http://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_Prob_Simul&feed=atom&action=historyAP Statistics Curriculum 2007 Prob Simul - Revision history2024-03-29T05:41:17ZRevision history for this page on the wikiMediaWiki 1.15.1http://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_Prob_Simul&diff=12315&oldid=prevIvoDinov: /* Card Experiment */2013-10-15T14:00:59Z<p><span class="autocomment">Card Experiment</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>====[[SOCR_EduMaterials_Activities_CardExperiment | Card Experiment]]====</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>====[[SOCR_EduMaterials_Activities_CardExperiment | Card Experiment]]====</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>This experiment is more involved because the sample space is significantly larger. It demonstrates the basic properties of dealing n cards at random from a standard deck of 52 cards. At every trial, n cards are randomly drawn and their denomination and suit are recorded in the result table below. We can use this simulation to estimate the probabilities of various hands (e.g., the odds of getting a pair of cards with the same denomination). Run the experiment 100 times and count the number of 5-card hands that had at least one pair in them (at least one pair of cars in the 5-card had had a matching denomination, i.e., <math>Y_i=Y_j</math>, for <math>1 \leq i < j \leq 5</math>). Dividing this number by 100 gives the simulation-based estimate of the probability of the complex event of interest (at least one pair).</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>This experiment is more involved because the sample space is significantly larger. It demonstrates the basic properties of dealing n cards at random from a standard deck of 52 cards. At every trial, n cards are randomly drawn and their denomination and suit are recorded in the result table below. We can use this simulation to estimate the probabilities of various hands (e.g., the odds of getting a pair of cards with the same denomination). Run the experiment 100 times and count the number of 5-card hands that had at least one pair in them (at least one pair of cars in the 5-card had had a matching denomination, i.e., <math>Y_i=Y_j</math>, for <math>1 \leq i < j \leq 5</math>). Dividing this number by 100 gives the simulation-based estimate of the probability of the complex event of interest (at least one pair)<ins class="diffchange diffchange-inline">. Also see the [[AP_Statistics_Curriculum_2007_Prob_Count#Poker_Game_Calculations|Poker Game Probability Calculations section]]</ins>.</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_EBook_Dinov_Probability_012908_Fig4.jpg|400px]]</center></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><center>[[Image:SOCR_EBook_Dinov_Probability_012908_Fig4.jpg|400px]]</center></div></td></tr>
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</table>IvoDinovhttp://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_Prob_Simul&diff=12172&oldid=prevIvoDinov: /* Chuck A Luck Experiment */2013-04-17T01:50:00Z<p><span class="autocomment">Chuck A Luck Experiment</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>| Outcome = Number of matching dice || None=loss || 1 || 2 || 3</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>| Outcome = Number of matching dice || None=loss || 1 || 2 || 3</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>|-</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>|-</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>| <del class="diffchange diffchange-inline">X </del>(profit/payoff) || -1 || 1 || 2 || 3</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>| <ins class="diffchange diffchange-inline">W </ins>(profit/payoff) || -1 || 1 || 2 || 3</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>|-</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>|-</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>| P(<del class="diffchange diffchange-inline">X</del>=<del class="diffchange diffchange-inline">x</del>) || \( \left( \frac{5}{6} \right)^3 \) || \(\binom{3}{1} \left( \frac{5}{6} \right)^2 \frac{1}{6}\) || \(\binom{3}{2}\frac{5}{6} \left( \frac{1}{6} \right)^2 \) || \( \left(\frac{1}{6}\right)^3 \)</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>| P(<ins class="diffchange diffchange-inline">W</ins>=<ins class="diffchange diffchange-inline">w</ins>) || \( \left( \frac{5}{6} \right)^3 \) || \(\binom{3}{1} \left( \frac{5}{6} \right)^2 \frac{1}{6}\) || \(\binom{3}{2}\frac{5}{6} \left( \frac{1}{6} \right)^2 \) || \( \left(\frac{1}{6}\right)^3 \)</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>|-</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>|-</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>| <del