[RPG] Randomizers

Most RPGs have some element of randomization to determine outcomes and abilities- a die roll, a card draw, rock, paper, scissors- something. Most of these elements are fairly well delineated as solely for things the player does (or wants to do).

Examples include:Want to rob a bank? Roll me an intimidation and/or firearms. Want to jump across the chasm between two tall buildings? Roll me athletics or leaping.Want to go first? Draw a card and tell me the number and suit. And so on, depending on what system you're using and what feel you're going for.

Some of the randomizers are more random than others- a D20 is automatically more random than a 6. If you use suits of cards to determine whether something works, you're effectively using a D4- sufficiently less random than a D10.

Some of the more "random" options pull down the curve some by giving ranges where success or failure is possible. The D20 system is a good example- you "win" when you roll OVER a certain number, rather than being limited to a specific number (eg: you need to roll over a 15; not you must roll a 20). Other systems allow "cheating" by use of chips, points, pushing, cards etc.

What if you don't want cheating to be an option but use a system that allows it? Do you just ignore the rules that exist for fudging things a little? 

Other questions I'm considering are - what if an ability ONLY activates on a "16" (and not higher or lower)? How effective or complex might a system where abilities were strictly determined by a preset randomizing table, without any real wiggle room? What if you're looking for a truly "random" possibility? How would you accomplish that?

And then, how do you do it without interfering in the mood too much?