Dice Probability - Wizard of Odds

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</div> <h1>Dice Probability</h1> <div class="d-none-below-lm"> <div class="sideNav-block short"> <h3>On this page<i class="fa-solid fa-caret-down"></i></h3> </div> </div> <h2>Introduction</h2> <div class="article-image"> <a class="trans" href="/media/modules/article/images/55748/Dice_Probability.jpg"> <img class="lcp-mobile" src="/media/modules/article/images/55748/Dice_Probability.jpg" alt="Dice Probability" width="300" height="228"> </a> </div> <body><p>Before you play any dice game it is good to know the probability of any given total to be thrown. First lets look at the possibilities of the total of two dice. The table below shows the six possibilities for die 1 along the left column and the six possibilities for die 2 along the top column. The body of the table shows the sum of die 1 and die 2.</p> <div class="box"> <h2>Two Dice Totals</h2>  <div class="box-content has-data"> <table border="0" cellpadding="0" cellspacing="0" class="data"> <tbody> <tr> <th class="data-heading" rowspan="2">Die 1</th> <th class="data-heading" colspan="6">Die 2</th> </tr> <tr> <th class="data-heading">1</th> <th class="data-heading">2</th> <th class="data-heading">3</th> <th class="data-heading">4</th> <th class="data-heading">5</th> <th class="data-heading">6</th> </tr> <tr> <td>1</td> <td style="background-color:#FF00FF;">2</td> <td style="background-color:#77FF00;">3</td> <td style="background-color:#0077FF;">4</td> <td style="background-color:#FF0077;">5</td> <td style="background-color:#7700FF;">6</td> <td style="background-color:#FF7700;">7</td> </tr> <tr> <td>2</td> <td style="background-color:#77FF00;">3</td> <td style="background-color:#0077FF;">4</td> <td style="background-color:#FF0077;">5</td> <td style="background-color:#7700FF;">6</td> <td style="background-color:#FF7700;">7</td> <td style="background-color:#FF0000;">8</td> </tr> <tr> <td>3</td> <td style="background-color:#0077FF;">4</td> <td style="background-color:#FF0077;">5</td> <td style="background-color:#7700FF;">6</td> <td style="background-color:#FF7700;">7</td> <td style="background-color:#FF0000;">8</td> <td style="background-color:#00FF00;">9</td> </tr> <tr> <td>4</td> <td style="background-color:#FF0077;">5</td> <td style="background-color:#7700FF;">6</td> <td style="background-color:#FF7700;">7</td> <td style="background-color:#FF0000;">8</td> <td style="background-color:#00FF00;">9</td> <td style="background-color:#0000FF;">10</td> </tr> <tr> <td>5</td> <td style="background-color:#7700FF;">6</td> <td style="background-color:#FF7700;">7</td> <td style="background-color:#FF0000;">8</td> <td style="background-color:#00FF00;">9</td> <td style="background-color:#0000FF;">10</td> <td style="background-color:#77DD77;">11</td> </tr> <tr> <td>6</td> <td style="background-color:#FF7700;">7</td> <td style="background-color:#FF0000;">8</td> <td style="background-color:#00FF00;">9</td> <td style="background-color:#0000FF;">10</td> <td style="background-color:#77DD77;">11</td> <td style="background-color:#FF00FF;">12</td> </tr> </tbody> </table> </div> </div>  <p>The colors of the body of the table illustrate the number of ways to throw each total. The probability of throwing any given total is the number of ways to throw that total divided by the total number of combinations (36). In the following table the specific number of ways to throw each total and the probability of throwing that total is shown.</p> <div class="box desk-50percent"> <div class="box-content"> <table border="0" cellpadding="0" cellspacing="0" class="data"> <tbody> <tr> <th class="data-heading">Total</th> <th class="data-heading">Number of<br> Combinations</th> <th class="data-heading">Probability</th> </tr> <tr> <td>2</td> <td>1</td> <td>2.78%</td> </tr> <tr> <td>3</td> <td>2</td> <td>5.56%</td> </tr> <tr> <td>4</td> <td>3</td> <td>8.33%</td> </tr> <tr> <td>5</td> <td>4</td> <td>11.11%</td> </tr> <tr> <td>6</td> <td>5</td> <td>13.89%</td> </tr> <tr> <td>7</td> <td>6</td> <td>16.67%</td> </tr> <tr> <td>8</td> <td>5</td> <td>13.89%</td> </tr> <tr> <td>9</td> <td>4</td> <td>11.11%</td> </tr> <tr> <td>10</td> <td>3</td> <td>8.33%</td> </tr> <tr> <td>11</td> <td>2</td> <td>5.56%</td> </tr> <tr> <td>12</td> <td>1</td> <td>2.78%</td> </tr> <tr> <td>Total</td> <td>36</td> <td>100%</td> </tr> </tbody> </table> </div> </div>  <p>The following shows the probability of throwing each total in a chart format. 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For example consider the field bet in <a href="/games/craps/">craps</a>. This bet pays 1:1 (even money) if the next throw is a 3, 4, 9, 10, or 11, 2:1 (double the bet) on the 2, and 3:1 (triple the bet) on the 12. Note that there are 7 totals that win and only 4 that lose which might cause someone who didn't know better to think it was a good gamble.</p> <p>The player's return can be defined as the sum of the products of the probability of each event and the net return of that event. The following table shows each possible total, the net return, the probability of throwing that total, and the average return. The average return is the product of the net return and the probability. The player's return is the sum of the average returns.</p> <div class="box"> <div class="box-content has-data"> <table border="0" cellpadding="0" cellspacing="0" class="data"> <tbody> <tr> <th class="data-heading">Total</th> <th class="data-heading">Net Return</th> <th class="data-heading">Probability</th> <th class="data-heading">Average Return</th> </tr> <tr> <td>2</td> <td>2</td> <td>0.0278</td> <td>0.0556</td> </tr> <tr> <td>3</td> <td>1</td> <td>0.0556</td> <td>0.0556</td> </tr> <tr> <td>4</td> <td>1</td> <td>0.0833</td> <td>0.0833</td> </tr> <tr> <td>5</td> <td>-1</td> <td>0.1111</td> <td>-0.1111</td> </tr> <tr> <td>6</td> <td>-1</td> <td>0.1389</td> <td>-0.1389</td> </tr> <tr> <td>7</td> <td>-1</td> <td>0.1667</td> <td>-0.1667</td> </tr> <tr> <td>8</td> <td>-1</td> <td>0.1389</td> <td>-0.1389</td> </tr> <tr> <td>9</td> <td>1</td> <td>0.1111</td> <td>0.1111</td> </tr> <tr> <td>10</td> <td>1</td> <td>0.0833</td> <td>0.0833</td> </tr> <tr> <td>11</td> <td>1</td> <td>0.0556</td> <td>0.0556</td> </tr> <tr> <td>12</td> <td>3</td> <td>0.0278</td> <td>0.0834</td> </tr> <tr> <td>Total</td> <td> </td> <td>1</td> <td>-0.0278</td> </tr> </tbody> </table> </div> </div>  <p>The last row shows the player's return to be -.0278, in other words for every $1 bet the player can expect to lose 2.78 cents. 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