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<blockquote data-quote="Ainamacar" data-source="post: 5920019" data-attributes="member: 70709">Since this topic will probably come up often, it can't hurt to have pictures. I've already been beaten to the punch on tables. <img src="https://cdn.jsdelivr.net/joypixels/assets/8.0/png/unicode/64/1f642.png" class="smilie smilie--emoji" loading="lazy" width="64" height="64" alt=":)" title="Smile&nbsp; &nbsp; :)"&nbsp; data-smilie="1"data-shortname=":)" />It is easiest to start with disadvantage.&nbsp; If a character has disadvantage the only way to succeed is if both dice meet or exceed the DC.&nbsp; If the probability of success when rolling a single die is p, then both dice succeed with probability p^2.&nbsp; Therefore the probability of success with disadvantage is just p^2.In the case of advantage, at least one die must beat the DC.&nbsp; That is equivalent, however, to saying that the check only fails if both dice fail.&nbsp; Well, the probability of failure with a single die is 1-p, so the probability of failing with both dice is (1-p)^2.&nbsp; Therefore the probability of success with advantage is 1-(1-p)^2.These results are shown below.<img src="http://www.pa.msu.edu/%7Egranlund/ENWorld/advantage1.png" alt="" class="fr-fic fr-dii fr-draggable " data-size="" style="" />How does that translate into equivalent bonuses?&nbsp; It does not do so directly, because the benefit or penalty depends on the original probability.&nbsp; That is shown on the graph below, where we can clearly see that advantage and disadvantage are mirror images.<img src="http://www.pa.msu.edu/%7Egranlund/ENWorld/advantage2.png" alt="" class="fr-fic fr-dii fr-draggable " data-size="" style="" />The peaks are clearly when the probability of success on the single roll is 0.5, in which case the increase or decrease in the probability of success is exactly equal to 0.25, equivalent to +/- 5 on a d20.&nbsp; At the edges this tapers, so advantage when one needs a 20 is still useful, but increases the probability of success by just 19/400=.0475, which is just less than the benefit of a +1 on a single roll.If in play DCs were presented such that all probability of success were equally common then the average increase or decrease in probability would be 19/120=.1583bar, which is better than a +3 and less than a +4. (These values are just the mean of p^2-p and 1-(1-p)^2-p, which turn out to be the same except for the sign, which isn't surprising considering the graph above.)In actual play, of course, certain probabilities of success are more likely to occur.&nbsp; I suppose one could attempt to model this, but it will vary from table to table anyhow.&nbsp; At most tables, however, the PCs probably tend to avoid rolling on challenges they almost certainly succeed at as well as challenges where they are very certain to fail.&nbsp; Therefore, the actual bonus at the table will be somewhat higher than .1583bar, but cannot be larger than .25 (which would mean every single roll always had p=0.5).&nbsp; It's probably slightly stronger than a +/- 4 in typical play.It is worth remember some cases not covered above.&nbsp; First of all, a bonus to a d20 roll itself can change whether certain tasks are possible or impossible, and advantage/disadvantage never does this.&nbsp; Secondly, in opposed rolls this math only applies after one party has resolved their roll (effectively setting the DC for the other person).&nbsp; Thirdly, not all cases are about meeting a DC, but involve exceeding an interval and getting different outcomes based on that.&nbsp; (In earlier editions the jump check is the classic example, where one simply jumps as far as one can roll.)&nbsp; In that case the probabilities above don't capture any notion about &quot;how well&quot; or &quot;how poorly&quot; the roll succeeds.Edit:&nbsp; I've added a table anyway.&nbsp; These are the exact probabilities of success in decimal form.[code]p&nbsp; &nbsp;&nbsp; Adv&nbsp; &nbsp; &nbsp; Dis&nbsp; &nbsp;&nbsp; Difference0.&nbsp; &nbsp; 0.&nbsp; &nbsp; &nbsp;&nbsp; 0.&nbsp; &nbsp; &nbsp; +/-0.0.05&nbsp; 0.0975&nbsp;&nbsp; 0.0025&nbsp; +/-0.04750.1&nbsp;&nbsp; 0.19&nbsp; &nbsp;&nbsp; 0.01&nbsp; &nbsp; +/-0.090.15&nbsp; 0.2775&nbsp;&nbsp; 0.0225&nbsp; +/-0.12750.2&nbsp;&nbsp; 0.36&nbsp; &nbsp;&nbsp; 0.04&nbsp; &nbsp; +/-0.160.25&nbsp; 0.4375&nbsp;&nbsp; 0.0625&nbsp; +/-0.18750.3&nbsp;&nbsp; 0.51&nbsp; &nbsp;&nbsp; 0.09&nbsp; &nbsp; +/-0.210.35&nbsp; 0.5775&nbsp;&nbsp; 0.1225&nbsp; +/-0.22750.4&nbsp;&nbsp; 0.64&nbsp; &nbsp;&nbsp; 0.16&nbsp; &nbsp; +/-0.240.45&nbsp; 0.6975&nbsp;&nbsp; 0.2025&nbsp; +/-0.24750.5&nbsp;&nbsp; 0.75&nbsp; &nbsp;&nbsp; 0.25&nbsp; &nbsp; +/-0.250.55&nbsp; 0.7975&nbsp;&nbsp; 0.3025&nbsp; +/-0.24750.6&nbsp;&nbsp; 0.84&nbsp; &nbsp;&nbsp; 0.36&nbsp; &nbsp; +/-0.240.65&nbsp; 0.8775&nbsp;&nbsp; 0.4225&nbsp; +/-0.22750.7&nbsp;&nbsp; 0.91&nbsp; &nbsp;&nbsp; 0.49&nbsp; &nbsp; +/-0.210.75&nbsp; 0.9375&nbsp;&nbsp; 0.5625&nbsp; +/-0.18750.8&nbsp;&nbsp; 0.96&nbsp; &nbsp;&nbsp; 0.64&nbsp; &nbsp; +/-0.160.85&nbsp; 0.9775&nbsp;&nbsp; 0.7225&nbsp; +/-0.12750.9&nbsp;&nbsp; 0.99&nbsp; &nbsp;&nbsp; 0.81&nbsp; &nbsp; +/-0.090.95&nbsp; 0.9975&nbsp;&nbsp; 0.9025&nbsp; +/-0.04751.&nbsp; &nbsp; 1.&nbsp; &nbsp; &nbsp;&nbsp; 1.&nbsp; &nbsp; &nbsp; +/-0.[/code]</blockquote>

