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556 lines
17 KiB
HTML
556 lines
17 KiB
HTML
<!doctype html>
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<meta http-equiv="content-type" content="text/html; charset=utf-8">
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<title>Activity: An Experiment with Dice</title>
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<article id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Activity: An Experiment with Dice</h1>
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<p>Let's throw two dice and add the scores ...</p>
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<table align="center" border="0">
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<tbody>
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<tr>
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<td>
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<p>You will need:</p>
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<ul>
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<li>Two <a href="../geometry/fair-dice.html">dice</a></li>
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</ul></td>
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<td width="50"> </td>
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<td><img src="../geometry/images/pair-dice2.svg" alt="pair dice"></td>
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</tr>
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</tbody></table>
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<div class="words">
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<h3>Interesting point</h3>
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<p>Many people think that one of these cubes is called "a dice". But no!</p>
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<p>The <b>plural is dice</b>, but the singular is <b>die</b>: i.e. 1 die, 2 dice.</p>
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</div>
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<p>The common die has six faces:</p>
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<p class="center"><img src="images/dice2.gif" alt="dice faces 1 to 6" height="61" width="297"></p>
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<p class="center">We usually call the faces 1, 2, 3, 4, 5 and 6.</p>
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<h2>Throwing Two Dice and Adding the Scores ...</h2>
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<div class="example">
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<p>Example: when one die shows 2 and the other shows 6 the total score is 2 + 6 = 8</p>
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<p class="center"><img src="images/dice3.gif" alt="dice 2 + 6 = 8" height="59" width="216"></p>
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</div>
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<p class="large">Question: Can you get a total of <b>8</b> any other way?</p>
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<div class="dotpoint">
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<p>What about <b>6 + 2 = 8</b> (the other way around), is that a different way?</p>
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<p>Yes! Because the two dice are different.</p>
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<div class="example">
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<p>Example: imagine one die is colored red and the other is colored blue.</p>
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<p>There are two possibilities:</p>
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<p class="center"><img src="images/dice4.gif" alt="2 blue + 6 red = 8 and 2 red + 6 blue = 8 " style="max-width:100%" height="81" width="383"></p>
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<p>So 2 + 6 and 6 + 2 are different.</p>
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</div>
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</div>
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<div class="dotpoint">
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<p>And you can get <b>8</b> with other numbers, such as <b>3 + 5 = 8</b> and <b>4 + 4 = 8</b></p>
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</div>
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<h2>High, Low, and Most Likely</h2>
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<p>Before we start, let's think about what might happen.</p>
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<p><b>Question: If you throw 2 dice together and add the two scores: </b></p>
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<ul>
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<li>1. What is the <b>least</b> possible total score?</li>
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<li>2. What is the <b>greatest</b> possible total score?</li>
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<li>3. What do you think is the <b>most likely</b> total score?</li>
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</ul>
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<div class="center80">
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<p>The first two questions are quite easy to answer:</p>
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<ul>
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<li>1. The <b>least</b> possible total score must be 1 + 1 = <b>2 </b></li>
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<li>2. The <b>greatest</b> possible total score must be 6 + 6 = <b>12</b></li>
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<li>3. The <b>most likely</b> total score is ... ???</li>
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</ul>
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<p>Are they all just as likely? Or will some happen more often?</p>
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</div>
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<p>To help answer the third question let us try an experiment.</p>
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<h2>The Experiment</h2>
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<div class="center80">
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<p class="center larger"><b>Throw</b> two dice together 108 times, <br>
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<b>add</b> the scores together each time, <br>
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<b>record</b> the scores in a tally table.</p>
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</div>
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<p>Why 108? That seems a strange number to choose. I will explain later.</p>
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<p>You can record the results in this table using <a href="../data/tally-marks.html">tally marks</a>:</p>
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<table align="center" cellpadding="5" border="1">
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<tbody>
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<tr align="center">
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<td class="larger">Added<br>
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Scores</td>
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<td class="larger" width="300">Tally</td>
