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<blockquote data-quote="Ovinomancer" data-source="post: 7522730" data-attributes="member: 16814"><p>I'm an electrical engineer who's made a few posts on this topic and I cannot follow what the chart in that post is doing. It's right at the ends, but very badly wrong in the middle. Whatever it's showing isn't a 'chance to roll this number <em>or higher</em>' which is the critical question of advantage (lower for disadvantage) and it's not chance to roll that exact number (which isn't interesting at all) so... </p><p></p><p>The basic question of advantage is can you roll higher that number N on 2 d20 rolls taking only the highest. This is figured by finding the odds of rolling higher than that number on a single d20 and then adding the product of those odds times the odds that you don't roll that number on a single d20. For example, if the target number is 11, the odds of rolling at least an 11 on a d20 is 10 in 20 or 50%. This, coincidentally, is the same odds of NOT rolling at least an 11. So, you add 10/20 to the product of (10/20 x 10/20) or 10/20+100/400. A little adding of fractions and you get 300/400 or 3/4 or 75% chance of rolling at least an 11 on 2d20 take highest. The equation form of this is a bit messy, and easier done in percentages:</p><p></p><p>.05(21-N)+.05(21-N)*.05(N-1) = p</p><p></p><p>To put this in a chart:</p><p></p><table style='width: 100%'><tr><td>Number</td><td>Base Chance <br /> (>= to N)</td><td>Advantage<br /> (>= to N)</td><td>Difference</td><td>Equivalent<br /> Bonus</td></tr><tr><td>1</td><td>100%</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>~1</td></tr><tr><td>3</td><td>90%</td><td>99%</td><td>9%</td><td>~2</td></tr><tr><td>4</td><td>85%</td><td>97.75%</td><td>12.75%</td><td>~3</td></tr><tr><td>5</td><td>80%</td><td>96%</td><td>16%</td><td>~3</td></tr><tr><td>6</td><td>75%</td><td>93.75%</td><td>18.75%</td><td>~4</td></tr><tr><td>7</td><td>70%</td><td>91%</td><td>21%</td><td>~4</td></tr><tr><td>8</td><td>65%</td><td>87.75%</td><td>22.75%</td><td>~5</td></tr><tr><td>9</td><td>60%</td><td>84%</td><td>24%</td><td>~5</td></tr><tr><td>10</td><td>55%</td><td>79.75%</td><td>24.75%</td><td>~5</td></tr><tr><td>11</td><td>50%</td><td>75%</td><td>25%</td><td>~5</td></tr><tr><td>12</td><td>45%</td><td>69.75%</td><td>24.75%</td><td>~5</td></tr><tr><td>13</td><td>40%</td><td>64%</td><td>24%</td><td>~5</td></tr><tr><td>14</td><td>35%</td><td>57.75%</td><td>22.75%</td><td>~5</td></tr><tr><td>15</td><td>30%</td><td>51%</td><td>21%</td><td>~4</td></tr><tr><td>16</td><td>25%</td><td>42.75%</td><td>18.75%</td><td>~4</td></tr><tr><td>17</td><td>20%</td><td>36%</td><td>16%</td><td>~3</td></tr><tr><td>18</td><td>15%</td><td>27.75%</td><td>12.75%</td><td>~3</td></tr><tr><td>19</td><td>10%</td><td>19%</td><td>9%</td><td>~2</td></tr><tr><td>20</td><td>5%</td><td>9.75%</td><td>4.75%</td><td>~1</td></tr></table><p></p><p></p><p>Disadvantage runs the inverse. 1 is included above for completeness above.</p><p></p><p>Now, please keep in mind that the equivalent bonus is a bad way to think about advantage/disadvantage because you can't reduce a normal distribution to a bonus to a flat distribution, but that's nerd math talk. Just know you're wrong to do it, but it's still kinda handy for a general rule of thumb.</p><p></p><p></p><p></p><p>I'm have a Bachelor's of Electrical Engineering (BSEE) from an ABET accredited institution with 10+ years experience in the field of communications technology and a particular interest in statistical analysis. Advantage/disadvantage is not a flat +/- 5, although it resembles such (if you squint and are okay being wrong) between 8 and 12. Given many rolls are in this range for bounded accuracy, it's probably why the designers chose to shorthand it as +5/-5 for passive checks. Makes it easy.</p><p></p><p>However, if the enemy needs an 18+ to hit you, they have a base 15% chance to score a hit. If you give them disadvantage, they'll have a 2.25% chance to hit you. In some circumstances, dodging is a smart move. </p><p></p><p>Besides, not every encounter is about reducing the other side's hitpoint to zero first.</p></blockquote><p></p>
