modeler] or [http://office.microsoft.com/excel MS Excel]). </div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Similarly, one may approximate the transcendental number <math>\pi</math>, using the [[SOCR_EduMaterials_Activities_BuffonNeedleExperiment#Buffon.27s_needle_experiment_and_estimation_of_the_constant_.CF.80 | SOCR Buffon’s Needle Experiment]]. Here, the LLN again provides the foundation for a better approximation of <math>\pi</math> by virtually dropping needles (many times) on a tiled surface and observing if the needle crosses a tile grid-line. For a tile grid of size 1, the odds of a needle-line intersection are <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.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Similarly, one may approximate the transcendental number <math>\pi</math>, using the [[SOCR_EduMaterials_Activities_BuffonNeedleExperiment#Buffon.27s_needle_experiment_and_estimation_of_the_constant_.CF.80 | SOCR Buffon’s Needle Experiment]]. Here, the LLN again provides the foundation for a better approximation of <math>\pi</math> by virtually dropping needles (many times) on a tiled surface and observing if the needle crosses a tile grid-line. For a tile grid of size 1, the odds of a needle-line intersection are <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.</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">== Hands-on activities==</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">The following practice problems will help students experiment with the SOCR LLN activity and understand the meaning, ramifications and limitations of the LLN.</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">* Run the [http://socr.ucla.edu/htmls/SOCR_Experiments.html SOCR Coin Toss LLN Experiment] twice with stop=100 and p=0.5. This corresponds to flipping a fair coin 100 times and observing the behavior of the proportion of heads across (discrete) time.</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">** What will be different in the outcomes of the 2 experiments?</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">** What properties of the 2 outcomes will be very similar?</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">** If we did this 10 times, what is expected to vary and what may be predicted accurately?</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">* Use the [http://socr.ucla.edu/htmls/SOCR_Experiments.html SOCR Uniform ''e''-Estimate Experiment] to obtain stochastic estimates of the natural number <math>e \approx 2.7182</math>.</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">** Try to explain in words, and support your argument with data/results from this simulation, why is the expected value of the variable ''U'' (defined above) equal to ''e'', <math>E(U)=e</math>.</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">** How does the LLN come into play in this experiment?</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">** How would you go about in practice if you had to estimate <math>e^2 \approx 7.38861124</math> ?</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">** Similarly, try to estimate <math>\pi \approx 3.141592</math> and <math>\pi^2 \approx 9.8696044</math> using the [[SOCR_EduMaterials_Activities_BuffonNeedleExperiment#Buffon.27s_needle_experiment_and_estimation_of_the_constant_.CF.80 | SOCR Buffon’s Needle Experiment]].</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">* Run the [http://socr.ucla.edu/htmls/SOCR_Experiments.html SOCR Roulette Experiment] and bet on 1-18 (out of the 38 possible numbers/outcomes).</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">** What is the probability of success (''p'')?</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">** What does the LLN imply about p and repeated runs of this experiment?</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">** Run this experiment 3 times. What is the sample estimate of p (<math>\hat{p}</math>)? What is the difference <math>p-\hat{p}</math>? Would this difference change if we ran the experiment 10 or 100 times? How?</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">** In 100 Roulette experiments, what can you say about the difference of the number of successes (outcome in 1-18) and the number of failures? How about the proportion of successes?</ins></div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>== Other SOCR LLN Activities==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>== Other SOCR LLN Activities==</div></td></tr>