class="diffchange diffchange-inline">x</del>*P(<del class="diffchange diffchange-inline">X</del>=<del class="diffchange diffchange-inline">x</del>) || (-1)x0.5787 || 0.34722 || 2x0.06944 || 3x0.00463</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>| <ins class="diffchange diffchange-inline">w</ins>*P(<ins class="diffchange diffchange-inline">W</ins>=<ins class="diffchange diffchange-inline">w</ins>) || (-1)x0.5787 || 0.34722 || 2x0.06944 || 3x0.00463</div></td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>Thus, if <del class="diffchange diffchange-inline">X </del>is the [[AP_Statistics_Curriculum_2007_Distrib_RV |random variable]] denoting the return of this game (player's profit), its expected value is:</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>Thus, if <ins class="diffchange diffchange-inline">W </ins>is the [[AP_Statistics_Curriculum_2007_Distrib_RV |random variable]] denoting the return of this game (player's profit), its expected value is:</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>: \( E[<del class="diffchange diffchange-inline">X</del>]=\<del class="diffchange diffchange-inline">sum_x</del>{<del class="diffchange diffchange-inline">xP</del>(<del class="diffchange diffchange-inline">X</del>=<del class="diffchange diffchange-inline">x</del>)} = -0.0787\).</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>: \( E[<ins class="diffchange diffchange-inline">W</ins>]=\<ins class="diffchange diffchange-inline">sum_w</ins>{<ins class="diffchange diffchange-inline">wP</ins>(<ins class="diffchange diffchange-inline">W</ins>=<ins class="diffchange diffchange-inline">w</ins>)} = -0.0787\).</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>Similarly, we can compute the [[AP_Statistics_Curriculum_2007_Distrib_MeanVar#Variance|variance]] and [[AP_Statistics_Curriculum_2007_Distrib_MeanVar#Standard_Deviation|standard deviation]] of the outcome/payoff variable X:</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Similarly, we can compute the [[AP_Statistics_Curriculum_2007_Distrib_MeanVar#Variance|variance]] and [[AP_Statistics_Curriculum_2007_Distrib_MeanVar#Standard_Deviation|standard deviation]] of the outcome/payoff variable X:</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>: \( Var[<del class="diffchange diffchange-inline">X</del>]= 1.23917\) and \( SD[<del class="diffchange diffchange-inline">X</del>]= 1.11318 = \sqrt{Var[<del class="diffchange diffchange-inline">X</del>]}\).</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>: \( Var[<ins class="diffchange diffchange-inline">W</ins>]= 1.23917\) and \( SD[<ins class="diffchange diffchange-inline">W</ins>]= 1.11318 = \sqrt{Var[<ins class="diffchange diffchange-inline">W</ins>]}\).</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>: '''Question''': What is your interpretation of this quantitative analysis of the [http://socr.ucla.edu/htmls/exp/Chuck_A_Luck_Experiment.html Chuck A Luck Experiment]? Would you agree to play (or host) this game?</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>: '''Question''': What is your interpretation of this quantitative analysis of the [http://socr.ucla.edu/htmls/exp/Chuck_A_Luck_Experiment.html Chuck A Luck Experiment]? Would you agree to play (or host) this game?</div></td></tr>
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</table>IvoDinovhttp://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_Prob_Simul&diff=12171&oldid=prevIvoDinov: /* Chuck A Luck Experiment */2013-04-17T01:48:04Z<p><span class="autocomment">Chuck A Luck Experiment</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>====[[SOCR_EduMaterials_Activities_ChuckALuckExperiment#Probability_calculations| Chuck A Luck Experiment]]====</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>====[[SOCR_EduMaterials_Activities_ChuckALuckExperiment#Probability_calculations| Chuck A Luck Experiment]]====</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>The [http://socr.ucla.edu/htmls/exp/Chuck_A_Luck_Experiment.html Chuck A Luck Experiment] shows how to compute the expectation of a real game, which indicated the odds of a player to win or loose money.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>The [http://socr.ucla.edu/htmls/exp/Chuck_A_Luck_Experiment.html Chuck A Luck Experiment] shows how to compute the expectation of a real game, which indicated the odds of a player to win or loose money<ins class="diffchange diffchange-inline">. In this game, a player/gambler chooses an integer from ''1'' to ''6''. Then three fair dice are rolled. If exactly ''k'' dice show the gambler's number, the payoff is ''k:1''. The random variable W represents the gambler's net profit on a ''$1'' bet</ins>.