Aeloric1976
First Post
This is a long text, and I'm not a native english speaker, so mistakes are bound to happen. Having said that, I wrote this with care and attention. Hope the form doesn't make the content worse.
Study on calculating odds and Mean Returns in D&D5ed.
-Attack odds
The term “attack odds” is a broad term that refers to the probability of occurrence of each of the three possible outcomes in an attack roll ("Atk") (which involves rolling one or more d20s - probabilistic event). The possible outcomes are:
The odds of each result stems from the quantity of results, in one or more d20 rolls, that produces either a 1) failure, a 2) simple success or, finally, a 3) critical success.
Let's say that for a simple success (“Hit”) you need a minimum of 10 ina d20 roll, or 10+ in a d20 rol, and to achieve critical success (“Crit”) you need a 20 in a d20 roll. In this example, the chance of a Hit is 50% (results 10 to 19 in d20, that is, 10 results in 20 possible results, that is, 50%) and the chance of critical success is 5% (result 20 in d20 , or 1 result in 20 possible outcomes, that is, 5%).
The chance of a failure (“Miss”) is equal to 1 (or 100%) minus the sum of the chance of simple success with the chance of critical success.
The minimum required value for a success is obtained in the manner explained in “The Player's Handbook” (PHB).
Summarizing the above in abbreviated terms (which will be used in this analysis):
Acronym for chance (in%) of normal success per attack: HitC/Atk
Acronym for chance (in%) of critical success per attack: CritC/Atk
Acronym for failure or miss chance per attack: MissC/Atk = 100% - (HitC/Atk + CritC/Atk)
In the example given above:
HitC/Atk = 50% (10 results - Values 10 to 19 - on 20)
CritC/Atk = 5% (1 result - value 20 - in 20)
MissC/Atk = 100% - (50% + 5%) = 45%
**Note: The CMiss of a single isolated roll of 1d20 can never be less than 5% since the result "1" in d20 always results infailure (it is called "critical failure" in the sense that it is always a failure)
- Mean Returns per attack (MR/Atk) by each outcome: success (MRHit/Atk); critical success (MRCrit/Atk); and failure (MRMiss/Atk)
As seen above, the attack odds encompasses three distinct probabilities, or outcome probabilities, namely: HitC/Atk, CritC/Atk and MissC/Atk.
HitC/Atk: odds (in %) of a simple success
CritC/Atk: odds (in %) of a critical success
MissC/Atk: odds (in %) of a failure
What results from each of these events is the so-called "damage"(“D”), which is the amount of damage points caused in each of the possible events (success, critical success, and failure).
The damage is determined, in simple terms, by the "instrument" (weapon, spell, etc.) used for the attack action plus relevant modifiers of the damage, as described in detail in the PHB.
Usually this damage is calculated by rolling one or more dice (with varying number of sides, i.e., d4, d6, d8, and so on), adding the results on the dice and then the static modifiers.
Damage, then, is composed by variable damage (“VD”) and static damage(“SD”).
Example: dagger causes 1d4, ou 1-4 VD, in a simple success event. 0 in miss event.
Inthe case of a critical success, the number of rolled dice is doubled,or the VD is doubled (no effect on SD). In the case of a miss event, damage is zero ("0").
Example: a dagger deals 2d4 VD in a critical success event. A Greatsword, on theother hand (two hands, actually, haha, lame joke) would deal 4d6 VD.
The average variable damage (“AVD”) produced in a normal hit ("HitAVD")is obtained by summing the values of all the possible damage results, dividing the sum by the number of possible damage results and, finally, adding the SD (static damage).
Thus, in a 1d4 damage, the sum of the possible values is 1 + 2 + 3 +4 = 10 while the number of possible outcomes is 4 (four-sided dice) resulting in a HitAVD, for the dagger, of 10/4 = 2,5.
In the case of a Crit, the dagger would give a CritAVD of 2 x 2.5 = 5.
