The problem - roll and pray
At least once every session a player will come across a situation that is very important to them - perhaps they are down to one last hit point and need to kill the enemy before they themselves are killed, or the character is preparing for a hack that is a culmination of the adventure and their paycheck depends on it. Those dramatic moments where the players wish there was anything they could do to improve the odds, but alas - they can only roll and pray.
I really don't like such situations of being powerless and not having some mechanic one can leverage to improve those odds. Luckily, there are a number of options that various systems have implemented.
Rerolls
The simplest approach to addressing the issue is letting the players have a reroll handy. Those often are a resource shared either between the party or given to the characters individually. Stars Without Number has an Expert class that can reroll one non-combat roll each scene, Savage Worlds gives players a few Bennys per session they can use as a reroll, Godbound has a Word of Luck that is all about manipulating rolls and causing rerolls in various ways.
In general, rerolls are a good option when the characters are somewhat competent at a task and should be able to normally succeed at it. It's a bit less useful when the character is trying to tackle something they are woefully unprepared for - if it's unlikely they will succeed, they will likely still fail with a reroll.
Improving the odds
Another approach that can be taken is improving the odds of the roll before it is made. Chronicles of Darkness gives you more dice to roll if you have the proper gear, and you can further improve your odds by spending Willpower - a limited, refreshable resource. The system also features a number of ways to fudge the dice math to imrove the chance of an exceptional success by using 9-again or 8-again rules. Star Trek Adventures makes you accrue Momentum by performing tasks you're good as so you can buy more dice for a roll when it's really needed. CoD, STA and Stars Without Number give the option of other characters assisting the primary character performing a roll - usually a success by the assistant gives a small boon to the primary roll.
Overall, improving the odds can be especially useful when the character isn't as competent - even a small boon to an unskilled character usually translates to much better odds than adding a simple reroll. They also usually are simpler to justify narratively - you have a clear explanation of why you're getting that +2 to the roll.
Success, but at a cost
Success in a role playing game is often seen as binary - you either succeed, or you don't. Maybe you also have a botch or an exceptional success, but either way - the line is drawn hard. However, there are a few systems where there is a gradient to the failure.
In The Veil (or Powered by the Apocalypse in general), you almost always have two or three gradients of a success. You can outright fail, succeed completely, or in the middle - succeed at a cost. Perhaps you don't get everything you wanted, or there is some complication that occurred as a result of your actions. Star Trek Adventures gives players an option to Succeed At a Cost instead of accepting a failed loss. This allows the task to be successful, but some complication will arise as a result of that roll.
This option is useful for fudging the numbers a little - if the player is just slightly off from succeeding. It's always useful to have it as an option to allow the player to decide whether their task is important enough to warrant the extra cost.
Controlled botches
Somewhat related to the previous section, although distinct enough to warrant its own. In a lot of games you will find rules for botching a roll - failing so badly it's causing some trouble. World of Darkness up to Chronicles of Darkness had a rule where rolling 1s meant subtracting from successes and causing botches if their amount got too big for example. While fun in their own right at times, it's perhaps best when this option is left to the player - letting them decide when they want to be discovered, mislead and so on to create a more interesting game.
Chronicles of Darkness is perhaps a good example of how to give players agency over their botches. When a roll is a failure, the player can opt to make it a dramatic failure to get a beat (a point of XP basically). Outside of attempting a roll with only a chance die, this is the only way to botch in the system, giving the players agency over when it occurs, and rewarding them for allowing it to happen (hopefully giving the GM a chance to make the game more interesting as a result).
Trumping the roll
Some games allow the characters to outright trump the roll with some usually high-end powers or skills. Stars Without Number have some Psychic and True AI powers that roll back an entire turn or a single roll, or go as far as dictating an entire turn and all of its rolls completely (max level True AI power). The same system also features a Warrior class that can auto-succeed at one combat roll per scene, or negate an incoming attack completely. Godbound has one Gift that once per character's entire life they can succeed at any undertaking, albeit only at a basic level.
Those powers are neat, but they need to be carefully balanced not to throw the entire game off balance.
Conclusions
There are many ways RPG systems have figured out how to allow the players to influence their rolls. It's good to have at least once option available to the players, and multiple of these examples can be used simultaneously without feeling like they're overlapping too much.
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