</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Let's first look at the outcomes of each experiment and compute the corresponding outcome probability values. Then, we can use the [[AP_Statistics_Curriculum_2007_Distrib_MeanVar#Expectation_.28Mean.29 |expectation formula]] to compute the expected return (expected value) for this experiment. A zero, positive or negative expected return indicates if the game is fair, or biased for the player or the house, respectively.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Let's first look at the outcomes of each experiment and compute the corresponding outcome probability values. Then, we can use the [[AP_Statistics_Curriculum_2007_Distrib_MeanVar#Expectation_.28Mean.29 |expectation formula]] to compute the expected return (expected value) for this experiment. A zero, positive or negative expected return indicates if the game is fair, or biased for the player or the house, respectively.</div></td></tr>
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</table>IvoDinovhttp://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_Prob_Simul&diff=11877&oldid=prevIvoDinov at 00:16, 2 November 20122012-11-02T00:16:43Z<p></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><center>[[Image:SOCR_EBook_Dinov_Probability_012908_Fig7.jpg|400px]]</center></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><center>[[Image:SOCR_EBook_Dinov_Probability_012908_Fig7.jpg|400px]]</center></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;">====[[SOCR_EduMaterials_Activities_ChuckALuckExperiment#Probability_calculations| Chuck A Luck 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;">The [http://socr.ucla.edu/htmls/exp/Chuck_A_Luck_Experiment.html Chuck A Luck Experiment] shows how to compute the expectation of a real game, which indicated the odds of a player to win or loose money.</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;">Let's first look at the outcomes of each experiment and compute the corresponding outcome probability values. Then, we can use the [[AP_Statistics_Curriculum_2007_Distrib_MeanVar#Expectation_.28Mean.29 |expectation formula]] to compute the expected return (expected value) for this experiment. A zero, positive or negative expected return indicates if the game is fair, or biased for the player or the house, respectively.</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;">The table below identifies the possible outcomes of this experiment and shows how to use the [[AP_Statistics_Curriculum_2007_Distrib_Binomial#Binomial_Random_Variables|Binomial Distribution]] to compute their corresponding probabilities. Note that the probability of a single die turning up and matching (or not matching, the [[AP_Statistics_Curriculum_2007_Prob_Basics|complementary event]]) the player betting number is \(\frac{1}{6}\) (or \(\frac{5}{6}\)).</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;"><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>
<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;">| Outcome = Number of matching dice || None=loss || 1 || 2 || 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;">|-</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;">| X (profit/payoff) || -1 || 1 || 2 || 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;">|-</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;">| P(X=x) || \( \left( \frac{5}{6} \right)^3 \) || \(\binom{3}{1} \left( \frac{5}{6} \right)^2 \frac{1}{6}\) || \(\binom{3}{2}\frac{5}{6} \left( \frac{1}{6} \right)^2 \) || \( \left(\frac{1}{6}\right)^3 \)</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;">|-</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;">| x*P(X=x) || (-1)x0.5787 || 0.34722 || 2x0.06944 || 3x0.00463</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></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;">Thus, if X is the [[AP_Statistics_Curriculum_2007_Distrib_RV |random variable]] denoting the return of this game (player's profit), its expected value is:</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;">: \( E[X]=\sum_x{xP(X=x)} = -0.0787\).</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;">Similarly, we can compute the [[AP_Statistics_Curriculum_2007_Distrib_MeanVar#Variance|variance]] and [[AP_Statistics_Curriculum_2007_Distrib_MeanVar#Standard_Deviation|standard deviation]] of the outcome/payoff variable X:</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;">: \( Var[X]= 1.23917\) and \( SD[X]= 1.11318 = \sqrt{Var[X]}\).</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;">: '''Question''': What is your interpretation of this quantitative analysis of the [http://socr.ucla.edu/htmls/exp/Chuck_A_Luck_Experiment.html Chuck A Luck Experiment]? Would you agree to play (or host) this game?</ins></div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>===[[EBook_Problems_Prob_Simul|Problems]]===</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>===[[EBook_Problems_Prob_Simul|Problems]]===</div></td></tr>