If the attacker in question has a +3 SD modifier, the average damage in a (“HitD”)would be 2.5 + 3 = 5.5, and the average damage in a critical success (CritD) would be 5 + 3 = 8.
Summarizing the above in abbreviated terms (which will be used in this analysis):
Average damage on a normal or simple hit = HitD
Average damage in critical hit = CritD
Mean damage in a miss or failure = MissD (0)
HitD = HitAVD + SD
CritD = CritAVD + SD = (2 x HitAVD) + SD
MissD = 0
-Mean Returns per attack (MR / Atk)
The Mean Return for a Hit per Attack ("HitMR/Atk") (that is, for a hit with "simple or normal success", that is, success that is not a critical success) is HitC/Atk times HitD.
The MRHit/Att is expressed in damage points and represents the mean amount of damage points caused by simple or normal hit per attack (remembering that each attack event has three possible outcomes - success, critical success, and failure – MRHit/Atk only indicates the mean result, per attack, of one of these results: succes).
Summarizing the above in abbreviated terms (which will be used in this analysis):
HitMR/ Atk = (HitC / Atk) x HitD
The Mean Return of a Crit ("CritMR/Atk") equals CritC/Atk times CritD. The CritMR/Atk is expressed in damage points and represents the mean amount of damage points caused by critical hits per attack (remembering that each attack event has three possible outcomes -success, critical success, and failure - with RMCrit/Atk only indicates the mean result per attack of one of these results: critical success).
Summarizing:
CritMR/ Atk = (CritC/Atk) x CritD
The Mean Miss Returns ("MissMR/Atk") (and failure is a mean miss, indeed, hahah, lame joke again) equals MissC/Atk times MissD, which is usually zero (but weird 4E-style rules may be created in the future).
Summarizing the above in abbreviated terms (which will be used in this analysis):
MissMR/ Atk = (MissC / Att) x DMiss
The Mean Return per Attack (MR/Atk) is the sum of all previous MRs, MissMR/Atk (usually zero), HitMR/Atk and CritMR/Atk.
Summarizing the above in abbreviated terms (which will be used in this analysis):
MR/Atk = HitMR/Atk + CritMR/Atk + MissMR/Atk
Note that MR/Atk refers to only one attack event or instance (MR/Atk). The attack action MR ("MR/AtkAct"), in the case of a character with multiple attack instances per AtkAct, equals MR/Atk times the number of attacks ("#Atk" where "#" is the number) that the character can perform per AtkAct.
Summarizingthe above in abbreviated terms (which will be used in this analysis):
MR/AtkAct = (MR/Atk) x (#Atk/AtkAct)
-Influence of Combat Advantage (CAdv)
Here it is necessary to make an explanation on how to calculate attack odds (MissC/Atk, HitC/Atk and CritC/Atk) in a CAdv context, in which there are two chances of obtaining at least one sufficient result (that is, roll 2d20 and use the best result).
Let's begin calculating the odds of a Critical in a Atk with CAdv (“CritC/CAdvAtk”). First we calculate the odds of NO critical success in either of the two events (NCritC/CAdvAtk) (and rememberthat each roll of a d20 is an independent event).
The odds of two independent events happening together equals the odds of one times the other. In this case, and in abbreviated terms:
NCritC/CAdvAtk = (NCritC/Atk)²
Since NCritC/Atk = (1 -- CritC/Atk), NCritC/CAdvAtk equals (1 – CritC/Atk) x (1 – CritC/Atk).
In abreviated terms:
NCritC/CAdvAtk = (1 – CritC/Atk)²
CritC/CAdvAtk= 1 – NcritC/CAdvAtk
or
CritC/CAdvAtk= 1 – (1 – CritC/Atk)²
Example:
If the chance to get critical success is 1 in 20 (rolling 20 on d20), the odds of NOT getting a critical success (NCritC/Atk) is 19 out of 20, or 95%. The square of this chance (95% x 95%) is 90.25%. Subtracting this value from 100% gives us a CritC/CAdvAtk of 9.75%.