[QUOTE="Ainamacar, post: 5920019, member: 70709"] Since this topic will probably come up often, it can't hurt to have pictures. I've already been beaten to the punch on tables. :) It is easiest to start with disadvantage. If a character has disadvantage the only way to succeed is if both dice meet or exceed the DC. If the probability of success when rolling a single die is p, then both dice succeed with probability p^2. Therefore the probability of success with disadvantage is just p^2. In the case of advantage, at least one die must beat the DC. That is equivalent, however, to saying that the check only fails if both dice fail. Well, the probability of failure with a single die is 1-p, so the probability of failing with both dice is (1-p)^2. Therefore the probability of success with advantage is 1-(1-p)^2. These results are shown below. [IMG]http://www.pa.msu.edu/%7Egranlund/ENWorld/advantage1.png[/IMG] How does that translate into equivalent bonuses? It does not do so directly, because the benefit or penalty depends on the original probability. That is shown on the graph below, where we can clearly see that advantage and disadvantage are mirror images. [IMG]http://www.pa.msu.edu/%7Egranlund/ENWorld/advantage2.png[/IMG] The peaks are clearly when the probability of success on the single roll is 0.5, in which case the increase or decrease in the probability of success is exactly equal to 0.25, equivalent to +/- 5 on a d20. At the edges this tapers, so advantage when one needs a 20 is still useful, but increases the probability of success by just 19/400=.0475, which is just less than the benefit of a +1 on a single roll. If in play DCs were presented such that all probability of success were equally common then the average increase or decrease in probability would be 19/120=.1583bar, which is better than a +3 and less than a +4. (These values are just the mean of p^2-p and 1-(1-p)^2-p, which turn out to be the same except for the sign, which isn't surprising considering the graph above.) In actual play, of course, certain probabilities of success are more likely to occur. I suppose one could attempt to model this, but it will vary from table to table anyhow. At most tables, however, the PCs probably tend to avoid rolling on challenges they almost certainly succeed at as well as challenges where they are very certain to fail. Therefore, the actual bonus at the table will be somewhat higher than .1583bar, but cannot be larger than .25 (which would mean every single roll always had p=0.5). It's probably slightly stronger than a +/- 4 in typical play. It is worth remember some cases not covered above. First of all, a bonus to a d20 roll itself can change whether certain tasks are possible or impossible, and advantage/disadvantage never does this. Secondly, in opposed rolls this math only applies after one party has resolved their roll (effectively setting the DC for the other person). Thirdly, not all cases are about meeting a DC, but involve exceeding an interval and getting different outcomes based on that. (In earlier editions the jump check is the classic example, where one simply jumps as far as one can roll.) In that case the probabilities above don't capture any notion about "how well" or "how poorly" the roll succeeds. Edit: I've added a table anyway. These are the exact probabilities of success in decimal form. [code] p Adv Dis Difference 0. 0. 0. +/-0. 0.05 0.0975 0.0025 +/-0.0475 0.1 0.19 0.01 +/-0.09 0.15 0.2775 0.0225 +/-0.1275 0.2 0.36 0.04 +/-0.16 0.25 0.4375 0.0625 +/-0.1875 0.3 0.51 0.09 +/-0.21 0.35 0.5775 0.1225 +/-0.2275 0.4 0.64 0.16 +/-0.24 0.45 0.6975 0.2025 +/-0.2475 0.5 0.75 0.25 +/-0.25 0.55 0.7975 0.3025 +/-0.2475 0.6 0.84 0.36 +/-0.24 0.65 0.8775 0.4225 +/-0.2275 0.7 0.91 0.49 +/-0.21 0.75 0.9375 0.5625 +/-0.1875 0.8 0.96 0.64 +/-0.16 0.85 0.9775 0.7225 +/-0.1275 0.9 0.99 0.81 +/-0.09 0.95 0.9975 0.9025 +/-0.0475 1. 1. 1. +/-0. [/code] [/QUOTE]

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