This is an analysis of the 4E DMG2 skill challenges as written.

These are the complexity cases listed on page 80.

Complexity 1 - 4 successes before 3 failures.

Complexity 2 - 6 successes before 3 failures.

Complexity 3 - 8 successes before 3 failures.

Complexity 4 - 10 successes before 3 failures.

Complexity 5 - 12 successes before 3 failures.

Basically it is a "three strikes and you're out" system.

Without going through the technical details, the probability of passing a skill challenge requiring "n successes before 3 failures" is:

{1 + n(1-p) + [(n+1)n/2] (1-p)^2} * p^n

where (for simplicity), we assume each individual roll for a skill check has an probability p of success.

Let's calculate the cases of a level 1 skill challenge with a "hard" DC of 15 (page 80 of 4E DMG2), where p is 0.75 to 0.85 (ie. each player is using their trained skills with their primary stat, and assisting one another).

Complexity 1 - prob of passing skill challenge is 0.83 to 0.95

Complexity 2 - prob of passing skill challenge is 0.68 to 0.89

Complexity 3 - prob of passing skill challenge is 0.53 to 0.82

Complexity 4 - prob of passing skill challenge is 0.39 to 0.74

(Exercise for the reader: show that for a skill challenge with "n successes before 4 failures", the probability of passing is

{1 + n(1-p) + [(n+1)n/2] (1-p)^2 + [(n+2)(n+1)n/6] (1-p)^3} * p^n

where p is defined as before).

## Friday, February 17, 2012

### 4E DMG1 Skill Challenges.

Details of probability of a succeeding a skill challenge from page 72 of the 4E DMG1 as written.

(For simplicity, assume each individual roll for a skill check, has an probability p of success).

- "complexity 1" (4 success before 2 failures)

prob of succeeding skill challenge = 5 p^4 - 4 p^5

- "complexity 2" (6 success before 3 failures)

prob of succeeding skill challenge = 21 p^8 - 48 p^7 + 28 p^6

(An exercise for the reader with lots of patience: calculate the higher "complexity" case).

Let's look at an example for a level 1 group of players. On page 42 of the 4E DMG1, the DC's for skill check rolls are:

10 - easy

15 - moderate

20 - hard

In the case where the players are all using their trained skills (+5 bonus) based on their primary stat (+4 or +5 bonus), with assistance from another player (+2 bonus), the bonus to the d20 skill check roll is already +10 to +12.

So easy skill checks (DC 10) are a done deal, while the probability of success for moderate skill checks (DC 15) are p = 0.75 to 0.85

For a level 1 "complexity 1" moderate (DC 15) skill challenge, the probability of succeeding the skill challenge is 0.63 to 0.84, where each player is using their trained skills with their primary stat, and assisting one another. (ie. +10 to +12 added to each skill roll).

For a level 1 "complexity 2" moderate (DC 15) skill challenge, the probability of succeeding the skill challenge is 0.68 to 0.89, where each player is using their trained skills with their primary stat, and assisting one another. (ie. +10 to +12 added to each skill roll).

(For simplicity, assume each individual roll for a skill check, has an probability p of success).

- "complexity 1" (4 success before 2 failures)

prob of succeeding skill challenge = 5 p^4 - 4 p^5

- "complexity 2" (6 success before 3 failures)

prob of succeeding skill challenge = 21 p^8 - 48 p^7 + 28 p^6

(An exercise for the reader with lots of patience: calculate the higher "complexity" case).

Let's look at an example for a level 1 group of players. On page 42 of the 4E DMG1, the DC's for skill check rolls are:

10 - easy

15 - moderate

20 - hard

In the case where the players are all using their trained skills (+5 bonus) based on their primary stat (+4 or +5 bonus), with assistance from another player (+2 bonus), the bonus to the d20 skill check roll is already +10 to +12.

So easy skill checks (DC 10) are a done deal, while the probability of success for moderate skill checks (DC 15) are p = 0.75 to 0.85

For a level 1 "complexity 1" moderate (DC 15) skill challenge, the probability of succeeding the skill challenge is 0.63 to 0.84, where each player is using their trained skills with their primary stat, and assisting one another. (ie. +10 to +12 added to each skill roll).

For a level 1 "complexity 2" moderate (DC 15) skill challenge, the probability of succeeding the skill challenge is 0.68 to 0.89, where each player is using their trained skills with their primary stat, and assisting one another. (ie. +10 to +12 added to each skill roll).