After obtaining the CritC/CAdvAtk, it's time to determine the HitC/CAdvAtk (odds of asimple success attacking with combat advantage).
As we did before, with the CritC/CAdvAtk, first we calculate the MissC/CAdvAtk (chance of a miss result when attacking with combat advantage).
In order to miss an Atk with CAdv, both d20 rolls must miss, that is:
MissC/CAdvAtk = (MissC/Atk)²
Once we have it, it's time to get the HitC/CAdvAtk, wich we get by subtracting, from 1, of the CritC/CAdvAtk with MissC/CAdvAtk. In abbreviated terms:
HitC/CAdvAtk = 1 – (CritC/CAdvAtk + MissC/CAdvAtk)
-Effect of feat Elven Accuracy ("ElvAcc")
The "Elven Accuracy" (“ElvAcc”) feat (described in XGtE) ammounts to rolling 3d20s, instead of 2d20s, when you attack with combat advantage.
This alters the probabilities in a manner similar to the change produced by CAdv but instead of squaring the probabilities of non-critical occurrence or of a Miss, you take it to the third power. Changing the relevant equations, we have:
CritC/ElvAccAtk= 1 – (1 – CritC/Atk)³
HitC/ElvAccAtk = 1 – (CritC/Atk + MissC/Atk)³
-Effects of the Great Weapon Fighter feat Bonus Action Attack ("BActAtk")
Several game elements give the option of performing a Bonus Action Attack (“BActAtk”)
TheGreat Weapon Fighter feat ("GWF"), described in the PHB, grants such an option ("GWFBActAtk") when a Crit is scored in any of the AtkAct attacks, when the AtkAct attacks are performed using a melee weapon.
This option, if taken, produces an increase in the Mean Returns per Turn (“MR/T”) (see more on that at the end of this post) that equals the chance of at least one critical success in any attack made on an attack action (“CritC/AtkAct”) times the MR/Atk.
In abbreviated terms:
MR/GWFBActAtk= (CritC/AtkAct) x (MR/Atk)
CritC/AtkAct equals the chance of rolling at least one Crit on any of the attacks made during an AtkAct.
Thus,the CritC/AtkAct equals 1 – (NCrit/Atk) elevated to a power equal to the number of d20s rolled during an AtkAct (“#d20s/AtkAct”)(since all is needed is one Crit to open up the BActAtk option granted by the GWF feat).
#d20s/AtkAtc is obtained by adding the numberof d20s rolled in each attack instances performed in an AtkAct, that is, 1 for an Atk, 2 for a CadvAtk and 3 for a ElvAccAtk.
In abbreviated terms:
CritC/AtkAct = 1 – (NCritC/Atk) ^ (#d20s/AtkAtc)
#d20s/AtkAtc = (1 x #Atk) + (2 x #CAdvAtk) + (3 x #ElvAccAtk)
-Effect of the Power Attack (PAtk)
TheGWF feat opens up the option of making an attack with – 5 penalty (– 25% in HitC/Atk), but with an increase of +10 to the SD (static damage). In abbreviated terms:
HitC/PAtk = (HitC/PAtk – 25%)
Its important to notice that HitC/Atk can't be lowered below 0% and PAtk has no effect on CritC/Atk (it does, of course, inscreases by 10 the SD of the CritD).
– MeanResults per Turn
TheMeans Results per Turn (MR/T), after all that was shown above, iscomposed by the Mean Results of the Attack Action attacks plus the Mean Results of the Bonus Action Attacks (always one, I think). In abbreviated terms:
MR/T= MR/AtkAct + MR/BActAtk.
That,of course, is just a partial analysis of the elements that influencethe MR/T. Various others can be taken into account as well, but these are enough to get the ball rolling.
So, what do you guys think?
