Advantage: Difference between revisions

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(→‎Math: Added subsection on Critical successes and failures (Natural 1s and 20s). Placed it under the Math section, but since it's less in-depth than other secitons, feel free to move it if appropriate.)
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==Math==
==Math==
=== Chances of succeeding a specific roll===
=== Chances of succeeding a specific roll===
The benefits of rolling with advantage (or the detriments of rolling with disadvantage) change depending on the target number you need on the 1d20 roll to succeed. The bonus from advantage can be as large as 24-25% when needing a 9, 10, 11, 12, or 13  on the 1d20 roll, and as small as 9% if one needs to roll a 19.
The benefits of rolling with advantage (or the detriments of rolling with disadvantage) change depending on the target number you need on the 1d20 roll to succeed. The bonus from advantage can be as large as 24-25% when needing a 9, 10, 11, 12, or 13  on the 1d20 roll, and as small as 9% if one needs to roll a 19.
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===Effects on the average of dice rolls===
=== Effects on the average of dice rolls ===
A more general way of looking at advantage/disadvantage is calculating the effect on the average of dice rolls. This makes it more broadly applicable than looking at specific rolls and makes it easier to compare to other bonuses and penalties which may apply to a roll.
A more general way of looking at advantage/disadvantage is calculating the effect on the average of dice rolls. This makes it more broadly applicable than looking at specific rolls and makes it easier to compare to other bonuses and penalties which may apply to a roll.


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To know what bonus having advantage gives to our roll, we calculate
To know what bonus having advantage gives to our roll, we calculate
{{math_block|1= \mathbb{E}[\text{D}n \text{ with advantage}] - \mathbb{E}[\text{D}n] = \frac{2n}{3} + \frac{1}{2} - \frac{1}{6n} - \frac{n + 1}{2} = \frac{1}{6}\left(n - \frac{1}{n}\right) }}
{{math_block|1= \mathbb{E}[\text{D}n \text{ with advantage}] - \mathbb{E}[\text{D}n] = \frac{2n}{3} + \frac{1}{2} - \frac{1}{6n} - \frac{n + 1}{2} = \frac{1}{6}\left(n - \frac{1}{n}\right) }}
When we apply this elegant expression to a D20 we get that '''having advantage is equivalent to an average bonus of +3.325'''.
When we apply this expression to a d20 we get that having advantage is equivalent to an average bonus of +3.325.


Because of symmetry, having disadvantage instead of advantage means we can simply make the permutation of {{math|\{1, \dots, n\} \to \{n, \dots, 1\} }} for the values of dice rolls and all the calculations will remain the same. Therefore the size of the bonus of advantage is equal to the size of the penalty of disadvantage.
Because of symmetry, having disadvantage instead of advantage means we can simply make the permutation of {{math|\{1, \dots, n\} \to \{n, \dots, 1\} }} for the values of dice rolls and all the calculations will remain the same. Therefore the size of the bonus of advantage is equal to the size of the penalty of disadvantage.


===Effects on Critical Successes and Failures===
=== Effects on critical successes and failures ===
In the mechanics of Baldur's Gate 3, rolling a 1 or a 20 for an {{Ability check}}, {{Attack roll}}, or {{Saving throw}} will ''always'' be treated as a Critical Failure or Critical Success (respectively), regardless of any other bonuses or penalties on the roll. On a normal dice roll, this effectively means there is a {{math|1/20}} (or 5%) chance of either a Critical Success or Failure.
When making an ability check, attack roll or saving throw, a 1 or a 20 will {{em|always}} be treated as a critical failure or success, respectively, regardless of the results after any potential modifiers are added. On a dice roll without advantage or disadvantage, this effectively means there is a {{math|1/20}} (or 5%) chance of either a critical success or failure.


Having Advantage or Disadvantage can drastically increase or reduce the chance of Critical Successes and Failures. For example, when rolling with Advantage, the only way to get a Critical Failure is to roll ''two'' 1s at the same time. The odds of this result is {{math|1=1/20 \cdot 1/20 = 1/400}} (or 0.25%). Conversely, rolling a Critical Success is far more likely - out of the 400 possible dice roll outcomes,  39 will result in a 20 (rolling 20 on the first die and 1, 2, 3, ... 20 on the second die, plus rolling 20 on the second die and 1, 2, 3, ... 20 on the first die, minus one so that the result of two 20s is not doubly-counted). The odds of this result is {{math|39/400}} (or 9.75%). The opposite is true for rolling with Disadvantage: a Critical Success has a 0.25% chance and a Critical Failure has a 9.75% chance.
Having advantage or disadvantage can drastically increase or reduce the chance of critical successes and Failures. For example, when rolling with advantage, the only way to get a Critical Failure is to roll {{em|two}} 1s at the same time. The odds of this result is {{math|1=1/20 \cdot 1/20 = 1/400}} (or 0.25%). Conversely, rolling a Critical Success is far more likely - out of the 400 possible dice roll outcomes,  39 will result in a 20 (rolling 20 on the first die and 1, 2, 3, ... 20 on the second die, plus rolling 20 on the second die and 1, 2, 3, ... 20 on the first die, minus one so that the result of two 20s is not doubly-counted). The odds of this result is {{math|39/400}} (or 9.75%). The opposite is true for rolling with Disadvantage: a Critical Success has a 0.25% chance and a Critical Failure has a 9.75% chance.


Effectively, rolling with '''Advantage means that Critical Failures are 20 times less likely and Critical Successes are almost twice as likely''', and the inverse is true for Disadvantage.
Effectively, rolling with advantage means that critical failures are 20 times {{em|less}} likely and critical successes are almost {{em|twice}} as likely, and the inverse is true for disadvantage.


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