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<td class="larger">Frequency</td>
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</tr>
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<tr align="center">
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<td class="larger">2</td>
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<td width="300"> </td>
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<td> </td>
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</tr>
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<tr align="center">
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<td class="larger">3</td>
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<td width="300"> </td>
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<td> </td>
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</tr>
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<tr align="center">
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<td class="larger">4</td>
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<td width="300"> </td>
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<td> </td>
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</tr>
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<tr align="center">
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<td class="larger">5</td>
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<td width="300"> </td>
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<td> </td>
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</tr>
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<tr align="center">
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<td class="larger">6</td>
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<td width="300"> </td>
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<td> </td>
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</tr>
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<tr align="center">
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<td class="larger">7</td>
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<td width="300"> </td>
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<td> </td>
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</tr>
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<tr align="center">
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<td class="larger">8</td>
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<td width="300"> </td>
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<td> </td>
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</tr>
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<tr align="center">
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<td class="larger">9</td>
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<td width="300"> </td>
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<td> </td>
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</tr>
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<tr align="center">
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<td class="larger">10</td>
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<td width="300"> </td>
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<td> </td>
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</tr>
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<tr align="center">
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<td class="larger">11</td>
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<td width="300"> </td>
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<td> </td>
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</tr>
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<tr align="center">
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<td class="larger">12</td>
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<td width="300"> </td>
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<td> </td>
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</tr>
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<tr align="center">
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<td> </td>
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<td align="right" width="300">Total Frequency = </td>
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<td class="larger">108</td>
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</tr>
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</tbody></table>
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<p class="largest">OK, Go!</p>
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<p class="center">... ...</p>
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<p class="center">... ...</p>
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<p class="center"> </p>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/die-results-empty.gif" alt="die results empty" height="208" width="300"></p>
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<p><b>Finished ...? </b></p>
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<p>Now draw a bar graph to illustrate your results.</p>
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<p>You can make your own.</p>
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<p>Or you can use <a href="../data/data-graph.html">Data Graphs (Bar, Line and Pie)</a> then print it out.</p>
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<div style="clear:both"></div>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/dice-results-thumb.gif" alt="dice results thumb" height="137" width="200"></p>
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<p>You may get something like this:</p>
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<div style="clear:both"></div>
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<div class="center80">
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<ul>
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<li>Are the bars all about the same height?</li>
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<li>If not ... why not?</li>
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</ul>
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</div>
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<h2>So Why Did We Get That Shape?</h2>
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<p>The explanation is simple:</p>
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<ul>
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<div class="bigul">
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<li>There is only <b>one</b> way to get a total of <b>2</b> (1 + 1),</li>
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<li>but there are <b>six</b> ways of getting a total of <b>7</b> (1 + 6, 2 + 5, 3 + 4, 4 + 3, 5 + 2 and 6 + 1)</li>
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</div>
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</ul>
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<p>Here is a table of all possibile outcomes, and the totals. I have also shown what adds to 7 in <b>bold</b>.</p>
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<table align="center" border="0">
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<tbody>
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<tr align="center">
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<td> </td>
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<td> </td>