Study on calculating odds and Mean Returns in D&D5ed.
-Attack odds
The term “attack odds” is a broad term that refers to the probability of occurrence of each of the three possible outcomes in an attack roll ("Atk") (which involves rolling one or more d20s - probabilistic event). The possible outcomes are:
- failure or miss;
- simple success; and
- critical success.
The odds of each result stems from the quantity of results, in one or more d20 rolls, that produces either a 1) failure, a 2) simple success or, finally, a 3) critical success.
Let's say that for a simple success (“Hit”) you need a minimum of 10 ina d20 roll, or 10+ in a d20 rol, and to achieve critical success (“Crit”) you need a 20 in a d20 roll. In this example, the chance of a Hit is 50% (results 10 to 19 in d20, that is, 10 results in 20 possible results, that is, 50%) and the chance of critical success is 5% (result 20 in d20 , or 1 result in 20 possible outcomes, that is, 5%).
The chance of a failure (“Miss”) is equal to 1 (or 100%) minus the sum of the chance of simple success with the chance of critical success.
The minimum required value for a success is obtained in the manner explained in “The Player's Handbook” (PHB).
Summarizing the above in abbreviated terms (which will be used in this analysis):
Acronym for chance (in%) of normal success per attack: HitC/Atk
Acronym for chance (in%) of critical success per attack: CritC/Atk
Acronym for failure or miss chance per attack: MissC/Atk = 100% - (HitC/Atk + CritC/Atk)
In the example given above:
HitC/Atk = 50% (10 results - Values 10 to 19 - on 20)
CritC/Atk = 5% (1 result - value 20 - in 20)
MissC/Atk = 100% - (50% + 5%) = 45%
**Note: The CMiss of a single isolated roll of 1d20 can never be less than 5% since the result "1" in d20 always results infailure (it is called "critical failure" in the sense that it is always a failure)
- Mean Returns per attack (MR/Atk) by each outcome: success (MRHit/Atk); critical success (MRCrit/Atk); and failure (MRMiss/Atk)
As seen above, the attack odds encompasses three distinct probabilities, or outcome probabilities, namely: HitC/Atk, CritC/Atk and MissC/Atk.
HitC/Atk: odds (in %) of a simple success
CritC/Atk: odds (in %) of a critical success
MissC/Atk: odds (in %) of a failure
What results from each of these events is the so-called "damage"(“D”), which is the amount of damage points caused in each of the possible events (success, critical success, and failure).
The damage is determined, in simple terms, by the "instrument" (weapon, spell, etc.) used for the attack action plus relevant modifiers of the damage, as described in detail in the PHB.
Usually this damage is calculated by rolling one or more dice (with varying number of sides, i.e., d4, d6, d8, and so on), adding the results on the dice and then the static modifiers.
Damage, then, is composed by variable damage (“VD”) and static damage(“SD”).
Example: dagger causes 1d4, ou 1-4 VD, in a simple success event. 0 in miss event.
Inthe case of a critical success, the number of rolled dice is doubled,or the VD is doubled (no effect on SD). In the case of a miss event, damage is zero ("0").
Example: a dagger deals 2d4 VD in a critical success event. A Greatsword, on theother hand (two hands, actually, haha, lame joke) would deal 4d6 VD.
The average variable damage (“AVD”) produced in a normal hit ("HitAVD")is obtained by summing the values of all the possible damage results, dividing the sum by the number of possible damage results and, finally, adding the SD (static damage).
Thus, in a 1d4 damage, the sum of the possible values is 1 + 2 + 3 +4 = 10 while the number of possible outcomes is 4 (four-sided dice) resulting in a HitAVD, for the dagger, of 10/4 = 2,5.
In the case of a Crit, the dagger would give a CritAVD of 2 x 2.5 = 5.
If the attacker in question has a +3 SD modifier, the average damage in a (“HitD”)would be 2.5 + 3 = 5.5, and the average damage in a critical success (CritD) would be 5 + 3 = 8.