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</table>IvoDinovhttp://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_Prob_Simul&diff=10957&oldid=prevIvoDinov: moved the Poker Game Counting and probablities section to Counting section2011-04-25T15:46:54Z<p>moved the Poker Game Counting and probablities section to Counting section</p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><center>[[Image:SOCR_EBook_Dinov_Probability_012908_Fig4.jpg|400px]]</center></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><center>[[Image:SOCR_EBook_Dinov_Probability_012908_Fig4.jpg|400px]]</center></div></td></tr>
<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;"></del></div></td><td colspan="2"> </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_PokerExperiment | Poker Game]]====</del></div></td><td colspan="2"> </td></tr>
<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;">There is a variety of events that may be of interest for 5-card (poker) games (see also [[AP_Statistics_Curriculum_2007_Prob_Count#Hands-on_combination_activity | this section on card games]]. Some of these are:</del></div></td><td colspan="2"> </td></tr>
<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;">* '''A single Pair Match''': For instance, the hand with the pattern AABCD, where A, B, C and D are from the distinct kinds (denominations) of cards: aces, twos, threes, tens, jacks, queens, and kings (there are 13 denominations, and four suits, in the standard 52 card deck). The number of such hands is <math>{13 \choose 1}{4\choose 2}{12\choose 3}{4\choose 1}^3</math>. If all hands are equally likely, the probability of a single pair is obtained by dividing this number by the total number of 5-card hands possible (<math>{52\choose 5}=2,598,960</math>). Thus, P(1 pair only) = 0.422569.</del></div></td><td colspan="2"> </td></tr>
<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;"></del></div></td><td colspan="2"> </td></tr>
<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;">* '''Two pairs''': For instance, the pattern AABBC where A, B, and C are from distinct kinds. The number of such hands is <math>{13\choose 2}{4\choose 2}{4\choose 2}{11\choose 1}{4\choose 1}</math>. And therefore, dividing by <math>{52\choose 5}=2,598,960</math>, the P(2 pairs) = 0.047539.</del></div></td><td colspan="2"> </td></tr>
<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;"></del></div></td><td colspan="2"> </td></tr>
<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;">* '''One triple''': For example, the pattern AAABC where A, B, and C are from distinct kinds. The number of such hands is <math>{13\choose 1}{4\choose 3}{12\choose 2}{4\choose 1}^2</math>. Thus, the P(1 triple) = 0.021128.</del></div></td><td colspan="2"> </td></tr>
<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;"></del></div></td><td colspan="2"> </td></tr>
<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;">* '''A Full House''': This includes patterns like AAABB where A and B are from distinct kinds. The number of such hands is <math>{13\choose 1}{4\choose 3}{12\choose 1}{4\choose 2}</math>. Thus, P(Full House)=0.001441.</del></div></td><td colspan="2"> </td></tr>
<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;"></del></div></td><td colspan="2"> </td></tr>
<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;">* '''Four of a kind''': For instance, the pattern AAAAB where A and B are from distinct kinds. The number of such hands is <math>{13\choose 1}{4\choose 4}{12\choose 1}{4\choose 1}</math>. The P(4-of-a-kind)=0.000240.</del></div></td><td colspan="2"> </td></tr>
<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;"></del></div></td><td colspan="2"> </td></tr>
<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;">* '''A straight''': Five cards in a sequence (e.g., 4,5,6,7,8), with aces allowed to be either 1 or 13 (low or high) and with the cards allowed to be of the same suit (e.g., all hearts) or from some different suits. The number of such hands is <math>10*{4\choose 1}^5</math>. Thus, P(A Straight)=0.003940. But if you exclude Straight-Flushes AND Royal Flushes, the number of such hands is <math>10*{4\choose 1}^5 - 36 - 4 = 10200</math>, the corresponding probability P(A Straight, but not a Straight or Royal Flush)=0.00392465.</del></div></td><td colspan="2"> </td></tr>
<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;"></del></div></td><td colspan="2"> </td></tr>
