(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).