Summarizing the above in abbreviated terms (which will be used in this analysis):
Average damage on a normal or simple hit = HitD
Average damage in critical hit = CritD
Mean damage in a miss or failure = MissD (0)
HitD = HitAVD + SD
CritD = CritAVD + SD = (2 x HitAVD) + SD
MissD = 0
-Mean Returns per attack (MR / Atk)
The Mean Return for a Hit per Attack ("HitMR/Atk") (that is, for a hit with "simple or normal success", that is, success that is not a critical success) is HitC/Atk times HitD.
The MRHit/Att is expressed in damage points and represents the mean amount of damage points caused by simple or normal hit per attack (remembering that each attack event has three possible outcomes - success, critical success, and failure – MRHit/Atk only indicates the mean result, per attack, of one of these results: succes).
Summarizing the above in abbreviated terms (which will be used in this analysis):
HitMR/ Atk = (HitC / Atk) x HitD
The Mean Return of a Crit ("CritMR/Atk") equals CritC/Atk times CritD. The CritMR/Atk is expressed in damage points and represents the mean amount of damage points caused by critical hits per attack (remembering that each attack event has three possible outcomes -success, critical success, and failure - with RMCrit/Atk only indicates the mean result per attack of one of these results: critical success).
Summarizing:
CritMR/ Atk = (CritC/Atk) x CritD
The Mean Miss Returns ("MissMR/Atk") (and failure is a mean miss, indeed, hahah, lame joke again) equals MissC/Atk times MissD, which is usually zero (but weird 4E-style rules may be created in the future).
Summarizing the above in abbreviated terms (which will be used in this analysis):
MissMR/ Atk = (MissC / Att) x DMiss
The Mean Return per Attack (MR/Atk) is the sum of all previous MRs, MissMR/Atk (usually zero), HitMR/Atk and CritMR/Atk.
Summarizing the above in abbreviated terms (which will be used in this analysis):
MR/Atk = HitMR/Atk + CritMR/Atk + MissMR/Atk
Note that MR/Atk refers to only one attack event or instance (MR/Atk). The attack action MR ("MR/AtkAct"), in the case of a character with multiple attack instances per AtkAct, equals MR/Atk times the number of attacks ("#Atk" where "#" is the number) that the character can perform per AtkAct.
Summarizingthe above in abbreviated terms (which will be used in this analysis):
MR/AtkAct = (MR/Atk) x (#Atk/AtkAct)
-Influence of Combat Advantage (CAdv)
Here it is necessary to make an explanation on how to calculate attack odds (MissC/Atk, HitC/Atk and CritC/Atk) in a CAdv context, in which there are two chances of obtaining at least one sufficient result (that is, roll 2d20 and use the best result).
Let's begin calculating the odds of a Critical in a Atk with CAdv (“CritC/CAdvAtk”). First we calculate the odds of NO critical success in either of the two events (NCritC/CAdvAtk) (and rememberthat each roll of a d20 is an independent event).
The odds of two independent events happening together equals the odds of one times the other. In this case, and in abbreviated terms:
NCritC/CAdvAtk = (NCritC/Atk)²
Since NCritC/Atk = (1 -- CritC/Atk), NCritC/CAdvAtk equals (1 – CritC/Atk) x (1 – CritC/Atk).
In abreviated terms:
NCritC/CAdvAtk = (1 – CritC/Atk)²
CritC/CAdvAtk= 1 – NcritC/CAdvAtk
or
CritC/CAdvAtk= 1 – (1 – CritC/Atk)²
Example:
If the chance to get critical success is 1 in 20 (rolling 20 on d20), the odds of NOT getting a critical success (NCritC/Atk) is 19 out of 20, or 95%. The square of this chance (95% x 95%) is 90.25%. Subtracting this value from 100% gives us a CritC/CAdvAtk of 9.75%.
