The previous thread on the pros and cons of botch mechanics got me curious about the more general topic of what "failing a roll" means in certain systems. This might help to clarify certain opinions about why odds of botching are too high in certain cases--I'm confused, because I'm hearing whiny powergamer arguments from people I don't think of as whiny powergamers. But I digress.
In every system I can recall, failure on a given roll doesn't automatically mean that you fail for all eternity, but that you didn't acheive the desired result on that particular attempt. Sometimes you can retry immediately (you fail to pick a lock), sometimes it takes a bit longer (your initial attempt to seduce the Countess failed, try again at tomorrow's masquerade), but sometimes you're just screwed (you didn't hear the thug sneaking up behind you, and now you're at knifepoint). Are there systems out there where you never get to retry anything?
In situations where you can retry, the expected number of attempts needed before you succeed is 1/p, where p is your probability of success on a single roll (assuming that subsequent attempts don't get more difficult). This means that linearity in success rate (which a few people have been griping about) isn't what it seems. Taking d20 as the roll in question, a completely incompetant character (1/20 chance for a natural 20) will, on average, succeed by dumb luck in about 20 attempts. Minimal training or ability (say a 3/20 chance) will dramatically decrease the expected number of attempts to about 6 or 7. A competent character needs about 2 or 3 tries on average. An expert will usually get it right the first time, and almost never need to try more than twice. In situations where it is not just important to succeed, but to succeed quickly, training and ability give the advantage.
This makes an awful lot of sense to me. It also shows why the Take 10/20 rules work prett well.