I realize that this might not be an easy question to answer, but I am curious if anyone else has thought about the relationship between time remaining on a task and obtaining a "successful" result from rushing.
Programing-wise, it makes the most sense to me that the probability of an incident rate (e.g., 30%) would be factored into the amount of time remaining somehow (so rushing a task that has 10 seconds left on it should be less likely to fail than rushing a task that has 10 minutes remaining). For example, it might be that each second has a 30% chance of failure, and the game does X tests to see if it fails (where X might be the number of seconds remaining). This would mean that more seconds remaining would lead to more tests, which would mean that more time remaining would be associated with more failures. This is kind of implied by the "fast-forwarding" animation in the game.
However, it also might be the case that when you click "rush", it simply determines the success/fail of the rush based upon the success rate percentage in terms of one evaluation at that percent – so it doesn't matter how much time is remaining when you rush. If you rush at 10 seconds or 10 minutes, in both cases it is simply the 30/100 chance of failing.
As mentioned below, there are factors we know affect success: SPECIAL levels, especially Luck, and previous attempts at rushing (with both successes and failures increasing the likelihood of failure).
So, my underlying question is: does amount of time remaining on a task affect the probability of successfully rushing?
Best Answer
No correlation between time remaining and rush failure percentage
The in-game help mentions that the probability of a rush failure decays over time. Attempting a rush, whether it succeeds or fails, will increase the probability of a rush failure in that room, but this is a one time increase and it starts to decay again after that. There also seems to be a lower limit on rush failure. likely based on Luck and the appropriate SPECIAL stat for the room. The lowest I've seen is 21% failure chance on a room that I haven't rushed in days. James reported seeing as low as 16% with a maxed out, very lucky water treatment plant.
In my experience, the failure decay rate is so small that you probably would not notice it over a few minutes (the normal task completion time for a well populated room), but it becomes noticeable over a few hours. After spiking the failure chance in a room to around 50% right before bed, I woke up the next morning with the failure chance back down to upper 20s/lower 30s. This seems to indicate that it is based on elapsed time rather than decaying after a room is harvested.
When you couple this with the fact that the timer is reset to max after a rush is attempted, rushing a room with a few seconds left is not as resource effective as harvesting the resources, then attempting the rush when the timer is full. In that case, you save a lot of time if you succeed and don't lose as much time if you fail.
tl;dr
It seems to be a percentage calculated when you click the rush button, regardless of how much time is left until the room produces resources. Rush failure percentage chances slowly decrease over time, but this is noticeable over tens of minutes and hours, rather than seconds. If you want to maximize the amount of time your rush saves, rush right after you collect, when the room timer has a lot of time left.