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<td colspan="6"><b>Score on One Die</b></td>
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</tr>
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<tr align="center">
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<td width="50"> </td>
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<td width="50"> </td>
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<td class="large" width="50">1</td>
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<td class="large" width="50">2</td>
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<td class="large" width="50">3</td>
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<td class="large" width="50">4</td>
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<td class="large" width="50">5</td>
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<td class="large" width="50">6</td>
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</tr>
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<tr align="center">
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<td rowspan="6"><b>Score<br>
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on the<br>
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Other<br>
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Die</b></td>
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<td class="large">1</td>
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<td bgcolor="#99CCFF">2</td>
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<td bgcolor="#FFCCCC">3</td>
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<td bgcolor="#CCCC99">4</td>
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<td bgcolor="#9966FF">5</td>
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<td bgcolor="#00CC99">6</td>
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<td bgcolor="#9999FF"><b>7</b></td>
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</tr>
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<tr align="center">
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<td class="large">2</td>
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<td bgcolor="#FFCCCC">3</td>
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<td bgcolor="#CCCC99">4</td>
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<td bgcolor="#9966FF">5</td>
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<td bgcolor="#00CC99">6</td>
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<td bgcolor="#9999FF"><b>7</b></td>
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<td bgcolor="#FFFFCC">8</td>
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</tr>
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<tr align="center">
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<td class="large">3</td>
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<td bgcolor="#CCCC99">4</td>
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<td bgcolor="#9966FF">5</td>
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<td bgcolor="#00CC99">6</td>
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<td bgcolor="#9999FF"><b>7</b></td>
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<td bgcolor="#FFFFCC">8</td>
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<td bgcolor="#99CCCC">9</td>
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</tr>
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<tr align="center">
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<td class="large">4</td>
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<td bgcolor="#9966FF">5</td>
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<td bgcolor="#00CC99">6</td>
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<td bgcolor="#9999FF"><b>7</b></td>
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<td bgcolor="#FFFFCC">8</td>
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<td bgcolor="#99CCCC">9</td>
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<td bgcolor="#FFFF66">10</td>
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</tr>
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<tr align="center">
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<td class="large">5</td>
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<td bgcolor="#00CC99">6</td>
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<td bgcolor="#9999FF"><b>7</b></td>
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<td bgcolor="#FFFFCC">8</td>
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<td bgcolor="#99CCCC">9</td>
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<td bgcolor="#FFFF66">10</td>
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<td bgcolor="#999900">11</td>
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</tr>
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<tr align="center">
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<td class="large">6</td>
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<td bgcolor="#9999FF"><b>7</b></td>
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<td bgcolor="#FFFFCC">8</td>
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<td bgcolor="#99CCCC">9</td>
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<td bgcolor="#FFFF66">10</td>
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<td bgcolor="#999900">11</td>
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<td bgcolor="#9933FF">12</td>
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</tr>
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</tbody></table>
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<p class="center">You can see there is only 1 way to get 2, there are 2 ways to get 3, and so on.</p>
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<p>Let us count the ways of getting each total and put them in a table:</p>
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<table align="center" border="0">
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<tbody>
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<tr align="center">
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<td width="100">Total<br>
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Score</td>
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<td width="100">Number of <br>
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Ways to Get <br>
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Score</td>
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</tr>
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<tr align="center">
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<td>2</td>
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<td><b>1</b></td>
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</tr>
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<tr align="center">
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<td>3</td>
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<td><b>2</b></td>
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</tr>
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<tr align="center">
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<td>4</td>
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<td><b>3</b></td>
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</tr>
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<tr align="center">
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<td>5</td>
