Entropi said:
The idea of analyzing all skill challenges mathematically is flawed. Every skill challenge is meant to be unique, and tailored to the individual party, per the DMG. There is no such thing as an 'average' skill challenge.
Saying that parties generally have a 8.73% chance to win an average skill challenge is like saying they have a 8.73% chance to win an average combat. What's an average combat? No such thing. The term is way too vague to have meaning, let alone be statistically analyzed. In this case, it simply means your concept of an 'average skill challenge' is flawed. By the DMG, an average skill challenge is one in which the the party has roughly the chance to win it that the DM desires for them to have (since he has prepared it custom, for that party).
I definitely agree that each skill challenge is different, and the tables in the DMG aren't a straitjacket - just a starting point.
However, as a DM, I like to have some idea of the odds when I tailor-make or adjust a challenge for my players. For combat, there's too many variables to measure, but for a simple skill check, it's an easy percentage.
That's what - in theory - game designers do. They actually check the math - that's their job. It's the reason, for 3.5, why Power Attack was changed for 2-handed weapons. It's the basis behind the CR systems. In theory, the math is done well and it becomes more transparent to us, the players. We, as DMs, are meant to work from there, adjusting the scales upwards or downwards.
The problem is that a lot of the math that goes into skill challenges is counter-intuitive. For example, I would not have guessed that a 70% success rate per-check swings the odds in the players' favor the longer the challenge goes on. That's surprising to me, but it's clear now that Stalker0 has pointed it out. By the same token, I need to know that - by default - a lot of the DCs in the DMG require me to give a lot of ad-hoc bonuses if they're going to work out in the players' favor.
Tables and discussions like this help DMs make exactly the kinds of rulings you're encouraging us to make.
-O