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The math of Advantage and Disadvantage
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<blockquote data-quote="Keravath" data-source="post: 7523115" data-attributes="member: 6916036"><p>There seems to be a lot of misinformation about the effects of advantage, disadvantage and elven accuracy (or the lucky feat) on the chances to hit, skill checks and saving throws. Some folks seem to consider them equivalent to +/-3 ... others +/-5. However, the sources some of them seem to reference are either incorrect or incomplete. </p><p></p><p>So in case anyone is interested <img src="data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" class="smilie smilie--sprite smilie--sprite1" alt=":)" title="Smile :)" loading="lazy" data-shortname=":)" /> ... the math of advantage <img src="data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" class="smilie smilie--sprite smilie--sprite1" alt=":)" title="Smile :)" loading="lazy" data-shortname=":)" /></p><p></p><p></p><p>----</p><p></p><p>Just for interest sake ... here is the math for advantage and disadvantage as a function of the number you need to roll to hit. </p><p></p><p>If the number you need to roll to hit is X ... e.g. X + to hit modifier = AC then the formulae are (X has a minimum value of 2 since a 1 always misses and a maximum value of 20 since 20 always hits). </p><p></p><p>1) Advantage</p><p></p><p>Advantage probability to hit is: 1.0 - the odds that both die rolls miss. </p><p></p><p>Advantage % to hit X = 100 * ( 1 - [(X-1)/20]^2) </p><p></p><p>e.g. AC=15, to hit modifier is +5, to hit = 10</p><p>Advantage % to hit = 79.8%</p><p></p><p>2) Disadvantage</p><p></p><p>Disadvantage probability to hit is: probability that both die rolls hit</p><p></p><p>Disadvantage % to hit X = 100 * [(20-X+1)/20]^2 </p><p></p><p>e.g. AC=15, to hit modifier is +5, to hit = 10</p><p>Disadvantage % to hit = 30.6%</p><p></p><p>3) Tri-vantage (elven accuracy or lucky feat applied to advantage)</p><p></p><p>Tri-vantage probability to hit is: 1.0 - the odds that all three die rolls miss. </p><p></p><p>Tri-advantage % to hit X = 100 * ( 1 - [(X-1)/20]^3) </p><p></p><p>e.g. AC=15, to hit modifier is +5, to hit = 10</p><p>Tri-vantage % to hit = 90.9%</p><p></p><p>---------------</p><p></p><p>The base probability to get a 10 or greater result without advantage or disadvantage is 55% </p><p></p><p>Thus in this case, disadvantage reduces the to hit by about 25% while advantage increases it by about 25%. </p><p></p><p>THIS is why passive skills are modified by +/-5 for advantage and disadvantage since for the average case advantage/disadvantage incur a +/-25% success probability change which is roughly equivalent to a static +/-5. </p><p></p><p>----------------</p><p></p><p>So I just thought I would add a table showing the % to hits for normal/advantage/disadvantage/trivantage vs target number along with the difference from the base case. </p><p></p><table style='width: 100%'><tr><td>X</td><td>Normal %</td><td>Advantage</td><td>% Diff</td><td>Disadvantage</td><td>% Diff</td><td>Tri-vantage</td><td>%Diff Base</td><td>%Diff Adv</td></tr><tr><td>1</td><td>100</td><td>100</td><td>0</td><td>100</td><td>0</td><td>100</td><td>0</td><td>0</td></tr><tr><td>2</td><td>95</td><td>99.75</td><td>4.75</td><td>90.25</td><td>-4.75</td><td>99.99</td><td>5.0</td><td>0.2</td></tr><tr><td>3</td><td>90</td><td>99</td><td>9</td><td>81</td><td>-9</td><td>99.9</td><td>9.9</td><td>0.9</td></tr><tr><td>4</td><td>85</td><td>97.75</td><td>12.75</td><td>72.75</td><td>-12.75</td><td>99.7<br /> </td><td>14.7</td><td>1.9</td></tr><tr><td>5</td><td>80</td><td>96</td><td>16</td><td>64</td><td>-16</td><td>99.2<br /> </td><td>19.2</td><td>3.2</td></tr><tr><td>6</td><td>75</td><td>93.75</td><td>18.75</td><td>56.25</td><td>-18.75</td><td>98.4<br /> </td><td>23.4</td><td>4.7</td></tr><tr><td>7</td><td>70</td><td>91</td><td>21</td><td>49</td><td>-21</td><td>97.3<br /> </td><td>27.3</td><td>6.3</td></tr><tr><td>8</td><td>65</td><td>87.75</td><td>22.75</td><td>42.25</td><td>-22.75</td><td>95.7<br /> </td><td>30.7</td><td>8.0</td></tr><tr><td>9</td><td>60</td><td>84</td><td>24</td><td>36</td><td>-24</td><td>93.6<br /> </td><td>33.6</td><td>9.6</td></tr><tr><td>10</td><td>55</td><td>79.75</td><td>24.75</td><td>30.25</td><td>-24.75</td><td>90.9<br /> </td><td>35.9</td><td>11.1</td></tr><tr><td>11</td><td>50</td><td>75</td><td>25</td><td>25</td><td>-25</td><td>87.5<br /> </td><td>37.5</td><td>12.5</td></tr><tr><td>12</td><td>45</td><td>69.75</td><td>24.75</td><td>20.25</td><td>-24.75</td><td>83.4<br /> </td><td>38.4</td><td>13.6</td></tr><tr><td>13</td><td>40</td><td>64</td><td>24</td><td>16</td><td>-24</td><td>78.4<br /> </td><td>38.4</td><td>14.4</td></tr><tr><td>14</td><td>35</td><td>57.75</td><td>22.75</td><td>12.25</td><td>-22.75</td><td>72.5<br /> </td><td>37.5</td><td>14.8</td></tr><tr><td>15</td><td>30</td><td>51</td><td>21</td><td>9</td><td>-21</td><td>65.7<br /> </td><td>35.7</td><td>14.7</td></tr><tr><td>16</td><td>25</td><td>43.75</td><td>18.75</td><td>6.25</td><td>-18.75</td><td>57.8<br /> </td><td>32.8</td><td>14.1</td></tr><tr><td>17</td><td>20</td><td>36</td><td>16</td><td>4</td><td>-16</td><td>48.8<br /> </td><td>28.8</td><td>12.8</td></tr><tr><td>18</td><td>15</td><td>27.75</td><td>12.75</td><td>2.25</td><td>-12.75</td><td>38.6<br /> </td><td>23.6</td><td>10.8</td></tr><tr><td>19</td><td>10</td><td>19</td><td>9</td><td>1</td><td>-9</td><td>27.1<br /> </td><td>17.1</td><td>8.1</td></tr><tr><td>20</td><td>5</td><td>9.75</td><td>4.75</td><td>0.25</td><td>-4.75</td><td>14.3<br /> </td><td>9.3</td><td>4.5</td></tr></table><p></p><p> </p><p></p><p>Notes:</p><p></p><p>This shows that over the range of the most common target numbers from about 8 to 14 (due to bounded accuracy) the effect of advantage/disadvantage varies from +/-21% to +/-25% (+4 to +5). </p><p></p><p>At the very extreme of the target numbers ... like needing a natural 20 to hit ... the effect is closer to that of a +1. However, the extremes do not come up as often as the middle of the distribution ... the game is balanced around typical target numbers in a standard encounter around 11. AC16 with +5 to hit at level 3 or maybe a typical AC20 with +9 to hit at level 11 ... sometimes the AC's are much easier or much harder to hit but then the creatures likely have varied hit points or other compensating abilities (like resistances). </p><p></p><p>Due to this, ascribing a static +/-3 to advantage/disadvantage isn't an accurate assessment.</p><p></p><p>The effect of elven accuracy "trivantage" is also interesting since for target numbers between 12 and 17 it is the equivalent of +2 to almost +3 to hit compared to regular advantage. </p><p></p><p>Finally, by comparing the target numbers for a base hit against target numbers with a +5 applied for advantage and trivantage you can assess the impact of using feats like GWM and SS. (eg. if the target number is normally 10 ... it will be 15 when using SS/GWM) </p><p></p><p></p><p>P.S. Keep in mind that the 1 line is only for skill checks (and saving throws?) ... to hit rolls auto miss on a 1 so it uses the "2" line in the table.</p></blockquote><p></p>
