Good example is abstract wealth systems (Burning Wheel and d20 modern have one, for example). The question “Can the NPC afford this and that?” can be quickly solved with a simple die roll or checking the wealth levels involved. PCs wanting to buy a certain item just roll wealth. with adjustments for obscure or specialised items. And failure means that I have the perfect excuse to add some fun complication, like the chars getting into a hostage scene or catching the attention of authorities/pickpockets or buying unreliable equipment. An invaluable feature.
Abstract contact systems work pretty much exactly the like: Roll to find whoever you are seeking, with factors such as character background and social station affecting the roll. Failure means that you get the attention of someone or that the person you find happens to hate your guts (called enmity clause in Burning Wheel). Again, great way to introduce new complications and conflicts to play and the players do part of the NPC design work (asking them to name the NPCs thus found may be useful trick, too).
Random encounters are kinda similar. A good random encounter table can be used when characters fail a roll in the wilderness. Maybe they are tracking the bugbear that slaughtered some villagers and fail. It’d be no fun for them simply to not find anything, so instead they trigger a random encounter (maybe the vily bugbear tricked them into territory dominated by whatever beasties they encounter, or maybe it is pure bad luck).