[QUOTE="Ovinomancer, post: 7522730, member: 16814"] I'm an electrical engineer who's made a few posts on this topic and I cannot follow what the chart in that post is doing. It's right at the ends, but very badly wrong in the middle. Whatever it's showing isn't a 'chance to roll this number [I]or higher[/I]' which is the critical question of advantage (lower for disadvantage) and it's not chance to roll that exact number (which isn't interesting at all) so... The basic question of advantage is can you roll higher that number N on 2 d20 rolls taking only the highest. This is figured by finding the odds of rolling higher than that number on a single d20 and then adding the product of those odds times the odds that you don't roll that number on a single d20. For example, if the target number is 11, the odds of rolling at least an 11 on a d20 is 10 in 20 or 50%. This, coincidentally, is the same odds of NOT rolling at least an 11. So, you add 10/20 to the product of (10/20 x 10/20) or 10/20+100/400. A little adding of fractions and you get 300/400 or 3/4 or 75% chance of rolling at least an 11 on 2d20 take highest. The equation form of this is a bit messy, and easier done in percentages: .05(21-N)+.05(21-N)*.05(N-1) = p To put this in a chart: [TABLE="class: grid, width: 500, align: center"] [TR] [TD]Number[/TD] [TD]Base Chance (>= to N)[/TD] [TD]Advantage (>= to N)[/TD] [TD]Difference[/TD] [TD]Equivalent Bonus[/TD] [/TR] [TR] [TD]1[/TD] [TD]100%[/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]~1[/TD] [/TR] [TR] [TD]3[/TD] [TD]90%[/TD] [TD]99%[/TD] [TD]9%[/TD] [TD]~2[/TD] [/TR] [TR] [TD]4[/TD] [TD]85%[/TD] [TD]97.75%[/TD] [TD]12.75%[/TD] [TD]~3[/TD] [/TR] [TR] [TD]5[/TD] [TD]80%[/TD] [TD]96%[/TD] [TD]16%[/TD] [TD]~3[/TD] [/TR] [TR] [TD]6[/TD] [TD]75%[/TD] [TD]93.75%[/TD] [TD]18.75%[/TD] [TD]~4[/TD] [/TR] [TR] [TD]7[/TD] [TD]70%[/TD] [TD]91%[/TD] [TD]21%[/TD] [TD]~4[/TD] [/TR] [TR] [TD]8[/TD] [TD]65%[/TD] [TD]87.75%[/TD] [TD]22.75%[/TD] [TD]~5[/TD] [/TR] [TR] [TD]9[/TD] [TD]60%[/TD] [TD]84%[/TD] [TD]24%[/TD] [TD]~5[/TD] [/TR] [TR] [TD]10[/TD] [TD]55%[/TD] [TD]79.75%[/TD] [TD]24.75%[/TD] [TD]~5[/TD] [/TR] [TR] [TD]11[/TD] [TD]50%[/TD] [TD]75%[/TD] [TD]25%[/TD] [TD]~5[/TD] [/TR] [TR] [TD]12[/TD] [TD]45%[/TD] [TD]69.75%[/TD] [TD]24.75%[/TD] [TD]~5[/TD] [/TR] [TR] [TD]13[/TD] [TD]40%[/TD] [TD]64%[/TD] [TD]24%[/TD] [TD]~5[/TD] [/TR] [TR] [TD]14[/TD] [TD]35%[/TD] [TD]57.75%[/TD] [TD]22.75%[/TD] [TD]~5[/TD] [/TR] [TR] [TD]15[/TD] [TD]30%[/TD] [TD]51%[/TD] [TD]21%[/TD] [TD]~4[/TD] [/TR] [TR] [TD]16[/TD] [TD]25%[/TD] [TD]42.75%[/TD] [TD]18.75%[/TD] [TD]~4[/TD] [/TR] [TR] [TD]17[/TD] [TD]20%[/TD] [TD]36%[/TD] [TD]16%[/TD] [TD]~3[/TD] [/TR] [TR] [TD]18[/TD] [TD]15%[/TD] [TD]27.75%[/TD] [TD]12.75%[/TD] [TD]~3[/TD] [/TR] [TR] [TD]19[/TD] [TD]10%[/TD] [TD]19%[/TD] [TD]9%[/TD] [TD]~2[/TD] [/TR] [TR] [TD]20[/TD] [TD]5%[/TD] [TD]9.75%[/TD] [TD]4.75%[/TD] [TD]~1[/TD] [/TR] [/TABLE] Disadvantage runs the inverse. 1 is included above for completeness above. Now, please keep in mind that the equivalent bonus is a bad way to think about advantage/disadvantage because you can't reduce a normal distribution to a bonus to a flat distribution, but that's nerd math talk. Just know you're wrong to do it, but it's still kinda handy for a general rule of thumb. I'm have a Bachelor's of Electrical Engineering (BSEE) from an ABET accredited institution with 10+ years experience in the field of communications technology and a particular interest in statistical analysis. Advantage/disadvantage is not a flat +/- 5, although it resembles such (if you squint and are okay being wrong) between 8 and 12. Given many rolls are in this range for bounded accuracy, it's probably why the designers chose to shorthand it as +5/-5 for passive checks. Makes it easy. However, if the enemy needs an 18+ to hit you, they have a base 15% chance to score a hit. If you give them disadvantage, they'll have a 2.25% chance to hit you. In some circumstances, dodging is a smart move. Besides, not every encounter is about reducing the other side's hitpoint to zero first. [/QUOTE]
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