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</table>IvoDinovhttp://wiki.stat.ucla.edu/socr/index.php?title=SOCR_EduMaterials_Activities_LawOfLargeNumbers&diff=5678&oldid=prevIvoDinov: /* Estimating ''e'' using SOCR simulation */2008-01-10T22:38:11Z<p><span class="autocomment">Estimating ''e'' using SOCR simulation</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Now, the expected value <math>E(U) = e \approx 2.7182</math>. Therefore, by LLN, taking averages of <math>\left \{ U_1, U_2, U_3, ..., U_k \right \}</math> values, each computed from random samples <math>X_1, X_2, ..., X_n \sim U(0,1)</math> as described above, will provide a more accurate estimate (as <math>k \rightarrow \infty</math>) of the natural number ''e''.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Now, the expected value <math>E(U) = e \approx 2.7182</math>. Therefore, by LLN, taking averages of <math>\left \{ U_1, U_2, U_3, ..., U_k \right \}</math> values, each computed from random samples <math>X_1, X_2, ..., X_n \sim U(0,1)</math> as described above, will provide a more accurate estimate (as <math>k \rightarrow \infty</math>) of the natural number ''e''.</div></td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>The '''Uniform E-Estimate Experiment''', part of [http://www.socr.ucla.edu/htmls/SOCR_Experiments.html SOCR Experiments] provides a hands-on demonstration of how the LLN facilitates stochastic simulation-based estimation of ''e''.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>The '''Uniform E-Estimate Experiment''', part of [http://www.socr.ucla.edu/htmls/SOCR_Experiments.html SOCR Experiments]<ins class="diffchange diffchange-inline">, </ins>provides a hands-on demonstration of how the LLN facilitates stochastic simulation-based estimation of ''e''.</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><center>[[Image:SOCR_Activities_Uniform_E_EstimateExperiment_Dinov_121907_Fig1.jpg|400px]]</center></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><center>[[Image:SOCR_Activities_Uniform_E_EstimateExperiment_Dinov_121907_Fig1.jpg|400px]]</center></div></td></tr>
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</table>IvoDinovhttp://wiki.stat.ucla.edu/socr/index.php?title=SOCR_EduMaterials_Activities_LawOfLargeNumbers&diff=5608&oldid=prevIvoDinov at 19:19, 18 December 20072007-12-18T19:19:18Z<p></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>* '''Misconception 1''': If we observe a streak of 10 consecutive heads (when p=0.5, say) the odds of the <math>11^{th}</math> trial being a Head is > p! This is of course, incorrect, as the coin tosses are independent trials (an example of a ''memoryless'' process).</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>* '''Misconception 1''': If we observe a streak of 10 consecutive heads (when p=0.5, say) the odds of the <math>11^{th}</math> trial being a Head is > p! This is of course, incorrect, as the coin tosses are independent trials (an example of a ''memoryless'' process).</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>* '''Misconception 2''': If run large number of coin tosses, the '''number of heads''' and '''number of tails''' become more and more equal. This is incorrect, as the LLN only guarantees that the sample proportion of heads will converge to the true population proportion (the p parameter that we selected). In fact, the difference |Heads - Tails| diverges!</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>* '''Misconception 2''': If run large number of coin tosses, the '''number of heads''' and '''number of tails''' become more and more equal. This is incorrect, as the LLN only guarantees that the sample proportion of heads will converge to the true population proportion (the p parameter that we selected). In fact, the difference |Heads - Tails| diverges!</div></td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">==[[SOCR_EduMaterials_Activities_LawOfLargeNumbers2 | Part II of this activity]]==</del></div></td><td colspan="2"> </td></tr>
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</table>IvoDinovhttp://wiki.stat.ucla.edu/socr/index.php?title=SOCR_EduMaterials_Activities_LawOfLargeNumbers&diff=5607&oldid=prevIvoDinov at 19:18, 18 December 20072007-12-18T19:18:36Z<p></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>This is part I of a heterogeneous activity that demonstrates the theory and applications of the Law of Large Numbers (LLN). [[SOCR_EduMaterials_Activities_LawOfLargeNumbers2 | Part II]] and [[SOCR_EduMaterials_Activities_LawOfLargeNumbersExperiment | Part III]] of this activity contain more examples and diverse experiments.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>This is part I of a heterogeneous activity that demonstrates the theory and applications of the Law of Large Numbers (LLN). [[SOCR_EduMaterials_Activities_LawOfLargeNumbers2 | Part II]] and [[SOCR_EduMaterials_Activities_LawOfLargeNumbersExperiment | Part III]] of this activity contain more examples and diverse experiments.</div></td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">=</del>=== Example <del class="diffchange diffchange-inline">=</del>===</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>===Example===</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The average weight of 10 students from a class of 100 students is most likely closer to the ''real average'' weight of all 100 students, compared to the average weight of 3 randomly chosen students from that same class. This is because the sample of 10 is a ''larger number'' than the sample of only 3 and better represents the entire class. At the extreme, a sample of 99 of the 100 students will produce a sample average almost exactly the same as the average for all 100 students. On the other extreme, sampling a single student will be an extremely variant estimate of the overall class average weight.