<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;">* '''A Straight Flush''': All 5 cards are from the same suit and they form a straight. The number of such hands is 4*10, and P(Straight Flush)=0.0000153908.</del></div></td><td colspan="2"> </td></tr>
<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;"></del></div></td><td colspan="2"> </td></tr>
<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;">* '''A Royal Flush''': This consists of the ten, jack, queen, king, and ace all in one suit. There are only 4 such hands. Thus, P(Royal Flush)=0.00000153908.</del></div></td><td colspan="2"> </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;"><center>[[Image:SOCR_EBook_Dinov_Probability_013008_Fig2.jpg|400px]]</center></del></div></td><td colspan="2"> </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_RouletteExperiment | Roulette Experiment]]====</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>====[[SOCR_EduMaterials_Activities_RouletteExperiment | Roulette Experiment]]====</div></td></tr>
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</table>IvoDinovhttp://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_Prob_Simul&diff=10279&oldid=prevJenny at 16:55, 28 June 20102010-06-28T16:55:44Z<p></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>===SOCR simulations===</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>===SOCR simulations===</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>There <del class="diffchange diffchange-inline">are </del>a large number of [http://socr.ucla.edu/htmls/SOCR_Experiments.html SOCR Simulations] that may be used to compute (approximately) probabilities of various processes and compare these empirical probabilities to their exact counterparts.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>There <ins class="diffchange diffchange-inline">is </ins>a large number of [http://socr.ucla.edu/htmls/SOCR_Experiments.html SOCR Simulations] that may be used to compute (approximately) probabilities of various processes and compare these empirical probabilities to their exact counterparts.</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_BallAndRunExperiment | Ball and Urn Experiment]]====</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>====[[SOCR_EduMaterials_Activities_BallAndRunExperiment | Ball and Urn Experiment]]====</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>This experiment allows sampling with or without replacement from a virtual urn that has one of two types of balls - red (successes) and green (failures). The user may control the total number of balls in the urn (N), the number of red balls (R) and the number of balls sampled from the urn (n). Depending on whether we sample with or without replacement, the chance (or probability) of getting m red balls (successes) in a sample of n balls changes. Note that when sampling ''without replacement'', <math>1 \leq n \leq N</math>, and when sampling ''with replacement'', <math>1 \leq n</math>. The applet records numerically the empirical outcomes and compares these to the theoretical expected counts using distribution tables and graphs. Suppose we set N=50, R=25 and n=10. How many successes (red ball) do we expect to get in the sample of 10 balls, if we sample without replacement? [[SOCR_EduMaterials_Activities_Binomial_Distributions | Hypergeometric distribution]] provides the theoretical model for experiment and allows us to compute this probability exactly. We can also run the experiment once and approximate this answer by dividing the number of observed red balls by 10 (the sample size). You can try this experiment gauge the accuracy of the simulation-based approximations relative to various setting, e.g., sample size. According to the [[SOCR_EduMaterials_Activities_LawOfLargeNumbers | Law of Large Numbers]], the accuracy of the estimation will rapidly increase with the increase of the sample-size (n). </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>This experiment allows sampling with or without replacement from a virtual urn that has one of two types of balls - red (successes) and green (failures). The user may control the total number of balls in the urn (N), the number of red balls (R) and the number of balls sampled from the urn (n). Depending on whether we sample with or without replacement, the chance (or probability) of getting m red balls (successes) in a sample of n balls changes. Note that when sampling ''without replacement'', <math>1 \leq n \leq N</math>, and when sampling ''with replacement'', <math>1 \leq n</math>. The applet records numerically the empirical outcomes and compares these to the theoretical expected counts using distribution tables and graphs. Suppose we set N=50, R=25 and n=10. How many successes (red ball) do we expect to get in the sample of 10 balls, if we sample