### 4E skill challenges analysis articles.

We'll be reposting the series of articles on an analysis of 4E skill challenges, from therpgsite.

## Tuesday, September 14, 2010

### Scaling of 4E Essentials fighter at-will powers.

Let's analyze the scaling behavior of the average number of hits it takes for a 4E Essentials fighter (both knight and slayer) to kill a monster, with some very egregious extrapolations. (We will ignore "Power Strike" in this analysis).

Recall that the average number of hits to kill a monster (N) is the ratio:

N = (monster hit points)/(average amount of damage per successful attack).

The damage done by at-will powers is typically:

[W] + stat mod + magic enhancement + misc.

Stat increases happen at levels 4 and 8, where +1 is added to two stats of choice. We assume one of the stats goes into the primary stat (STR), and the other goes into the secondary stat (DEX).

At the paragon and epic tiers, the stat mod increases happen at levels 11, 14, 18 (paragon) and levels 21, 24, 28 (epic). At levels 14, 18, 24, 28, the stat mod increases are similar to the ones at levels 4 and 8. At levels 11 and 21, the stat mod increases are +1 to every stat.

The magic enhancement for different levels assumes the table:

1 -5 -> +1

6 - 10 -> +2

11 - 15 -> +3

16 - 20 -> +4

21 - 25 -> +5

26 - 30 -> +6

(ie. Magic enhancement increases by +1 every five levels).

To make things simple, we will examine the levels 11-30 of paragon and epic tiers as one entity. Over levels 11-30, the total additional damage contributed by the stat increases and magic enhancement is +7, due to +3 from stat increases and +4 from magic enhancement, by the time one reaches level 30. (Heroic tier by level 10, typically already has a +6 to +8 contributed to the damage, where: +3 to +5 is from the stat mod, +1 from the two stat increases, and +2 from magic enhancement).

One egregious assumption we will make, is that this pattern of stat increases and magic enhancement remains the same every 20 levels as one goes to higher levels beyond level 30. For example, stat increases at levels 31, 34, 38, 41, 44, 48, etc ... and magic enhancement increases of +1 every five levels.

So above level 10, the increase to damage from stat increases and magic enhancement scales approximately as: 7*(level-10)/20

Another egregious assumption we will make, is that the damage of at-will weapon attacks increase as follows.

(extrapolating the heroic, paragon, epic "mighty slayer" class ability)

level 5 --> 2 + dex mod damage

level 15 --> 5 + dex mod damage

level 25 --> 8 + dex mod damage

level 35 ---> 11 + dex mod damage

level 45 ---> 14 + dex mod damage

etc ...

(extrapolating the "battle wrath" stance)

level 1 --> +2 damage

level 11 --> +3 damage

level 21 --> +4 damage

level 31 --> +5 damage

level 41 --> +6 damage

etc ...

So for a Slayer always using the "battle wrath" stance, the average damage scales approximately as (above level 10):

avg[W] + level*14/27 + 1 + D + 7*(level-10)/20

where D is the dex mod at level 1.

(extrapolating the heroic, paragon, epic "weapon mastery" class ability)

level 5 --> +1 damage

level 15 --> +2 damage

level 25 --> +3 damage

level 35 ---> +4 damage

level 45 ---> +5 damage

etc ...

(extrapolating the "battle wrath" stance)

level 1 --> +2 damage

level 11 --> +3 damage

level 21 --> +4 damage

level 31 --> +5 damage

level 41 --> +6 damage

etc ...

So for a Knight always using the "battle wrath" stance, the average damage scales approximately as (above level 10):

avg[W] + level*5/24 + 2 + 7*(level-10)/20

On the monster side, the hit points of various monsters from page 184 of the 4E DMG1 are (assuming this is the same in the 4E Essentials DM Kit box set):

ROLE*(level +1) + CON

where ROLE is:

Artillery, Lurker --> ROLE = 6

Skirmisher, Soldier, Controller --> ROLE = 8

Brute --> ROLE = 10

(Elites double the hit points, while Solos quadruple the hit points).

Now the average number of hits to kill a monster being attacked by a player (of the same level as the monster) is approximately the ratio (for a high level):

N = [ROLE*(level+1) + CON]/[avg[W] + level*14/27 + 1 + D + 7*(level-10)/20]

N = [ROLE*(level+1) + CON]/[avg[W] + level*5/24 + 2 + 7*(level-10)/20]

Taking the level to infinity, the ratios approach the limits of:

N -> ROLE/[14/27 + 7/20] = 1.15*ROLE

N -> ROLE/[5/24 + 7/20] = 1.79*ROLE

For a skirmisher monster (ROLE=8), on average it will take a Slayer around 9 hits and a Knight around 14 hits (without "power strike") to kill the monster as the level goes to infinity.