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The average weight of 10 students from a class of 100 students is most likely closer to the ''real average'' weight of all 100 students, compared to the average weight of 3 randomly chosen students from that same class. This is because the sample of 10 is a ''larger number'' than the sample of only 3 and better represents the entire class. At the extreme, a sample of 99 of the 100 students will produce a sample average almost exactly the same as the average for all 100 students. On the other extreme, sampling a single student will be an extremely variant estimate of the overall class average weight.</div></td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">=</del>=== Statement of the Law of Large Numbers <del class="diffchange diffchange-inline">=</del>===</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>===Statement of the Law of Large Numbers===</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>If an event of probability p is observed repeatedly during '''independent repetitions''', the ratio of the observed frequency of that event to the total number of repetitions converges towards p as the number of repetitions becomes arbitrarily large.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>If an event of probability p is observed repeatedly during '''independent repetitions''', the ratio of the observed frequency of that event to the total number of repetitions converges towards p as the number of repetitions becomes arbitrarily large.</div></td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">=</del>=== The theory behind the LLN <del class="diffchange diffchange-inline">=</del>===</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>===The theory behind the LLN===</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Complete details about the ''weak'' and ''strong'' laws of large numbers may be found [http://en.wikipedia.org/wiki/Law_of_large_numbers here].</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Complete details about the ''weak'' and ''strong'' laws of large numbers may be found [http://en.wikipedia.org/wiki/Law_of_large_numbers here].</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>* Remember that the more experiments you run the closer the theoretical and sample proportions will be (by LLN). Go in '''Continuous run mode''' and watch the convergence of the sample proportion to <math>p</math>. Can you explain in words, why can't we expect the second variable of interest (the differences of Heads and Tails) to converge? [[Image:SOCR_Activities_LLN_Dinov_022007_Fig2.jpg|200px]]</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>* Remember that the more experiments you run the closer the theoretical and sample proportions will be (by LLN). Go in '''Continuous run mode''' and watch the convergence of the sample proportion to <math>p</math>. Can you explain in words, why can't we expect the second variable of interest (the differences of Heads and Tails) to converge? [[Image:SOCR_Activities_LLN_Dinov_022007_Fig2.jpg|200px]]</div></td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>== <del class="diffchange diffchange-inline">'''</del>Exercise 2<del class="diffchange diffchange-inline">'''</del>==</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>==Exercise 2==</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The second SOCR demonstration of the law of large numbers will be quite different and practically useful. Here we show how the LLN implies practical algorithms for estimation of [http://en.wikipedia.org/wiki/Transcendental_number transcendental numbers]. The two most popular transcendental numbers are [http://en.wikipedia.org/wiki/Pi <math>\pi</math>] and [http://en.wikipedia.org/wiki/E_%28mathematical_constant%29 ''e''].</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The second SOCR demonstration of the law of large numbers will be quite different and practically useful. Here we show how the LLN implies practical algorithms for estimation of [http://en.wikipedia.org/wiki/Transcendental_number transcendental numbers]. The two most popular transcendental numbers are [http://en.wikipedia.org/wiki/Pi <math>\pi</math>] and [http://en.wikipedia.org/wiki/E_%28mathematical_constant%29 ''e''].</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>* [[SOCR_EduMaterials_Activities_LawOfLargeNumbersExperiment | Part III of this activity]]</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>* [[SOCR_EduMaterials_Activities_LawOfLargeNumbersExperiment | Part III of this activity]]</div></td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>== <del class="diffchange diffchange-inline">'''</del>Common Misconceptions regarding the LLN<del class="diffchange diffchange-inline">'''</del>==</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>== Common Misconceptions regarding the LLN==</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>* '''Misconception 1''': If we observe a streak of 10 consecutive heads (when p=0.5, say) the odds of the <math>11^{th}</math> trial being a Head is > p! This is of course, incorrect, as the coin tosses are independent trials (an example of a ''memoryless'' process).</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>* '''Misconception 1''': If we observe a streak of 10 consecutive heads (when p=0.5, say) the odds of the <math>11^{th}</math> trial being a Head is > p! This is of course, incorrect, as the coin tosses are independent trials (an example of a ''memoryless'' process).