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D&D Older Editions
Archmage's Ascension - The 4E Wizard's Handbook (RuinsFate)
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<blockquote data-quote="tiornys" data-source="post: 8407450" data-attributes="member: 17633"><p>Decided to hunt down my old post about the expected damage increase from Evocation Apprentice. Here's the essential information:</p><p></p><p>This is an analysis of the increase to expected damage granted by the Evocation Apprentice ability to reroll a single damage die that comes up as a 1. Die size is listed on the right. Numbers across the top are the number of dice. Intersections are the gain in expected damage.</p><p style="text-align: right"></p> <p style="text-align: right"> <table style='width: 100%'><tr><td></td><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td><td>8</td></tr><tr><td>d4</td><td>0.38</td><td>0.66</td><td>0.87</td><td>1.03</td><td>1.14</td><td>1.23</td><td>1.3</td><td>1.35</td></tr><tr><td>d6</td><td>0.42</td><td>0.76</td><td>1.05</td><td>1.29</td><td>1.5</td><td>1.66</td><td>1.8</td><td>1.92</td></tr><tr><td>d8</td><td>0.44</td><td>0.82</td><td>1.16</td><td>1.45</td><td>1.7</td><td>1.93</td><td>2.13</td><td>2.3</td></tr><tr><td>d10</td><td>0.45</td><td>0.86</td><td>1.22</td><td>1.55</td><td>1.84</td><td>2.11</td><td>2.35</td><td>2.56</td></tr><tr><td>d12</td><td>0.46</td><td>0.88</td><td>1.26</td><td>1.62</td><td>1.94</td><td>2.24</td><td>2.51</td><td><p style="text-align: right">2.76</p> </td></tr></table> </p><p></p><p>Math breakdown: for the purpose of an expected damage calculation, rerolling a 1 is replacing 1 with the average value of the die. That replacement results in an increase of (N-1)/2 on an N-sided die (the average value of an N-sided die is (N+1)/2, then subtract 1 since we're replacing a 1). We get this increase whenever we roll at least one 1 in the damage roll. On X dice with N sides, the probability of rolling at least one 1 is 1-[(N-1/N)^X]. Therefore, the expected damage gain is [(N-1)/2]*[1-((N-1/N)^X)].</p></blockquote><p></p>
[QUOTE="tiornys, post: 8407450, member: 17633"] Decided to hunt down my old post about the expected damage increase from Evocation Apprentice. Here's the essential information: This is an analysis of the increase to expected damage granted by the Evocation Apprentice ability to reroll a single damage die that comes up as a 1. Die size is listed on the right. Numbers across the top are the number of dice. Intersections are the gain in expected damage. [RIGHT] [TABLE] [TR] [TD][/TD] [TD]1[/TD] [TD]2[/TD] [TD]3[/TD] [TD]4[/TD] [TD]5[/TD] [TD]6[/TD] [TD]7[/TD] [TD]8[/TD] [/TR] [TR] [TD]d4[/TD] [TD]0.38[/TD] [TD]0.66[/TD] [TD]0.87[/TD] [TD]1.03[/TD] [TD]1.14[/TD] [TD]1.23[/TD] [TD]1.3[/TD] [TD]1.35[/TD] [/TR] [TR] [TD]d6[/TD] [TD]0.42[/TD] [TD]0.76[/TD] [TD]1.05[/TD] [TD]1.29[/TD] [TD]1.5[/TD] [TD]1.66[/TD] [TD]1.8[/TD] [TD]1.92[/TD] [/TR] [TR] [TD]d8[/TD] [TD]0.44[/TD] [TD]0.82[/TD] [TD]1.16[/TD] [TD]1.45[/TD] [TD]1.7[/TD] [TD]1.93[/TD] [TD]2.13[/TD] [TD]2.3[/TD] [/TR] [TR] [TD]d10[/TD] [TD]0.45[/TD] [TD]0.86[/TD] [TD]1.22[/TD] [TD]1.55[/TD] [TD]1.84[/TD] [TD]2.11[/TD] [TD]2.35[/TD] [TD]2.56[/TD] [/TR] [TR] [TD]d12[/TD] [TD]0.46[/TD] [TD]0.88[/TD] [TD]1.26[/TD] [TD]1.62[/TD] [TD]1.94[/TD] [TD]2.24[/TD] [TD]2.51[/TD] [TD][RIGHT]2.76[/RIGHT][/TD] [/TR] [/TABLE][/RIGHT] Math breakdown: for the purpose of an expected damage calculation, rerolling a 1 is replacing 1 with the average value of the die. That replacement results in an increase of (N-1)/2 on an N-sided die (the average value of an N-sided die is (N+1)/2, then subtract 1 since we're replacing a 1). We get this increase whenever we roll at least one 1 in the damage roll. On X dice with N sides, the probability of rolling at least one 1 is 1-[(N-1/N)^X]. Therefore, the expected damage gain is [(N-1)/2]*[1-((N-1/N)^X)]. [/QUOTE]
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