without replacement? [[SOCR_EduMaterials_Activities_Binomial_Distributions | Hypergeometric distribution]] provides the theoretical model for experiment and allows us to compute this probability exactly. We can also run the experiment once and approximate this answer by dividing the number of observed red balls by 10 (the sample size). You can try this experiment <ins class="diffchange diffchange-inline">to </ins>gauge the accuracy of the simulation-based approximations relative to various setting, e.g., sample size. According to the [[SOCR_EduMaterials_Activities_LawOfLargeNumbers | Law of Large Numbers]], the accuracy of the estimation will rapidly increase with the increase of the sample-size (n). </div></td></tr>
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</table>Jennyhttp://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_Prob_Simul&diff=9644&oldid=prevIvoDinov: /* Poker Game */2009-11-19T17:02:15Z<p><span class="autocomment">Poker Game</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>====[[SOCR_EduMaterials_Activities_PokerExperiment | Poker Game]]====</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>====[[SOCR_EduMaterials_Activities_PokerExperiment | Poker Game]]====</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>There is a variety of events that may be of interest for 5-card (poker) games. Some of these are:</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>There is a variety of events that may be of interest for 5-card (poker) games <ins class="diffchange diffchange-inline">(see also [[AP_Statistics_Curriculum_2007_Prob_Count#Hands-on_combination_activity | this section on card games]]</ins>. Some of these are:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>* '''A single Pair Match''': For instance, the hand with the pattern AABCD, where A, B, C and D are from the distinct kinds (denominations) of cards: aces, twos, threes, tens, jacks, queens, and kings (there are 13 denominations, and four suits, in the standard 52 card deck). The number of such hands is <math>{13 \choose 1}{4\choose 2}{12\choose 3}{4\choose 1}^3</math>. If all hands are equally likely, the probability of a single pair is obtained by dividing this number by the total number of 5-card hands possible (<math>{52\choose 5}=2,598,960</math>). Thus, P(1 pair only) = 0.422569.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>* '''A single Pair Match''': For instance, the hand with the pattern AABCD, where A, B, C and D are from the distinct kinds (denominations) of cards: aces, twos, threes, tens, jacks, queens, and kings (there are 13 denominations, and four suits, in the standard 52 card deck). The number of such hands is <math>{13 \choose 1}{4\choose 2}{12\choose 3}{4\choose 1}^3</math>. If all hands are equally likely, the probability of a single pair is obtained by dividing this number by the total number of 5-card hands possible (<math>{52\choose 5}=2,598,960</math>). Thus, P(1 pair only) = 0.422569.</div></td></tr>
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</table>IvoDinovhttp://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_Prob_Simul&diff=9488&oldid=prevIvoDinov: added a link to the Problems set2009-10-26T04:50:29Z<p>added a link to the Problems set</p>
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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;"></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;">===[[EBook_Problems_Prob_Simul|Problems]]===</ins></div></td></tr>
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</table>IvoDinovhttp://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_Prob_Simul&diff=7221&oldid=prevIvoDinov: /* Ball and Urn Experiment */ fixed typo2008-04-22T22:13:44Z<p><span class="autocomment"><a href="/socr/index.php/SOCR_EduMaterials_Activities_BallAndRunExperiment" title="SOCR EduMaterials Activities BallAndRunExperiment"> Ball and Urn Experiment</a>: </span> fixed typo</p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>====[[SOCR_EduMaterials_Activities_BallAndRunExperiment | Ball and Urn Experiment]]====</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>====[[SOCR_EduMaterials_Activities_BallAndRunExperiment | Ball and Urn Experiment]]====</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>This experiment allows sampling with or without replacement from a virtual urn that has one of two types of balls - red (successes) and green (failures). The user may control the total number of balls in the urn (N), the number of red balls (R) and the number of balls sampled from the urn (n). Depending on whether we sample with or without replacement, the chance (or probability) of getting m red balls (successes) in a sample of n balls <del class="diffchange diffchange-inline">(</del><math>1 \<del class="diffchange diffchange-inline">geq </del>n \<del class="diffchange diffchange-inline">geq </del>N</math><del class="diffchange diffchange-inline">) changes</del>. The applet records numerically the empirical outcomes and compares these to the theoretical expected counts using distribution tables and graphs. Suppose we set N=50, R=25 and n=10. How many successes (red ball) do we expect to get in the sample of 10 balls, if we sample without replacement? [[SOCR_EduMaterials_Activities_Binomial_Distributions | Hypergeometric distribution]] provides the theoretical model for experiment and allows us to compute this probability exactly. We can also run the experiment once and approximate this answer by dividing the number of observed red balls by 10 (the sample size). You can try this experiment gauge the accuracy of the simulation-based approximations relative to various setting, e.g., sample size. According to the [[SOCR_EduMaterials_Activities_LawOfLargeNumbers | Law of Large Numbers]], the accuracy of the estimation will rapidly increase with the increase of the sample-size (n). </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>This experiment allows sampling with or without replacement from a virtual urn that has one of two types of balls - red (successes) and green (failures). The user may control the total number of balls in the urn (N), the number of red balls (R) and the number of balls sampled from the urn (n). Depending on whether we sample with or without replacement, the chance (or probability) of getting m red balls (successes) in a sample of n balls <ins class="diffchange diffchange-inline">changes. Note that when sampling ''without replacement'', </ins><math>1 \<ins class="diffchange diffchange-inline">leq </ins>n \<ins class="diffchange diffchange-inline">leq </ins>N</math><ins class="diffchange diffchange-inline">, and when sampling ''with replacement'', <math>1 \leq n</math></ins>. The applet records numerically the empirical outcomes and compares these to the theoretical expected counts using distribution tables and graphs. Suppose we set N=50, R=25 and n=10. How many successes (red ball) do we expect to get in the sample of 10 balls, if we sample without replacement? [[SOCR_EduMaterials_Activities_Binomial_Distributions | Hypergeometric distribution]] provides the theoretical model for experiment and allows us to compute this probability exactly. We can also run the experiment once and approximate this answer by dividing the number of observed red balls by 10 (the sample size). You can try this experiment gauge the accuracy of the simulation-based approximations relative to various setting, e.g., sample size. According to the [[SOCR_EduMaterials_Activities_LawOfLargeNumbers | Law of Large Numbers]], the accuracy of the estimation will rapidly increase with the increase of the sample-size (n). </div></td></tr>
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</table>IvoDinovhttp://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_Prob_Simul&diff=7206&oldid=prevIvoDinov: fixed typos2008-04-21T01:32:27Z<p>fixed typos</p>
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<td colspan='2' style="background-color: white; color:black;">Revision as of 01:32, 21 April 2008</td>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>A very simple example is the case of trying to estimate the area of a region, A, embedded in a square of size 1. The area of the region depends on the demarcation of its boundary, as a simple closed curve shown on the picture. This problem relates to the problem of computing the probability of the event A as a subset of the sample-space S - square of size 1. In other words, if we were to throw a dart at the square, S, what would be the chance that the dart <del class="diffchange diffchange-inline">land </del>inside A <del class="diffchange diffchange-inline">(</del>under certain conditions<del class="diffchange diffchange-inline">, </del>e.g., the dart must land in S and each location of S is equally likely to be hit by the dart)?</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>A very simple example is the case of trying to estimate the area of a region, A, embedded in a square of size 1. The area of the region depends on the demarcation of its boundary, as a simple closed curve shown on the picture. This problem relates to the problem of computing the probability of the event A as a subset of the sample-space S - square of size 1. In other words, if we were to throw a dart at the square, S, what would be the chance that the dart <ins class="diffchange diffchange-inline">lands </ins>inside A<ins class="diffchange diffchange-inline">, </ins>under certain conditions <ins class="diffchange diffchange-inline">(</ins>e.g., the dart must land in S and each location of S is equally likely to be hit by the dart)?