[QUOTE="Keravath, post: 7523115, member: 6916036"] There seems to be a lot of misinformation about the effects of advantage, disadvantage and elven accuracy (or the lucky feat) on the chances to hit, skill checks and saving throws. Some folks seem to consider them equivalent to +/-3 ... others +/-5. However, the sources some of them seem to reference are either incorrect or incomplete. So in case anyone is interested :) ... the math of advantage :) ---- Just for interest sake ... here is the math for advantage and disadvantage as a function of the number you need to roll to hit. If the number you need to roll to hit is X ... e.g. X + to hit modifier = AC then the formulae are (X has a minimum value of 2 since a 1 always misses and a maximum value of 20 since 20 always hits). 1) Advantage Advantage probability to hit is: 1.0 - the odds that both die rolls miss. Advantage % to hit X = 100 * ( 1 - [(X-1)/20]^2) e.g. AC=15, to hit modifier is +5, to hit = 10 Advantage % to hit = 79.8% 2) Disadvantage Disadvantage probability to hit is: probability that both die rolls hit Disadvantage % to hit X = 100 * [(20-X+1)/20]^2 e.g. AC=15, to hit modifier is +5, to hit = 10 Disadvantage % to hit = 30.6% 3) Tri-vantage (elven accuracy or lucky feat applied to advantage) Tri-vantage probability to hit is: 1.0 - the odds that all three die rolls miss. Tri-advantage % to hit X = 100 * ( 1 - [(X-1)/20]^3) e.g. AC=15, to hit modifier is +5, to hit = 10 Tri-vantage % to hit = 90.9% --------------- The base probability to get a 10 or greater result without advantage or disadvantage is 55% Thus in this case, disadvantage reduces the to hit by about 25% while advantage increases it by about 25%. THIS is why passive skills are modified by +/-5 for advantage and disadvantage since for the average case advantage/disadvantage incur a +/-25% success probability change which is roughly equivalent to a static +/-5. ---------------- So I just thought I would add a table showing the % to hits for normal/advantage/disadvantage/trivantage vs target number along with the difference from the base case. [TABLE="class: cms_table"] [TR] [TD]X[/TD] [TD]Normal %[/TD] [TD]Advantage[/TD] [TD]% Diff[/TD] [TD]Disadvantage[/TD] [TD]% Diff[/TD] [TD]Tri-vantage[/TD] [TD]%Diff Base[/TD] [TD]%Diff Adv[/TD] [/TR] [TR] [TD]1[/TD] [TD]100[/TD] [TD]100[/TD] [TD]0[/TD] [TD]100[/TD] [TD]0[/TD] [TD]100[/TD] [TD]0[/TD] [TD]0[/TD] [/TR] [TR] [TD]2[/TD] [TD]95[/TD] [TD]99.75[/TD] [TD]4.75[/TD] [TD]90.25[/TD] [TD]-4.75[/TD] [TD]99.99[/TD] [TD]5.0[/TD] [TD]0.2[/TD] [/TR] [TR] [TD]3[/TD] [TD]90[/TD] [TD]99[/TD] [TD]9[/TD] [TD]81[/TD] [TD]-9[/TD] [TD]99.9[/TD] [TD]9.9[/TD] [TD]0.9[/TD] [/TR] [TR] [TD]4[/TD] [TD]85[/TD] [TD]97.75[/TD] [TD]12.75[/TD] [TD]72.75[/TD] [TD]-12.75[/TD] [TD]99.7 [/TD] [TD]14.7[/TD] [TD]1.9[/TD] [/TR] [TR] [TD]5[/TD] [TD]80[/TD] [TD]96[/TD] [TD]16[/TD] [TD]64[/TD] [TD]-16[/TD] [TD]99.2 [/TD] [TD]19.2[/TD] [TD]3.2[/TD] [/TR] [TR] [TD]6[/TD] [TD]75[/TD] [TD]93.75[/TD] [TD]18.75[/TD] [TD]56.25[/TD] [TD]-18.75[/TD] [TD]98.4 [/TD] [TD]23.4[/TD] [TD]4.7[/TD] [/TR] [TR] [TD]7[/TD] [TD]70[/TD] [TD]91[/TD] [TD]21[/TD] [TD]49[/TD] [TD]-21[/TD] [TD]97.3 [/TD] [TD]27.3[/TD] [TD]6.3[/TD] [/TR] [TR] [TD]8[/TD] [TD]65[/TD] [TD]87.75[/TD] [TD]22.75[/TD] [TD]42.25[/TD] [TD]-22.75[/TD] [TD]95.7 [/TD] [TD]30.7[/TD] [TD]8.0[/TD] [/TR] [TR] [TD]9[/TD] [TD]60[/TD] [TD]84[/TD] [TD]24[/TD] [TD]36[/TD] [TD]-24[/TD] [TD]93.6 [/TD] [TD]33.6[/TD] [TD]9.6[/TD] [/TR] [TR] [TD]10[/TD] [TD]55[/TD] [TD]79.75[/TD] [TD]24.75[/TD] [TD]30.25[/TD] [TD]-24.75[/TD] [TD]90.9 [/TD] [TD]35.9[/TD] [TD]11.1[/TD] [/TR] [TR] [TD]11[/TD] [TD]50[/TD] [TD]75[/TD] [TD]25[/TD] [TD]25[/TD] [TD]-25[/TD] [TD]87.5 [/TD] [TD]37.5[/TD] [TD]12.5[/TD] [/TR] [TR] [TD]12[/TD] [TD]45[/TD] [TD]69.75[/TD] [TD]24.75[/TD] [TD]20.25[/TD] [TD]-24.75[/TD] [TD]83.4 [/TD] [TD]38.4[/TD] [TD]13.6[/TD] [/TR] [TR] [TD]13[/TD] [TD]40[/TD] [TD]64[/TD] [TD]24[/TD] [TD]16[/TD] [TD]-24[/TD] [TD]78.4 [/TD] [TD]38.4[/TD] [TD]14.4[/TD] [/TR] [TR] [TD]14[/TD] [TD]35[/TD] [TD]57.75[/TD] [TD]22.75[/TD] [TD]12.25[/TD] [TD]-22.75[/TD] [TD]72.5 [/TD] [TD]37.5[/TD] [TD]14.8[/TD] [/TR] [TR] [TD]15[/TD] [TD]30[/TD] [TD]51[/TD] [TD]21[/TD] [TD]9[/TD] [TD]-21[/TD] [TD]65.7 [/TD] [TD]35.7[/TD] [TD]14.7[/TD] [/TR] [TR] [TD]16[/TD] [TD]25[/TD] [TD]43.75[/TD] [TD]18.75[/TD] [TD]6.25[/TD] [TD]-18.75[/TD] [TD]57.8 [/TD] [TD]32.8[/TD] [TD]14.1[/TD] [/TR] [TR] [TD]17[/TD] [TD]20[/TD] [TD]36[/TD] [TD]16[/TD] [TD]4[/TD] [TD]-16[/TD] [TD]48.8 [/TD] [TD]28.8[/TD] [TD]12.8[/TD] [/TR] [TR] [TD]18[/TD] [TD]15[/TD] [TD]27.75[/TD] [TD]12.75[/TD] [TD]2.25[/TD] [TD]-12.75[/TD] [TD]38.6 [/TD] [TD]23.6[/TD] [TD]10.8[/TD] [/TR] [TR] [TD]19[/TD] [TD]10[/TD] [TD]19[/TD] [TD]9[/TD] [TD]1[/TD] [TD]-9[/TD] [TD]27.1 [/TD] [TD]17.1[/TD] [TD]8.1[/TD] [/TR] [TR] [TD]20[/TD] [TD]5[/TD] [TD]9.75[/TD] [TD]4.75[/TD] [TD]0.25[/TD] [TD]-4.75[/TD] [TD]14.3 [/TD] [TD]9.3[/TD] [TD]4.5[/TD] [/TR] [/TABLE] Notes: This shows that over the range of the most common target numbers from about 8 to 14 (due to bounded accuracy) the effect of advantage/disadvantage varies from +/-21% to +/-25% (+4 to +5). At the very extreme of the target numbers ... like needing a natural 20 to hit ... the effect is closer to that of a +1. However, the extremes do not come up as often as the middle of the distribution ... the game is balanced around typical target numbers in a standard encounter around 11. AC16 with +5 to hit at level 3 or maybe a typical AC20 with +9 to hit at level 11 ... sometimes the AC's are much easier or much harder to hit but then the creatures likely have varied hit points or other compensating abilities (like resistances). Due to this, ascribing a static +/-3 to advantage/disadvantage isn't an accurate assessment. The effect of elven accuracy "trivantage" is also interesting since for target numbers between 12 and 17 it is the equivalent of +2 to almost +3 to hit compared to regular advantage. Finally, by comparing the target numbers for a base hit against target numbers with a +5 applied for advantage and trivantage you can assess the impact of using feats like GWM and SS. (eg. if the target number is normally 10 ... it will be 15 when using SS/GWM) P.S. Keep in mind that the 1 line is only for skill checks (and saving throws?) ... to hit rolls auto miss on a 1 so it uses the "2" line in the table. [/QUOTE]
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