The probability is based on the mission's "percent chance" stat, multiplied by the number of intel points collected past the required minimum. Once the minimum is reached there is a random chance (1/4 of discovery chance) to discover the mission at every update. A mission cannot be detected until you generate intel equal to the "required intel" value. The difficulty of discovering a certain type of mission is controlled by two numbers: "required intel" and "discovery chance". Each mission keeps a pool of collected intel points.
These points are applied to undiscovered missions in the region at certain update intervals, currently 4 times a day (every 6 hours). Havens each generate "intel points" from having personnel assigned to intel gathering. A mission's expiration timer starts when it is created, not when it is discovered most infiltration missions have a base duration of 18 days, which is randomized by a few days each time a mission is created. Assigning more resistance members to the "intel" job can make missions considerably easier by discovering missions earlier and allowing more time to infiltrate. The mission is not visible at this time the local haven must collect enough intel to reveal the mission. Missions spawn automatically as the days pass, and expire after a given window of time.
You can still complete a mission after such a timer reaches zero but it will become harder with each wave. Reinforcement timers are delays before increasingly strong reinforcements start dropping.
When the timer reaches 1, the last turn start (there is no "0 turns left" turn). Any soldier still on the map when the timer reaches 0 is lost (captured).