Let's compare these results to the older Heinsoo 4E classes.

It turns out the 4E Essentials Slayer using "battle wrath" without using "Power Strike", is approximately equivalent damage-wise to a non-striker Heinsoo 4E class using a weapon with damage dice [W]=d10. The 4E Essentials Knight using "battle wrath" without using "Power Strike", is approximately equivalent damage-wise to a non-striker Heinsoo 4E class using a weapon with damage dice [W]=d4.

Recall that the average number of hits to kill a monster (N) is the ratio:

N = (monster hit points)/(average amount of damage per successful attack).

The damage done by at-will powers is typically:

[W] + stat mod + magic enhancement + misc.

Stat increases happen at levels 4 and 8, where +1 is added to two stats of choice. We assume one of the stats goes into the primary stat (STR), and the other goes into the secondary stat (DEX).

At the paragon and epic tiers, the stat mod increases happen at levels 11, 14, 18 (paragon) and levels 21, 24, 28 (epic). At levels 14, 18, 24, 28, the stat mod increases are similar to the ones at levels 4 and 8. At levels 11 and 21, the stat mod increases are +1 to every stat.

The magic enhancement for different levels assumes the table:

1 -5 -> +1

6 - 10 -> +2

11 - 15 -> +3

16 - 20 -> +4

21 - 25 -> +5

26 - 30 -> +6

(ie. Magic enhancement increases by +1 every five levels).

To make things simple, we will examine the levels 11-30 of paragon and epic tiers as one entity. Over levels 11-30, the total additional damage contributed by the stat increases and magic enhancement is +7, due to +3 from stat increases and +4 from magic enhancement, by the time one reaches level 30. (Heroic tier by level 10, typically already has a +6 to +8 contributed to the damage, where: +3 to +5 is from the stat mod, +1 from the two stat increases, and +2 from magic enhancement).

One egregious assumption we will make, is that this pattern of stat increases and magic enhancement remains the same every 20 levels as one goes to higher levels beyond level 30. For example, stat increases at levels 31, 34, 38, 41, 44, 48, etc ... and magic enhancement increases of +1 every five levels.

So above level 10, the increase to damage from stat increases and magic enhancement scales approximately as: 7*(level-10)/20

Another egregious assumption we will make, is that the damage of at-will weapon attacks increase as follows.

__Slayer__(extrapolating the heroic, paragon, epic "mighty slayer" class ability)

level 5 --> 2 + dex mod damage

level 15 --> 5 + dex mod damage

level 25 --> 8 + dex mod damage

level 35 ---> 11 + dex mod damage

level 45 ---> 14 + dex mod damage

etc ...

(extrapolating the "battle wrath" stance)

level 1 --> +2 damage

level 11 --> +3 damage

level 21 --> +4 damage

level 31 --> +5 damage

level 41 --> +6 damage

etc ...

So for a Slayer always using the "battle wrath" stance, the average damage scales approximately as (above level 10):

avg[W] + level*14/27 + 1 + D + 7*(level-10)/20

where D is the dex mod at level 1.

__Knight__(extrapolating the heroic, paragon, epic "weapon mastery" class ability)

level 5 --> +1 damage

level 15 --> +2 damage

level 25 --> +3 damage

level 35 ---> +4 damage

level 45 ---> +5 damage

etc ...

(extrapolating the "battle wrath" stance)

level 1 --> +2 damage

level 11 --> +3 damage

level 21 --> +4 damage

level 31 --> +5 damage

level 41 --> +6 damage

etc ...

So for a Knight always using the "battle wrath" stance, the average damage scales approximately as (above level 10):

avg[W] + level*5/24 + 2 + 7*(level-10)/20

On the monster side, the hit points of various monsters from page 184 of the 4E DMG1 are (assuming this is the same in the 4E Essentials DM Kit box set):

ROLE*(level +1) + CON

where ROLE is:

Artillery, Lurker --> ROLE = 6

Skirmisher, Soldier, Controller --> ROLE = 8

Brute --> ROLE = 10

(Elites double the hit points, while Solos quadruple the hit points).

