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.

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