After obtaining the CritC/CAdvAtk, it's time to determine the HitC/CAdvAtk (odds of asimple success attacking with combat advantage).
As we did before, with the CritC/CAdvAtk, first we calculate the MissC/CAdvAtk (chance of a miss result when attacking with combat advantage).
In order to miss an Atk with CAdv, both d20 rolls must miss, that is:
MissC/CAdvAtk = (MissC/Atk)²
Once we have it, it's time to get the HitC/CAdvAtk, wich we get by subtracting, from 1, of the CritC/CAdvAtk with MissC/CAdvAtk. In abbreviated terms:
HitC/CAdvAtk = 1 – (CritC/CAdvAtk + MissC/CAdvAtk)
-Effect of feat Elven Accuracy ("ElvAcc")
The "Elven Accuracy" (“ElvAcc”) feat (described in XGtE) ammounts to rolling 3d20s, instead of 2d20s, when you attack with combat advantage.
This alters the probabilities in a manner similar to the change produced by CAdv but instead of squaring the probabilities of non-critical occurrence or of a Miss, you take it to the third power. Changing the relevant equations, we have:
CritC/ElvAccAtk= 1 – (1 – CritC/Atk)³
HitC/ElvAccAtk = 1 – (CritC/Atk + MissC/Atk)³
-Effects of the Great Weapon Fighter feat Bonus Action Attack ("BActAtk")
Several game elements give the option of performing a Bonus Action Attack (“BActAtk”)
TheGreat Weapon Fighter feat ("GWF"), described in the PHB, grants such an option ("GWFBActAtk") when a Crit is scored in any of the AtkAct attacks, when the AtkAct attacks are performed using a melee weapon.
This option, if taken, produces an increase in the Mean Returns per Turn (“MR/T”) (see more on that at the end of this post) that equals the chance of at least one critical success in any attack made on an attack action (“CritC/AtkAct”) times the MR/Atk.
In abbreviated terms:
MR/GWFBActAtk= (CritC/AtkAct) x (MR/Atk)
CritC/AtkAct equals the chance of rolling at least one Crit on any of the attacks made during an AtkAct.
Thus,the CritC/AtkAct equals 1 – (NCrit/Atk) elevated to a power equal to the number of d20s rolled during an AtkAct (“#d20s/AtkAct”)(since all is needed is one Crit to open up the BActAtk option granted by the GWF feat).
#d20s/AtkAtc is obtained by adding the numberof d20s rolled in each attack instances performed in an AtkAct, that is, 1 for an Atk, 2 for a CadvAtk and 3 for a ElvAccAtk.
In abbreviated terms:
CritC/AtkAct = 1 – (NCritC/Atk) ^ (#d20s/AtkAtc)
#d20s/AtkAtc = (1 x #Atk) + (2 x #CAdvAtk) + (3 x #ElvAccAtk)
-Effect of the Power Attack (PAtk)
TheGWF feat opens up the option of making an attack with – 5 penalty (– 25% in HitC/Atk), but with an increase of +10 to the SD (static damage). In abbreviated terms:
HitC/PAtk = (HitC/PAtk – 25%)
Its important to notice that HitC/Atk can't be lowered below 0% and PAtk has no effect on CritC/Atk (it does, of course, inscreases by 10 the SD of the CritD).
– MeanResults per Turn
TheMeans Results per Turn (MR/T), after all that was shown above, iscomposed by the Mean Results of the Attack Action attacks plus the Mean Results of the Bonus Action Attacks (always one, I think). In abbreviated terms:
MR/T= MR/AtkAct + MR/BActAtk.
That,of course, is just a partial analysis of the elements that influencethe MR/T. Various others can be taken into account as well, but these are enough to get the ball rolling.
So, what do you guys think?
The only other reason I can think of for doing this much math is just out of curiosity. That's cool. But as @Harsel said above...what was you goal?
