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).