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<td><b>4</b></td>
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</tr>
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<tr align="center">
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<td>6</td>
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<td><b>5</b></td>
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</tr>
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<tr align="center">
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<td>7</td>
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<td><b>6</b></td>
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</tr>
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<tr align="center">
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<td>8</td>
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<td><b>5</b></td>
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</tr>
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<tr align="center">
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<td>9</td>
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<td><b>4</b></td>
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</tr>
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<tr align="center">
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<td>10</td>
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<td><b>3</b></td>
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</tr>
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<tr align="center">
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<td>11</td>
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<td><b>2</b></td>
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</tr>
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<tr align="center">
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<td>12</td>
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<td><b>1</b></td>
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</tr>
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<tr align="center">
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<td> </td>
|
|
<td>Total = 36</td>
|
|
</tr>
|
|
</tbody></table>
|
|
<p>Can you see the <a href="../geometry/symmetry-reflection.html">Symmetry</a> in this table?</p>
|
|
<ul>
|
|
<li>2 and 12 have the same number of ways = 1 each</li>
|
|
<li>3 and 11 have the same number of ways = 2 each</li>
|
|
<li>4 and 10 have the same number of ways = 3 each</li>
|
|
<li>5 and 9 have the same number of ways = 4 each</li>
|
|
<li>6 and 8 have the same number of ways = 5 each</li>
|
|
</ul>
|
|
<h2>108 Throws</h2>
|
|
<p>OK, why 108 throws? Well, 36 throws isn't enough for good results, 360 throws is great but takes a long time. So 108 (which is <b>3 lots of 36)</b> seems about right.</p>
|
|
<p>So let's multiply all these numbers by 3 to match our total of 108:</p>
|
|
<table align="center" border="0">
|
|
<tbody>
|
|
<tr align="center">
|
|
<td width="100">Total<br>
|
|
Score</td>
|
|
<td width="100">Number of <br>
|
|
Ways to Get <br>
|
|
Score</td>
|
|
</tr>
|
|
<tr align="center">
|
|
<td>2</td>
|
|
<td>3</td>
|
|
</tr>
|
|
<tr align="center">
|
|
<td>3</td>
|
|
<td>6</td>
|
|
</tr>
|
|
<tr align="center">
|
|
<td>4</td>
|
|
<td>9</td>
|
|
</tr>
|
|
<tr align="center">
|
|
<td>5</td>
|
|
<td>12</td>
|
|
</tr>
|
|
<tr align="center">
|
|
<td>6</td>
|
|
<td>15</td>
|
|
</tr>
|
|
<tr align="center">
|
|
<td>7</td>
|
|
<td>18</td>
|
|
</tr>
|
|
<tr align="center">
|
|
<td>8</td>
|
|
<td>15</td>
|
|
</tr>
|
|
<tr align="center">
|
|
<td>9</td>
|
|
<td>12</td>
|
|
</tr>
|
|
<tr align="center">
|
|
<td>10</td>
|
|
<td>9</td>
|
|
</tr>
|
|
<tr align="center">
|
|
<td>11</td>
|
|
<td>6</td>
|
|
</tr>
|
|
<tr align="center">
|
|
<td>12</td>
|
|
<td>3</td>
|
|
</tr>
|
|
<tr align="center">
|
|
<td> </td>
|
|
<td>Total = 108</td>
|
|
</tr>
|
|
</tbody></table>
|
|
<p>Those are the <b>theoretical</b> values, as opposed to the <b>experimental</b> ones you got from your experiment.</p>
|
|
<p>The <b>theoretical</b> values look like this in a bar graph:</p>
|
|
<p class="center"><img src="images/dice5.jpg" alt="dice results" height="243" width="350"></p>
|
|
<div class="center80">
|
|
<p class="center larger">How do these theoretical results compare with your experimental results?</p>
|
|
</div>
|
|
<p>This graph and your graph should be quite similar, but they are not likely to be exactly the same, as your experiment relied on <b>chance</b>, and the number of times you did it was fairly small.</p>
|
|
<p>If you did the experiment a very large number of times, you should get results much closer to the theoretical ones.</p>
|
|
<p>And, by the way, we've now answered the question from near the beginning of the experiment:</p>
|
|
<div class="center80">
|
|
<p class="larger">What is the most likely total score?</p>
|
|
<ul>
|
|
<li><b>7</b> has the highest bar, so 7 is the most likely total score.</li>
|
|
</ul>
|
|
<p>Hey, is that why people talk about <b>Lucky 7</b> ... ?</p>
|
|
</div>
|
|
<h2>Probability</h2>
|
|
<p>On the page <a href="../data/probability.html">Probability</a> you will find a formula:</p>
|
|
<p class="center">Probability of an event happening = <span class="intbl">
|
|
<em>Number of ways it can happen</em>
|
|
<strong>Total number of outcomes</strong>
|
|
</span></p>
|
|
<div class="example">
|
|
<h3>Example: Probability of a total of 2</h3>
|
|
<p>We know there are 36 possible outcomes.</p>
|
|
<p>And there is only 1 way to get a total score of 2.</p>
|
|
<p>So the probability of getting 2 is:</p>
|
|
<p class="center larger">Probability of a 2 = <span class="intbl">
|
|
<em>1</em>
|
|
<strong>36</strong>
|
|
</span></p>
|
|
|
|
</div>
|
|
<p>Doing that for each score gets us:</p>
|
|
<table align="center" border="0">
|
|
<tbody>
|
|
<tr align="center">
|
|
<td width="100">Total<br>
|
|
Score</td>
|
|
<td width="100">Probability</td>
|
|
</tr>
|
|
<tr align="center">
|
|
<td>2</td>
|
|
<td>1/36</td>
|
|
</tr>
|
|
<tr align="center">
|
|
<td>3</td>
|
|
<td>2/36</td>
|
|
</tr>
|
|
<tr align="center">
|
|
<td>4</td>
|
|
<td>3/36</td>
|
|
</tr>
|
|
<tr align="center">
|
|
<td>5</td>
|
|
<td>4/36</td>
|
|
</tr>
|
|
<tr align="center">
|
|
<td>6</td>
|
|
<td>5/36</td>
|
|
</tr>
|
|
<tr align="center">
|
|
<td>7</td>
|
|
<td>6/36</td>
|
|
</tr>
|
|
<tr align="center">
|
|
<td>8</td>
|
|
<td>5/36</td>
|
|
</tr>
|
|
<tr align="center">
|
|
<td>9</td>
|
|
<td>4/36</td>
|
|
</tr>
|
|
<tr align="center">
|
|
<td>10</td>
|
|
<td>3/36</td>
|
|
</tr>
|
|
<tr align="center">
|
|
<td>11</td>
|
|
<td>2/36</td>
|
|
</tr>
|
|
<tr align="center">
|
|
<td>12</td>
|
|
<td>1/36</td>
|
|
</tr>
|
|
<tr align="center">
|
|
<td> </td>
|
|
<td class="larger">Total = 1</td>
|
|
</tr>
|
|
</tbody></table>
|
|
<div><i>(Note: I didn't simplify the fractions)
|
|
</i></div>
|
|
<p class="center larger">The sum of all the probabilities is <b>1</b></p>
|
|
<div class="center80">
|
|
<p class="center">For any experiment:</p>
|
|
<p class="center">The sum of the probabilities of <b>all </b>possible outcomes is always equal to <b>1</b></p>
|
|
</div>
|
|
<p class="center"> </p>
|
|
<div class="related">
|
|
<a href="dice-experiment-1.html">An Experiment with 1 Die</a>
|
|
<a href="../data/index.html">Data Index</a>
|
|
<a href="../geometry/fair-dice.html">Fair Dice</a>
|
|
</div>
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