</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>This problem may be solved exactly by using integration, but an easier approximate solution would be throwing 100 darts at the board and recording the proportion of darts that landed inside A. This proportion will be a good simulation-based approximation to the real size (or probability) of the set (or event) A. For the instance of throwing 15 darts and having 7 land inside of A, the simulation-based estimate of the area (or probability) of A is <math>P(A) \approx {7 \over 15}</math>.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>This problem may be solved exactly by using integration, but an easier approximate solution would be throwing 100 darts at the board and recording the proportion of darts that landed inside A. This proportion will be a good simulation-based approximation to the real size (or probability) of the set (or event) A. For the instance of throwing 15 darts and having 7 land inside of A, the simulation-based estimate of the area (or probability) of A is <math>P(A) \approx {7 \over 15}</math>.</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_BallAndRunExperiment | Ball and Urn Experiment]]====</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>====[[SOCR_EduMaterials_Activities_BallAndRunExperiment | Ball and Urn Experiment]]====</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>This experiment allows sampling with or without replacement <del class="diffchange diffchange-inline">sampling </del>from a virtual urn that has one of two types of balls - red (successes) and green (failures). The user may control the total number of balls in the urn (N), the <del class="diffchange diffchange-inline">proportion </del>of red <del class="diffchange diffchange-inline">to green </del>balls (R) and the number of balls sampled from the urn (n). Depending on whether we sample with or without replacement, the chance (or probability) of getting <del class="diffchange diffchange-inline">n </del>red balls (successes) in a sample of n balls (<math>1 \geq n \geq N</math>) changes. The applet records numerically the empirical outcomes and compares these to the theoretical expected counts using distribution tables and graphs. Suppose we set N=50, R=25 and n=10. How many successes (red ball) do we expect to get in the sample of 10 balls, if we sample without replacement? [[SOCR_EduMaterials_Activities_Binomial_Distributions | Hypergeometric distribution]] provides the theoretical model for experiment and allows us to compute this probability exactly. We can also run the experiment once and approximate this answer by dividing the number of observed red balls by 10 (the sample size). You can try this experiment gauge the accuracy of the simulation-based approximations relative to various setting, e.g., sample size. According to the [[SOCR_EduMaterials_Activities_LawOfLargeNumbers | Law of Large Numbers]], the accuracy of the estimation will rapidly increase with the increase of the sample-size (n). </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>This experiment allows sampling with or without replacement from a virtual urn that has one of two types of balls - red (successes) and green (failures). The user may control the total number of balls in the urn (N), the <ins class="diffchange diffchange-inline">number </ins>of red balls (R) and the number of balls sampled from the urn (n). Depending on whether we sample with or without replacement, the chance (or probability) of getting <ins class="diffchange diffchange-inline">m </ins>red balls (successes) in a sample of n balls (<math>1 \geq n \geq N</math>) changes. The applet records numerically the empirical outcomes and compares these to the theoretical expected counts using distribution tables and graphs. Suppose we set N=50, R=25 and n=10. How many successes (red ball) do we expect to get in the sample of 10 balls, if we sample without replacement? [[SOCR_EduMaterials_Activities_Binomial_Distributions | Hypergeometric distribution]] provides the theoretical model for experiment and allows us to compute this probability exactly. We can also run the experiment once and approximate this answer by dividing the number of observed red balls by 10 (the sample size). You can try this experiment gauge the accuracy of the simulation-based approximations relative to various setting, e.g., sample size. According to the [[SOCR_EduMaterials_Activities_LawOfLargeNumbers | Law of Large Numbers]], the accuracy of the estimation will rapidly increase with the increase of the sample-size (n). </div></td></tr>
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