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General Tabletop Discussion
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The math of Advantage and Disadvantage
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<blockquote data-quote="Ovinomancer" data-source="post: 7523628" data-attributes="member: 16814"><p>Okay, I'm not beating on you, but this is... wrong. Understandably so, but wrong. Soapbox time!</p><p></p><p>Converting the difference in odds that occurs when using ad/disadvantage into a flat bonus leads to erroneous thinking. It's (for nerdy math reasons) just wrong to do this. It is, however, sometimes useful as a rough model to gauge effect. Like all models, though, it's still wrong, and will lead you into the thinking above where you assume the translation to the largest flat bonus represents the actual effect. It doesn't. Here's a good example of this:</p><p></p><p>Say you need to roll a 20 to succeed at a task. You have a 1/20 chance of success (or 5%). Cool. Now, let's say you have disadvantage on this roll. If you go with the relative flat bonus, the assumption is that at 20 the effect is very nearly a -1. But, if you need to roll a 20 to succeed and now have an additional -1 to that check added, then you cannot ever succeed. This, however, isn't true of actual disadvantage. Your chance of success just become much, much less.</p><p></p><p>And therein lies the point the makes your statement incorrect. Yes, the largest divergence occurs at 11, but not really the largest <em>effect</em>. Example, again. If you need an 11 to hit, you have a 10/20 chance of hitting. Suffer disadvantage, and that drops to 5/20. Your chance is halved. But, if you need a 20 to hit, that's a 1/20 chance. With disadvantage, your chance is 1/400, or <em>twenty times less.</em> The largest effect is actually at the ends, where it appears that the bonus translation is smallest. But, again, the bonus translation says that you can't hit a 20, ever, which means the impact of disadvantage is infinitely larger than the smaller success chance at 11.</p><p></p><p>To show this another way, let's assume you're attacked 5 times. Firstly, a case where your foe needs an 11 to hit you and secondly where your foe needs an 18 to hit you.</p><p></p><p>Case 1 (11 needed): The base chance that you are hit at least once in five attacks is 1 - p(all attacks miss). That's 1 - (10/20)^5 or 96.9%. If the enemy has disadvantage, the chance to be missed increases from 10/20 to 15/20, so disadvantage changes the chance you're hit at least once to 1-(15/20)^5 or 76.3%. That's a big change. But, let's look at case 2.</p><p></p><p>Case 2 (18 needed): Again, the base chance is 1-p(all attacks miss). For 18, that's 1-(18/20)^5 or 41.0% chance to be hit at least once. With disadvantage, though, that becomes 1-((1-3/20)^2)^5 or 10.8%. That's a Case 1 chance of 20%, but a case 2 change of 30% in the odds of disadvantage causing all 5 attacks to miss.</p><p></p><p>Again, converting the probability change due to ad/disadvantage is often a good rough gauge, but it's wrong, and relying on that shift (or even just the raw difference in probability) will lead you to believe that the biggest numbers mean the most impact. That's not exactly true, you just have to go a step further in exploring the impact to see it.</p></blockquote><p></p>
[QUOTE="Ovinomancer, post: 7523628, member: 16814"] Okay, I'm not beating on you, but this is... wrong. Understandably so, but wrong. Soapbox time! Converting the difference in odds that occurs when using ad/disadvantage into a flat bonus leads to erroneous thinking. It's (for nerdy math reasons) just wrong to do this. It is, however, sometimes useful as a rough model to gauge effect. Like all models, though, it's still wrong, and will lead you into the thinking above where you assume the translation to the largest flat bonus represents the actual effect. It doesn't. Here's a good example of this: Say you need to roll a 20 to succeed at a task. You have a 1/20 chance of success (or 5%). Cool. Now, let's say you have disadvantage on this roll. If you go with the relative flat bonus, the assumption is that at 20 the effect is very nearly a -1. But, if you need to roll a 20 to succeed and now have an additional -1 to that check added, then you cannot ever succeed. This, however, isn't true of actual disadvantage. Your chance of success just become much, much less. And therein lies the point the makes your statement incorrect. Yes, the largest divergence occurs at 11, but not really the largest [I]effect[/I]. Example, again. If you need an 11 to hit, you have a 10/20 chance of hitting. Suffer disadvantage, and that drops to 5/20. Your chance is halved. But, if you need a 20 to hit, that's a 1/20 chance. With disadvantage, your chance is 1/400, or [I]twenty times less.[/I] The largest effect is actually at the ends, where it appears that the bonus translation is smallest. But, again, the bonus translation says that you can't hit a 20, ever, which means the impact of disadvantage is infinitely larger than the smaller success chance at 11. To show this another way, let's assume you're attacked 5 times. Firstly, a case where your foe needs an 11 to hit you and secondly where your foe needs an 18 to hit you. Case 1 (11 needed): The base chance that you are hit at least once in five attacks is 1 - p(all attacks miss). That's 1 - (10/20)^5 or 96.9%. If the enemy has disadvantage, the chance to be missed increases from 10/20 to 15/20, so disadvantage changes the chance you're hit at least once to 1-(15/20)^5 or 76.3%. That's a big change. But, let's look at case 2. Case 2 (18 needed): Again, the base chance is 1-p(all attacks miss). For 18, that's 1-(18/20)^5 or 41.0% chance to be hit at least once. With disadvantage, though, that becomes 1-((1-3/20)^2)^5 or 10.8%. That's a Case 1 chance of 20%, but a case 2 change of 30% in the odds of disadvantage causing all 5 attacks to miss. Again, converting the probability change due to ad/disadvantage is often a good rough gauge, but it's wrong, and relying on that shift (or even just the raw difference in probability) will lead you to believe that the biggest numbers mean the most impact. That's not exactly true, you just have to go a step further in exploring the impact to see it. [/QUOTE]
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