Now the average number of hits to kill a monster being attacked by a player (of the same level as the monster) is approximately the ratio (for a high level):

__Slayer__N = [ROLE*(level+1) + CON]/[avg[W] + level*14/27 + 1 + D + 7*(level-10)/20]

__Knight__N = [ROLE*(level+1) + CON]/[avg[W] + level*5/24 + 2 + 7*(level-10)/20]

Taking the level to infinity, the ratios approach the limits of:

__Slayer__N -> ROLE/[14/27 + 7/20] = 1.15*ROLE

__Knight__N -> ROLE/[5/24 + 7/20] = 1.79*ROLE

For a skirmisher monster (ROLE=8), on average it will take a Slayer around 9 hits and a Knight around 14 hits (without "power strike") to kill the monster as the level goes to infinity.

Let's compare these results to the older Heinsoo 4E classes.

It turns out the 4E Essentials Slayer using "battle wrath" without using "Power Strike", is approximately equivalent damage-wise to a non-striker Heinsoo 4E class using a weapon with damage dice [W]=d10. The 4E Essentials Knight using "battle wrath" without using "Power Strike", is approximately equivalent damage-wise to a non-striker Heinsoo 4E class using a weapon with damage dice [W]=d4.

### 4E Essentials D&D scaling analysis articles.

We'll be reposting the series 4E Essentials D&D scaling analysis articles from therpgsite.

## Sunday, September 12, 2010

### In-game probability in practice - part 5.

Here are the 95% and 99% confidence intervals for the practical in-game probabilities "Y" after rolling a die n times, with an underlying theoretical probability of 50% (p = 0.5).

(They were calculated using a program which implemented the Clopper-Pearson method, when the normal approximation is no longer valid for n < 40).

n = 2 --> (0.0126, 0.9874)

n = 4 --> (0.0676, 0.9324)

n = 6 --> (0.1181, 0.8819)

n = 8 --> (0.1507, 0.8430)

n = 10 --> (0.1871, 0.8129)

n = 12 --> (0.2109, 0.7891)

n = 14 --> (0.2304, 0.7696)

n = 16 --> (0.2465, 0.7535)

n = 18 --> (0.2602, 0.7381)

n = 20 --> (0.2720, 0.7280)

n = 22 --> (0.2822, 0.7178)

n = 24 --> (0.2912, 0.7088)

n = 26 --> (0.2993, 0.7007)

n = 28 --> (0.3065, 0.6935)

n = 30 --> (0.3130, 0.6870)

n = 32 --> (0.3189, 0.6811)

n = 34 --> (0.3243, 0.6757)

n = 36 --> (0.3292, 0.6708)

n = 38 --> (0.3338, 0.6662)

n = 40 --> (0.3380, 0.6620)

n = 2 --> (0.0025, 0.9975)

n = 4 --> (0.0294, 0.9706)

n = 6 --> (0.0663, 0.9337)

n = 8 --> (0.0999, 0.9001)

n = 10 --> (0.1283, 0.8717)

n = 12 --> (0.1522, 0.8478)

n = 14 --> (0.1724, 0.8276)

n = 16 --> (0.1897, 0.8103)

n = 18 --> (0.2047, 0.7953)

n = 20 --> (0.2177, 0.7823)

n = 22 --> (0.2293, 0.7707)

n = 24 --> (0.2396, 0.7604)

n = 26 --> (0.2489, 0.7511)

n = 28 --> (0.2572, 0.7428)

n = 30 --> (0.2649, 0.7351)

n = 32 --> (0.2718, 0.7282)

n = 34 --> (0.2782, 0.7281)

n = 36 --> (0.2841, 0.7152)

n = 38 --> (0.2895, 0.7105)

n = 40 --> (0.2946, 0.7054)

As one can see for less than 4 or 5 rolls of a d20, with a 95% to 99% certainty, the practical in-game probabilities "Y" include the cases of 5% (ie. only hitting on a d20 roll of 20) and 95% (ie. hit on any roll of a d20, except on a 1).

Also in a generic DnD combat encounter which lasts 8 rounds or less, with 95% to 99% certainty, it should not be surprising at all to see a player which only hits once on all of their d20 rolls during the encounter (ie. Pinky's cousin), or a player which only misses once on all their d20 rolls during the encounter (ie. hot hand).

(Pinky is a "cursed" d20 which never rolls a success).

For completion, the number of rounds "n" where it is not surprising to see one with "Pinky's cousin" or a "hot hand", with 95% to 99% certainty.(They were calculated using a program which implemented the Clopper-Pearson method, when the normal approximation is no longer valid for n < 40).

