When a Character attempts an action that is risky or has an unclear outcome, the GM will call for a roll. They name an Attribute, an Ability, and a Difficulty, and the Player rolls a number of ten-sided dice (d10s) equal to the sum of their Character's ratings in the named Attribute and Ability. Each die that lands on a number equal to or greater than the Difficulty is considered a Success. The Player adds up how many Successes they scored. A die showing '1' is called a "botch" and subtracts a Success. A die showing a '10' is called a "double" and is worth two Successes instead of one. The total number of Successes is called the Outcome and determines the result.
GMs should not reference the above visualization at game-time when calling for a roll. If a roll must be made, default to the standard Difficulty of 6 (or the weapon's Difficulty for an attack). If the context of the action seems like it would significantly affect the odds, add or subtract one or two Difficulty as desired.
When you call for a roll, you are not judging the difficulty of the task at hand so much as you are entering an agreement with the Player about how much discretion you will have over the outcome. Notice that botches, partial successes, and exceptional successes all allow the GM to determine the specifics of the outcome. A partial success (the most common Outcome) can be anything from a near-failure to a success with a complication. The Contract's emphasis on outside-the-box problem solving necessitates putting such trust and power in the hands of the GM. A more rigid resolution system would feel restrictive and arbitrary when things inevitably go off the rails. GMs should use the leeway afforded to guide the action of the Contract. Tastes vary from GM to GM and World to World, but Players should never feel cheated. A complete success is a complete success.
This system gives the GM some control over the "tone" of the game during the standard play, however, during Combat-- when Contractors' lives are on the line-- the GM's discretion is greatly reduced. Partial and exceptional successes have specific outcomes determined by the Combat rules. This makes The Contract's quick, deadly Combat more tactical and fair.
The Contract's Design Philosophy de-emphasises advancement through dice bonuses and Difficulty modifiers. Dice bonuses from multiple sources do not stack. This means that dice pools remain relatively small. Beyond 12 dice, consider success a near-certainty even on high Difficulty rolls.
Dice pool probabilities that account for botches (and doubles) seem fairly difficult to calculate, but it is essentially a standard multinomial distribution problem. See this Github repo for the exact code used to generate the probabilities. There may be some very slight abnormalities resulting from rounding and computers' difficulty encoding fractional values.