</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>* '''Misconception 2''': If run large number of coin tosses, the '''number of heads''' and '''number of tails''' become more and more equal. This is incorrect, as the LLN only guarantees that the sample proportion of heads will converge to the true population proportion (the p parameter that we selected). In fact, the difference |Heads - Tails| diverges!</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>* '''Misconception 2''': If run large number of coin tosses, the '''number of heads''' and '''number of tails''' become more and more equal. This is incorrect, as the LLN only guarantees that the sample proportion of heads will converge to the true population proportion (the p parameter that we selected). In fact, the difference |Heads - Tails| diverges!</div></td></tr>
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</table>IvoDinovhttp://wiki.stat.ucla.edu/socr/index.php?title=SOCR_EduMaterials_Activities_LawOfLargeNumbers&diff=5606&oldid=prevIvoDinov: /* '''Exercise 1''' */2007-12-18T19:17:31Z<p><span class="autocomment">'''Exercise 1'''</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Complete details about the ''weak'' and ''strong'' laws of large numbers may be found [http://en.wikipedia.org/wiki/Law_of_large_numbers here].</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Complete details about the ''weak'' and ''strong'' laws of large numbers may be found [http://en.wikipedia.org/wiki/Law_of_large_numbers here].</div></td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>== <del class="diffchange diffchange-inline">'''</del>Exercise 1<del class="diffchange diffchange-inline">'''</del>==</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>== Exercise 1==</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>This exercise illustrates the statement and validity of the LLN in the situation of tossing (biased or fair) coins repeatedly. Suppose we let H and T denote Heads and Tails, the probabilities of observing a Head or a Tail at each trial are <math>0<p<1</math> and <math>0<1-p<1</math>, respectfully. The sample space of this experiment consist of sequences of H's and Ts. For example, an outcome may be <math>\{H, H, T, H, H, T, T, T, ....\}</math>. If we toss a coin n times, the size of the sample-space is <math>2^n</math>, as the coin tosses are independent. [[About_pages_for_SOCR_Distributions | Binomial Distribution]] governs the probability of observing <math>0\le k\le n</math> Heads in <math>n</math> experiments, which is evaluated by the binomial density at <math>k</math>. </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>This exercise illustrates the statement and validity of the LLN in the situation of tossing (biased or fair) coins repeatedly. Suppose we let H and T denote Heads and Tails, the probabilities of observing a Head or a Tail at each trial are <math>0<p<1</math> and <math>0<1-p<1</math>, respectfully. The sample space of this experiment consist of sequences of H's and Ts. For example, an outcome may be <math>\{H, H, T, H, H, T, T, T, ....\}</math>. If we toss a coin n times, the size of the sample-space is <math>2^n</math>, as the coin tosses are independent. [[About_pages_for_SOCR_Distributions | Binomial Distribution]] governs the probability of observing <math>0\le k\le n</math> Heads in <math>n</math> experiments, which is evaluated by the binomial density at <math>k</math>. </div></td></tr>
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</table>IvoDinovhttp://wiki.stat.ucla.edu/socr/index.php?title=SOCR_EduMaterials_Activities_LawOfLargeNumbers&diff=5605&oldid=prevIvoDinov at 19:16, 18 December 20072007-12-18T19:16:54Z<p></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>== [[SOCR_EduMaterials_Activities | SOCR Educational Materials - Activities]] - SOCR Law of Large Numbers Activity ==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>== [[SOCR_EduMaterials_Activities | SOCR Educational Materials - Activities]] - SOCR Law of Large Numbers Activity ==</div></td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>== This is part I of a heterogeneous activity that demonstrates the theory and applications of the Law of Large Numbers (LLN). [[SOCR_EduMaterials_Activities_LawOfLargeNumbers2 | Part II of this activity<del class="diffchange diffchange-inline">]] contains </del>more examples and diverse experiments. <del class="diffchange diffchange-inline">==</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>== <ins class="diffchange diffchange-inline">Overview==</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>This is part I of a heterogeneous activity that demonstrates the theory and applications of the Law of Large Numbers (LLN). [[SOCR_EduMaterials_Activities_LawOfLargeNumbers2 | Part II<ins class="diffchange diffchange-inline">]] and [[SOCR_EduMaterials_Activities_LawOfLargeNumbersExperiment | Part III]] </ins>of this activity <ins class="diffchange diffchange-inline">contain </ins>more examples and diverse experiments.</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==== Example ====</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==== Example ====</div></td></tr>
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