__95%__n = 2 --> (0.0126, 0.9874)

n = 4 --> (0.0676, 0.9324)

n = 6 --> (0.1181, 0.8819)

n = 8 --> (0.1507, 0.8430)

n = 10 --> (0.1871, 0.8129)

n = 12 --> (0.2109, 0.7891)

n = 14 --> (0.2304, 0.7696)

n = 16 --> (0.2465, 0.7535)

n = 18 --> (0.2602, 0.7381)

n = 20 --> (0.2720, 0.7280)

n = 22 --> (0.2822, 0.7178)

n = 24 --> (0.2912, 0.7088)

n = 26 --> (0.2993, 0.7007)

n = 28 --> (0.3065, 0.6935)

n = 30 --> (0.3130, 0.6870)

n = 32 --> (0.3189, 0.6811)

n = 34 --> (0.3243, 0.6757)

n = 36 --> (0.3292, 0.6708)

n = 38 --> (0.3338, 0.6662)

n = 40 --> (0.3380, 0.6620)

__99%__n = 2 --> (0.0025, 0.9975)

n = 4 --> (0.0294, 0.9706)

n = 6 --> (0.0663, 0.9337)

n = 8 --> (0.0999, 0.9001)

n = 10 --> (0.1283, 0.8717)

n = 12 --> (0.1522, 0.8478)

n = 14 --> (0.1724, 0.8276)

n = 16 --> (0.1897, 0.8103)

n = 18 --> (0.2047, 0.7953)

n = 20 --> (0.2177, 0.7823)

n = 22 --> (0.2293, 0.7707)

n = 24 --> (0.2396, 0.7604)

n = 26 --> (0.2489, 0.7511)

n = 28 --> (0.2572, 0.7428)

n = 30 --> (0.2649, 0.7351)

n = 32 --> (0.2718, 0.7282)

n = 34 --> (0.2782, 0.7281)

n = 36 --> (0.2841, 0.7152)

n = 38 --> (0.2895, 0.7105)

n = 40 --> (0.2946, 0.7054)

As one can see for less than 4 or 5 rolls of a d20, with a 95% to 99% certainty, the practical in-game probabilities "Y" include the cases of 5% (ie. only hitting on a d20 roll of 20) and 95% (ie. hit on any roll of a d20, except on a 1).

Also in a generic DnD combat encounter which lasts 8 rounds or less, with 95% to 99% certainty, it should not be surprising at all to see a player which only hits once on all of their d20 rolls during the encounter (ie. Pinky's cousin), or a player which only misses once on all their d20 rolls during the encounter (ie. hot hand).

(Pinky is a "cursed" d20 which never rolls a success).

__One hit only (or only one miss on a "hot hand")__

95% certainty --> n < 8

99% certainty --> n < 9

__Two hits only (or only two misses on a "hot hand")__

95% certainty --> n < 11

99% certainty --> n < 13

__Three hits only (or only three misses on a "hot hand")__

95% certainty --> n < 14

99% certainty --> n < 16

__Four hits only (or only four misses on a "hot hand")__

95% certainty --> n < 17

99% certainty --> n < 19

__Five hits only (or only five misses on a "hot hand")__

95% certainty --> n < 19

99% certainty --> n < 22

## Sunday, September 5, 2010

### In-game probability in practice - part 4.

For completion, the remaining |Y-p| cases for determining with 95% certainty that the underlying theoretical probability is not 50% (ie. p != 0.5). This was done by laboriously going through cumulative binomial distribution calculations, where the normal approximation is no longer valid.

|Y-p| = 0.20 --> n > 29

|Y-p| = 0.25 --> n > 19

|Y-p| = 0.30 --> n > 13

|Y-p| = 0.35 --> n > 10

|Y-p| = 0.40 --> n > 8

|Y-p| = 0.45 --> n > 7

Within a generic four-five hour DnD game with a few dozen or so d20 rolls made, one would certainly notice a bonus/penalty of 5 or greater to the d20 rolls.

|Y-p| = 0.20 --> n > 29

|Y-p| = 0.25 --> n > 19

|Y-p| = 0.30 --> n > 13

|Y-p| = 0.35 --> n > 10

|Y-p| = 0.40 --> n > 8

|Y-p| = 0.45 --> n > 7

Within a generic four-five hour DnD game with a few dozen or so d20 rolls made, one would certainly notice a bonus/penalty of 5 or greater to the d20 rolls.

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