In Tabletop RPGs skills are modelled using dice and a "skill" parameter. You state your intention to attempt a task. The Game Master (GM) asks you to roll. The mechanics of the system ensure that people who are good at a skill are more likely to succeed than those who are not.
Dungeons and dragons mostly relies on a roll of a 20 sided die. Some games use percentile dice, or sets of polyhedrals. But how do we acurately model skill?
Let's compare two archers
John Klutz (skill level: "incompetent") has barely a clue. He holds the bow incorrectly, takes a bad stance and tries pulling the arrow rather than drawing the string. The arrow will still go in a direction approximating forwards. But he'll rarely get a bullseye.
At the other end of the scale, we have the legendary archer (Robin Hood, or Katniss Everdeen). They never miss!
A fluke for John Klutz is hitting the centre ring. A calamity is hitting someone behind him.
For the legendary archer though, we expect that. Even a mere Olympic archer will hit yellow more often than not
. A fluke result for a legend is Robin Hood hitting the arrow that's already in the target. A calamity is Katniss not hitting the centre when it mattered most.
Flukes and calamities are what makes things exciting. They add more than a simple pass/fail. Robin Hood doesn't just win! He passes into legend! Katniss needs to do something incredible to make up for the failure. These are million to one to one shots, and in fiction, according to Terry Pratchett, "million to one chances crop up nine times out of ten"! And they should, but in games, it should be nine games
out of ten. Not nine attempts! We do want it to be possible for this to happen.
We can simply roll a die. A standard six sided die (d6 in gaming parlance). This makes the extremes come up a little too often. 1 in 6 attempts are flukes. 1 in 6 are calamities. That's far too frequent. You can make them less common with 20 sided dice (d20), or even percentile dice.
It lacks something though. All results are as likely as each other. In a skill based challenge they're not. In my archery example, the target has a yellow circle, a red circle, a blue one and a black one each of the same thickness. But because the diameter is larger, the area is larger. The red circle has 3 times the area of the yellow centre circle. John Klutz is more or less hitting random points on the target so he'll get 3 times as many reds as yellows. The expert is aiming for the centre, so should get substantially more yellows. ( At this point, things get longer and more mathematical. )
My conclusion here: The roll and keep mechanic is probably the best, both for elegant modelling and for general nice results. The next best is probably roll a bunch of six sided dice and add the modifiers. Noit sure which oif these I go for in my homebrew game but they're certainly my favourites.